1 can be important. The pseudopotentials are expected to give accurate predictions for R * ≪ l ⊥ .\n\nWhen R * ≫ l ⊥ , the pseudopotential approach is not applicable at all, and one has to resort to numerical calculations. As an example, we present in Fig. 6 values for the short-range phases for l ⊥ = 0.1R * . We note the presence of several resonances in the dependence of ϕ e and ϕ o . This behavior is related to the contribution of several partial waves for R * ≫ l ⊥ , leading to resonances when the energy of a bound state in the combined harmonic and r -4 potential becomes equal to the energy of the scattered wave. To analyze this issue more carefully -0.4 -0.2 0.0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 0.6 e l = R * numerical energy-dependent pseudopotential / l = R * o numerical energy-dependent pseudopotential / FIG. 5: (Color online) Even and odd short-range phases ϕe and ϕo calculated for l ⊥ = R * . Numerical results (red solid lines) are compared with predictions of the model replacing r -4 with the energy dependent pseudopotential.\n\nwe have calculated the energies of some particular bound states in in the combined harmonic and r -4 potential. Fig. 7 shows the relation between the bound-state energy and the short-range phase for few bound states that are responsible for the scattering resonances close to φ = 0 for φ e (cf. Fig. 6 ). For comparison we include the bound states of pure r -4 potential and we label them by the angular momentum l. We note that, due to the limited resolution of our numerical calculations, Fig. 6 does not allow to resolve the single resonances close to φ = 0, however, one can observe the rapid changes of φ e in this region.\n\nFor the numerical calculations we have transformed (16) into cylindrical coordinates and we were solving 2D elliptic partial differential equation using a finiteelement method. To fix the short-range phase we impose the boundary conditions on the logarithmic derivative of Ψ(r), ∂ r Ψ(r)/Ψ(r), at r min , where r min = 0.09R * (r min = 0.022R * ) for l ⊥ = R * (l ⊥ = 0.1R * ) . At large distances we impose the Dirichlet boundary condition on the rectangle with boundaries |z| = z max , ρ = ρ max , assuming that the wave function has a Gaussian transverse profile at |z| = z max and vanishes at ρ = ρ max . For l ⊥ = R * we took z max = 6.3R * , ρ max = 3R * , while for l ⊥ = 0.1R * we have used z max = R * , ρ max = 0.4R * .\n\nTo determine the values of ϕ e and ϕ o , we fit at large distances (|z| ≫ R 1D ) the symmetric and antisymmetric solutions of (16) for E = ω ⊥ , to the asymptotic formula (17).\n\nRelative motion in a 1D system: adiabatic eigenenergies and eigenstates\n\nIn this section we consider the 1D motion of an atom and an ion, assuming ω i = ω a . In this case we can focus only on the relative motion described by the Hamiltonian (9), with the boundary conditions at z → 0, stated by (10) and (11) . To find eigenenergies and eigenfunc-9 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 (b) e (a) o FIG. 6: (Color online) Even short-range phase ϕe (upper panel) and odd short-range phase ϕo (lower panel) calculated numerically for l ⊥ = 0.1R * , versus 3D short-range phase ϕ.\n\n-50 -5 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.0 0.5 1.0 l=6 l=8 E l=10 FIG. 7: (Color online) 3D short-range phase ϕ versus energy of bound states in the presence of the transverse confinement with l ⊥ = 0.1R * . The bound states in the combined harmonic and r -4 potentials (solid lines) are compared with the bound states of pure r -4 interaction (dashed lines).\n\ntions for arbitrary value of d we diagonalize the Hamiltonian (9) numerically in the basis of its eigenstates for d = 0, that are found by numerical integration of the 1D Schrödinger equation. In the basis we include the lowest 50 odd and even eigenfunctions, which is sufficient to perform the diagonalization for d/R * 2.5. Fig. 8 present the adiabatic energy spectrum for some example parameters: ϕ e = -π/4, ϕ o = π/4 and R * = 3.48l. The latter value corresponds to the system of 40 Ca + and 87 Rb in the trap with ω = 2π × 100kHz, while the particular choice of the short-range phases ϕ e and ϕ o is explained later in this section. Points with labels correspond to the wave functions shown in Fig. 9 presented together with their eigenenergies and potential energy curves.\n\nThe three panels shown in Fig. 9 illustrate three different regimes, where the system exhibits qualitatively different behavior. In the first regime, represented by the eigenstate Ψ a and realized at large distances between traps, d ≫ R rel with R rel defined in (5), the main effect of the atom-ion interaction is the distortion of the trapping potential, and in this limit the Hamiltonian can be diagonalized analytically in the model of two coupled harmonic oscillators (see Section III E for more details). The value of the short-range phase is not important for this model. At large separations the eigenstate Ψ a is typically only weakly perturbed with respect to the eigenstate of the harmonic oscillator.\n\nIn the second regime, represented by the eigenstates Ψ b and Ψ c , and realized at distances d ∼ R rel , the system exhibits resonances between vibrational (Ψ b ) and molecular (Ψ c ) states, manifesting themselves as avoided crossings in the adiabatic energy spectrum. The resonances appear when the energies of the two eigenstates become equal, and the energy splitting at the avoided crossing is proportional to the tunneling rate through the potential barrier separating the two regions of the potential. In section III D we calculate this splitting in the framework of a semiclassical approximation, and describe the dynamics in the vicinity of the avoided crossing. The value of the short-range phase determines the energy of the molecular states, but it is practically not important for the atom-ion vibrational states. Since the point of the avoided crossing depends on the short-range phases, the controlled collisions can provide some information on the short-range interaction potential.\n\nFinally, at distances d ≪ R rel the barrier separating the external trap from the well given by the atom-ion attraction disappears and all eigenstates have some shortrange component behaving at short distances according to Eqs. (10) and (11) . Hence, in this regime all eigenstates depend on the value of the short-range phases. The molecular states with energies below the dissociation threshold (E = 0) are mainly localized in the well of the atom-ion attractive potential, however for the tight traps considered here, they can be strongly affected by the external trapping (e.g. state Ψ e ). The other type of states, with energies E > 0, are analogs of the vibrational states and they are localized mainly at distances where the external trapping potential dominates (e.g. state Ψ d ).\n\nSo far we have discussed the properties of the energy spectrum for some particular choice of ϕ e and ϕ o . It turns out that a qualitatively similar behavior can be observed in all the systems with |ϕ e -ϕ o | = π/2. In the general case, however, the adiabatic energy spectrum has a slightly more complicated structure, as it is illustrated in Fig. 10 , showing the eigenenergies for ϕ e = ϕ o = -π/4. We note that by going from d = 0 10 0.0 0.5 1.0 1.5 2.0 2.5 -10 -5 0 5 e d c b o = + / 4 E/ * e = -/ 4 a FIG. 8: (Color online) Energy spectrum of the relative motion for an atom and an ion in 1D traps versus the distance d between traps, calculated for ϕe = -π/4, ϕo = π/4 and R * = 3.48l (see text for details). The dashed curve shows the line 1 2 µω 2 d 2 , giving the approximate shift of the bound state in the trapping potential (see section III C for derivation).\n\nto positive d = 0 all the energies split into two branches.\n\nTo understand the nature of this splitting in Fig. 11 we present the wave functions for d = 0.1R * , with the corresponding eigenvalues marked with stars in Fig. 10 . We observe that the branches represent the eigenstates localized on the left-hand side and right-hand side of the point z = 0. The left-localized eigenstates correspond to the rising branches, because they are mainly localized in the region of strong atom-ion attraction. We note that this behavior is similar to the properties of eigenstates in a double-well potential, where appropriate wave functions are constructed by taking symmetric and antisymmetric combination of the states in two wells. In the particular case of ϕ e = ϕ o , the symmetric and antisymmetric combination leads to the states localized on the positive and negative z semi axes, respectively.\n\nC. Relative motion in 3D for spherical ωi = ωa traps: adiabatic eigenenergies and eigenstates\n\nIn this section we extend our analysis to 3D, and consider spherically symmetric traps for the atom and the ion with the same trapping frequencies: ω i = ω a . In this case the COM and relative motion can be separated, and in the following we focus only on the relative motion governed by the Hamiltonian (4).\n\nWe have diagonalized the Hamiltonian (4) taking different values of the short-range phase ϕ, and we have observed that the adiabatic energy spectra exhibit qualitatively the same features as in 1D. In the numerical calculations we first calculated the eigenstates for d = 0, by solving the radial Schrödinger equation, for angular momenta l ≤ 75 and for energies E ≤ 75 ω. These states 0 10 20 -4 0 4 -3 -2 -1 0 1 2 3 4 -4 0 4 c e d b d = 2.5 R * e = -/4 o = + /4 E/ a d = 1.2 R * e = -/4 o = + /4 E/ d = 0 e = -/4 o = + /4 E/ * FIG. 9: (Color online) Eigenstates of the relative motion for an atom and an ion in 1D traps at different separations d between the traps, calculated for ϕe = -π/4, ϕo = π/4 and R * = 3.48l. Presented eigenstates correspond to labeled points in Fig. 8 . The horizontal lines present the corresponding eigenenergies, and the thick black line shows the potential energy.\n\n0.0 0.5 1.0 1.5 2.0 2.5 -10 -5 0 5 o = -/ 4 E/ * e = -/ 4 FIG. 10: (Color online) Energy spectrum of the relative motion for trapped atom and ion in 1D traps versus distance d between traps, calculated for ϕe = ϕo = -π/4 and R * = 3.48l (see text for details).\n\nwere used as a basis in the numerical diagonalization of the Hamiltonian (4). A sample adiabatic energy spectrum is shown in Fig. 12 , presenting the results for ϕ = -π/4 and R * = 3.48l. At d 0 the angular momentum l is a good quantum number and the states have definite angular 11 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -2 0 2 4 6 8 d = 0.1 R * e = -/4 o = -/4 E/ z/R * FIG. 11: (Color online) Eigenstates of the relative motion for trapped atom and ion in 1D at trap separation d = 0.1R * , calculated for ϕe = ϕo = -π/4 and R * = 3.48l. The horizontal lines present the corresponding eigenenergies, and the thick black line shows the potential energy.\n\n0.0 0.5 1.0 1.5 2.0 -2 0 2 4 6 l = 0 l = 1 l = 2 l = 3 l = 4 E/ * = -/4 FIG. 12: (Color online) Energy spectrum of the relative motion for an atom and an ion confined in spherically symmetric traps versus the distance d between traps, calculated for ϕ = -π/4, and R * = 3.48lr (see text for details). The blue dashed line shows the shift of the bound states with d, given approximately by 1 2 µω 2 d 2 .\n\nsymmetry, which is depicted by the appropriate symbols in Fig. 12 . For nonzero d, angular momentum is not conserved, hence in the dynamics the state can change its angular symmetry. We note that the energies of the bound states shift to a good approximation according to µω 2 d 2 /2. This behavior can be explained by noting that bound states |Ψ mol are concentrated around r = 0, and\n\nΨ mol (d)|H rel (d)|Ψ mol (d) ≈ E b + 1 2 µω 2 d 2 ,\n\nwhere E b is the binding energy at d = 0."},{"section_type":"OTHER","section_title":"D. Avoided crossings: semiclassical analysis","text":"In the considered setup, level anticrossings reflect resonances between molecular and trap states. A simple picture of such an avoided crossing is shown in Fig. 13 . If one passes an avoided crossing from the direction of the asymptotic trap state, then for adiabatic evolution the system at small distances evolves into a molecular state. On the other hand, for diabatic passage, the particles remain in their traps, and the state basically does not change, apart from the modification due to the smaller trap separation. In addition, for fast changes of the trap positions, the particles can be excited to higher motional states. To describe quantitatively the dynamics in the vicinity of the avoided crossing one can apply the Landau-Zener theory. Assuming that close to the avoided crossing the eigenenergies are linear in d, and that d(t) varies linearly in time, the probability that the crossing is traversed diabatically is given by [14, 15]\n\np |1 →|1 ′ = exp -2π | Ψ 1 |H|Ψ 2 | 2 | ḋ||∂E 12 /∂d| , ( 18\n\n)\n\nwhere the labels |1 , |2 (|1 ′ , |2 ′ ) respectively refer to the vibrational and molecular states before (after) passing the avoided crossing (cf. Fig. 13) , and E\n\n12 (d) = E 1 (d) -E 2 (d). For | Ψ 1 |H|Ψ 2 | 2 ≪ | ḋ||∂E 12 /∂d|,\n\nthe probability (18) is close to unity, and the avoided crossing is passed diabatically. In the opposite case:\n\n| Ψ 1 |H|Ψ 2 | 2 ≫ | ḋ||∂E 12 /∂d|\n\n, p is small, and the avoided crossing is traversed adiabatically. The matrix element Ψ 1 |H|Ψ 2 can be related to the energy gap at the avoided crossing, and in this way we obtain the following constraint on the adiabaticity of the transfer close to the avoided crossing:\n\nḋ ∂E 1 ∂d - ∂E 2 ∂d ≪ (E 1 (d) -E 2 (d)) 2 ( 19\n\n)\n\nThe position of the avoided crossings is directly related to the energies of the bound states; thus, a measurement of the final state after controlled collisions, provides technique for spectroscopy of the trapped atom-ion complex. Since the avoided crossings become weaker as the separation between traps increases, this scheme allows to probe only excited molecular states, having sufficiently small binding energies, comparable to the energy scales of the trapping potentials.\n\nIn the case when the tunneling barrier is sufficiently large, the energy splitting at the avoided crossings can be estimated using a semiclassical approximation. We first focus on the relative motion in the 1D system, assuming ω i = ω a = ω. Using the standard WKB method, one can derive the following result for the energy splitting ∆E [14]\n\n∆E = EΨ mol (x 1 )Ψ vib (x 2 ) v(x 1 )v(x 2 )T e -W/ . ( 20\n\n)\n\nHere x 1 , x 2 are some arbitrary points located in the classically forbidden region close to the classical turning 12 FIG. 13: (Color online) Schematic drawing of the avoided crossing between vibrational and molecular states in the system of a trapped atom and an ion. An adiabatic change of the distance d between traps induces a transition from the vibrational to the molecular states, while a diabatic process leaves the particles in the vibrational states."},{"section_type":"OTHER","section_title":"points","text":", W (x 1 , x 2 , E) = x2 x1 dx 2m(V (x) -E)\n\nis the action along the classical trajectory from x 1 to x 2 , T = ∂W ∂E is the tunneling time, Ψ mol (x 1 ), Ψ vib (x 2 ) denote, respectively, the molecular and vibrational states (cf. Fig. 13 ) with the same eigenvalues E (the resonance case), and v(x) = 2(V (x) -E)/m is the velocity of a particle with energy -E in the inverted potential -V (x). In principle the choice of x 1 and x 2 is arbitrary, but in our calculations we take x 1 and x 2 at fixed distance from the location of the molecular (x = 0) and vibrational states (x = d), close to the classical turning points. In this way at sufficiently large separations Ψ mol (x 1 ), Ψ vib (x 2 ) becomes independent of d.\n\nWe turn now to the case of relative motion in 3D traps. Similar to the analysis for 1D systems, the energy splitting at the avoided crossings can be calculated semiclassically by applying the instanton technique. In our calculation we adopt the formulation based on the path decomposition expansion developed by Auerbach and Kivelson [16] . We obtain the following formula describing the level splitting:\n\n∆E = EAΨ mol (x 1 )Ψ vib (x 2 ) v(x 1 )v(x 2 )T e -W/ (21)\n\nThe meaning of all the quantities is the same as in the 1D case, the only difference is the prefactor A which accounts for the fluctuations around the instanton path (see [16] for details). In the case ω i = ω a the instanton path that minimizes the classical action is simply a straight line connecting the trap centers.\n\nFig. 14 compares the semiclassical formulas Eqs. (20) and (21) with the exact energy splitting determined from the numerical energy spectra, as those presented in Fig. 8 . The presented results correspond to the avoided crossings between the vibrational ground state molecular states. In 1D the energy splitting at the avoided crossing does not 0.8 1.0 1.2 1.4 1.6 1.8 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 WKB 1D instanton 3D (s-wave) 1D numerical 3D numerical (s-wave) E/ d/R * R * = 3.48 l\n\nFIG. 14: (Color online) Energy splitting at the avoided crossing between vibrational ground-state and the molecular state in 1D and 3D systems. Semiclassical results (solid lines) are compared to the numerical values extracted from the adiabatic energy spectra for different combinations of ϕe and ϕo.\n\n3D results present avoided crossings widths for s-wave bound states.\n\ndepend on the symmetry of the molecular state, while in 3D it does depend on its angular momentum l, and we present here only the case l = 0. In the first approximation, the splitting depends on the short-range phases only through the critical trap separation at which the resonance occurs. The numerical data are obtained for different combinations of the short-range phases, for which the adiabatic spectra exhibit avoided crossings at different values of d. In our approach, instead of calculating Ψ mol (x 1 ) and Ψ vib (x 2 ), we fix the overall amplitude in (20) by fitting to the single point at the largest trap separation, where we expect the WKB approximation to be most accurate. The semiclassical curves stop at the distances where the potential barrier disappears. We observe that for the same ratio of R * to l the splittings in 1D are larger than in 3D. Fig. 14 shows the prediction of the semiclassical formula Eq. ( 20 ), and the energy splittings calculated numerically for molecular states with different l. We note that separations at the avoided crossing are largest for the spherically symmetric molecular states, with l = 0. The knowledge of the energy splitting ∆E can be used to calculate the probabilities of an adiabatic and diabatic passage of avoided crossings. Assuming that the avoided crossing is traversed at constant rate, we apply the formula (18) 12) , which leads to\n\nwith ∂E 1 (d)/∂d ≈ 0 and ∂E 2 (d)/∂d = Ψ mol (d)|∂H rel (d)/∂d|Ψ mol (d) ≈ µω 2 d (c.f. Figs. 8 and\n\np |1 →|1 ′ = exp - π 2 (∆E) 2 | ḋ|µω 2 d , ( 22\n\n)\n\nwhere we use the same notation as in Fig. 13 . Analyzing Figs. 14 and 15 we can now estimate the rates ḋ required 13 1.0 1.2 1.4 1.6 10 -4 10 -3 10 -2 10 -1 s-wave semiclassical s-wave numerical p-wave semiclassical p-wave numerical d-wave semiclassical d-wave numerical E/ d/R * R * = 3.48 l FIG. 15: (Color online) Energy splitting at the avoided crossing between vibrational ground state and molecular states with different angular momenta, for a 3D spherically symmetric trap. Semiclassical results obtained by means of the instanton technique (solid lines) are compared to the numerical values extracted from the adiabatic energy spectra. Semiclassical calculations stop at the distance d = 1.1R * where the potential barrier disappears.\n\nfor adiabatic and diabatic transitions. For instance in 3D, for the parameters of Fig. 14 , the diabatic transfer of particles across the avoided crossings up to distances d ≈ R * (∆E 0.1 ω) can be realized by keeping ḋ/R * ≫ 0.001ω . In summary, the analysis carried out here provides the basis for a description of the dynamics of various processes of interaction between an atom and an ion manipulated through external trapping potentials, and it gives a way to estimate quantitatively with simple analytical means the outcome of controlled interaction experiments in the different regimes described in Sect. II.\n\nE. Center of mass coupled to the relative motion: 1D analysis for ωi = ωa\n\nIn this section we consider the effects of coupling between COM and relative motions in 1D, that appear when the trapping frequencies for atom and ion are not equal.\n\nFig. 16 shows the adiabatic levels as a function of trap separation for some example parameters: ω i = 5.5ω a and l a = 0.9R * . This choice corresponds to the interaction of 40 Ca + and 87 Rb in the traps with ω a = 2π × 10kHz and ω i = 2π × 55kHz. To obtain the adiabatic spectrum presented in Fig. 16 , we performed the diagonalization of the Hamiltonian (6) in the product basis of the COM and relative motion eigenstates evaluated at d = 0. In the calculations we consider all the states with a total energy E ≤ 460 ω a , which leads to about 7100 states in the basis.\n\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -5 0 5\n\n10 |2 a |0 i |1 a |0 i |0 a |0 i CM |1 a |1 i |0 a |1 i E/ a d/R * = 5.5\n\na e = -/ 4 o = + / 4 CM FIG. 16: Energy spectrum for atom and ion confined in harmonic traps with ωi = 5.5ωa, as a function of the distance d. Calculations are performed for ϕe = -π/4, ϕo = π/4 and la = 0.9R * (see text for details).\n\nThe arrows on the right-hand side of Fig. 16 indicate the asymptotic states for large separations, which can be labeled by the number of phonons for the atom and ion trap, respectively. In comparison to the case where relative and COM motions are decoupled, we observe the following new features: (i) the molecular spectrum contains states with different numbers of excitations in the COM degree of freedom. This can be observed at d = 0, when the molecular levels are with a good approximation equally separated by ω CM where\n\nω 2 CM = (m a ω 2 a + m i ω 2 i )/M . [26];\n\n(ii) the avoided crossings between molecular and vibrational states are weaker for the states involving vibrational excitations of the ion. This can be observed by comparing the avoided crossings for the state |0 a |1 i with the avoided crossings for the neighboring states. This behavior can be understood when we notice that for weaker atom trap (for ω a < ω i ), the particles have larger probability to tunnel when the atom is excited.\n\nF. Large distances between traps: two coupled oscillators approximation Finally we turn to the limit of large distances: d ≫ R i , R a . Expansion of the interaction up to second-order terms in the distance r leads to\n\nH 1D ≃ ν=i,a p 2 ν 2m ν + 1 2 m ν ω 2 ν (z ν -zν ) 2 + 10 αe 2 d 6 (z i -zi )(z a -za ), ( 23\n\n)\n\nwhere zν denotes the equilibrium position of the particles and we neglect the second-order terms modifying the trapping frequencies. The Hamiltonian (23) describes the 14 system of two coupled harmonic oscillators, which can be written in the form\n\nH 1D = ω a a † a + ω i b † b + ω c (a + a † )(b + b † ), ( 24\n\n)\n\nwhere we have introduced the usual annihilation and creation operators\n\na = m i ω i 2 z i + i p i m i ω i ( 25\n\n) b = m a ω a 2 z a + i p a m a ω a . ( 26\n\n)\n\nHere ω c denotes the coupling frequency: ω c = 10E * (R * ) 4 l a l i /d 6 . The validity of the considered model is limited by the assumption that a stable equilibrium position exists, which is fulfilled for R i , R a 0.57d, as it can be easily verified. Typical, maximal values of ω c for atoms in optical potentials and ions in rf traps are of the order of 10kHz. In practice the model ceases to be valid already at weaker conditions, when the terms higher than the second order cannot be neglected. In any case, this model is relevant for an important class of processes involving coherent transfer of quanta between the atom and the ion, of direct application in a quantum information processing context."},{"section_type":"CONCLUSION","section_title":"IV. CONCLUSIONS AND OUTLOOK","text":"In this paper we analyzed in detail the interaction between a single atom and a single ion guided by external trapping potentials. This kind of work is motivated by recently opened experimental possibilities within combined systems -currently being built in several groups worldwide -where magneto-optical traps or optical lattices for neutral atoms coexist with electromagnetic traps for ions.\n\nTight confinement of single particles, associated with independent control of the respective confining potentials, allows for exploring different regimes of the twobody dynamics involving one atom and one ion at a time. At large distances, the interaction is dominated by the inverse quartic term arising from the attraction between the ion's charge and the electric dipole induced by it on the atomic electron wave function. A pseudopotential approximation, similar to that familiar from ultracoldatom collision theory, is not applicable in our case, as the characteristic range of the atom-ion molecular potential often exceeds the size of the tight traps experimentally available. To describe the interaction at short distances, smaller than that range, we employ quantum-defect theory, which allows to deal with different geometries from isotropic three-dimensional traps to very elongated quasione-dimensional ones.\n\nA good description of adiabatic dynamics, involving processes where the traps are moved toward or across each other at a rate much slower than the trapping frequencies, can be obtained from quasistatic eigenenergy curves, which we calculate for various trapping configurations based on the methods outlined above. A remarkable feature displayed by the system is the presence of resonances between molecular-ion bound states and motional excitations within the trap. These trap-induced resonances are similar in nature to Feshbach resonances driven by external fields, and they could as well be spectroscopically probed in simple experiments where the interaction is controlled via the external guiding potentials.\n\nIn addition to the above aspects, an important motivation for the interest in systems of trapped atoms and ions and their trap-induced resonances resides in possible applications to quantum information processing. In this context, one can utilize controlled atom-ion interactions to effect coherent transfer of qubits, thereby creating interfaces between atoms and ions. By storing quantum information in internal atomic states, and performing gate operations with ions, one would combine the advantages of both: (i) long decoherence times for neutral atoms and (ii) short gate-operation times for charged particles due to the relatively strong interactions. In the present work we have focused on the motional degrees of freedom, which can serve as auxiliary degrees of freedom for quantum gates involving internal-state qubits, provided an appropriate coupling mechanism between internal and external degrees of freedom is employed -e.g., sideband excitation via a laser.\n\nAnother potential application of our results is cooling of the atomic motion. In typical neutral-atom quantum computation schemes with qubits stored in internal states of atoms trapped in optical-lattice sites, motional excitations constitute a serious source of errors, and the atomic motion needs to be cooled in a state-insensitive manner between computational steps in order to avoid qubit decoherence. Since the long-range part of the atom-ion interaction is not sensitive to the internal state [27] , our setup can be applied for sympathetic cooling of atoms, through the exchange of energy with laser cooled ions [12] .\n\nBeside these applications in the context of quantum information processing, our work opens broader perspectives for the study of new interesting collisional physics in a physical situation never explored before. In principle, our scheme allows for production of ultracold trapped atom-ion molecules (molecular complexes), when the trapping potentials are lowered adiabatically at the stage when the particles remain close to each other. In this way our method can be regarded as a way to perform cold chemical reactions, where the final state of the molecule can be well controlled. Beyond sufficiently large separation between traps, however, the survival probability of such molecules in the final state is negligible. Indeed, in the calculations presented here, we have only regarded processes leading to a final state in which the atom and the ion are still separated, and no molecule has been formed as an outcome of the interaction. In other words, the Hamiltonian describing our single-channel model does not include the possibility of 15 transitions to other Born-Oppenheimer curves. While this is a justified assumption for the dynamics considered here, it is in a sense a limitation of our current approach. Investigation of molecular formation in traps beyond this approximation is certainly among the interesting developments that can arise from our work. Its long-term motivation is really to open, beyond the present examples, a new paradigm for cold collision physics, which can be described as the mechanical control of single-particle chemical reactions."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"The authors thank J. Denschlag, P. Schmidt, R. Stock, and A. Simoni for helpful discussions. This work was supported by the Austrian Science Foundation (FWF), the European Union projects OLAQUI (FP6-013501-OLAQUI), CONQUEST (MRTN-CT-2003-505089), the SCALA network (IST-15714), the Institute for Quantum Information GmbH, the EU Marie Curie Outgoing International Fellowship QOQIP, and the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory.\n\nAPPENDIX A: DERIVATION OF THE EFFECTIVE HAMILTONIAN (1)\n\nWhile the Hamiltonian (1) is intuitively obvious, we find it nonetheless worthwhile to summarize the microscopic derivation in an adiabatic approximation, and to discuss its validity. For simplicity we consider an atom with a single outer-shell electron (alkali atom) and an ion with single positive charge +e. In addition we do not consider the internal structure of the ion. The total Hamiltonian can be written as\n\nH = p 2 1 2m 1 + p 2 2 2m 2 + H e + H las + H rf . ( A1\n\n)\n\nHere the labels 1 and 2 correspond, respectively, to atomic nucleus and ion, and H e is the Hamiltonian of the electron, which includes the Coulomb interactions,\n\nH e = p 2 e 2m e - e 2 |x e -x 1 | - e 2 |x e -x 2 | + e 2 |x 2 -x 1 | . (A2)\n\nHere, for simplicity, we omit the contributions of the core regions for both the atom and ion. In this way our model refers in fact to H + 2 molecule. For alkali atoms and alkaline-earth ions one should treat the complete structure of core regions, however, our approach can be readily generalized to this more complicated case.\n\nThe term H las describes the interaction of the atom with a laser beam creating an optical potential, written in the electric dipole representation\n\nH las = -dE ⊥ (x 1 , t), ( A3\n\n)\n\nwhere d = e(x 1x e ) is the dipole moment of the atom, and for simplicity we neglected the influence of the laser on the ion, which has typically a different electronic structure that the atom. In addition we have applied the long-wavelength approximation, neglecting changes of the electric field on the scale of the atom. In the case of the optical lattice the transverse part of the electric field E ⊥ can be assumed to have the form of a standing wave\n\nE ⊥ (x, t) = E 0 cos(ω L t) [cos(k L x + φ x ) + (x → y, z)] , (A4)\n\nwhere for simplicity we have assumed the same amplitude E 0 , the same wave vector k L , and the same frequency of the laser light ω L for all three laser beams creating the optical lattice potential. The abbreviation (x → y, z) denotes sum of the terms with x replaced by y and z, and ϕ k for k = x, y, z is the phase factor characterizing the standing wave. Finally, H rf is the electric potential creating the rf-trap\n\nH rf = eΦ(x 1 , t) + eΦ(x 2 , t) -eΦ(x e , t), ( A5\n\n)\n\nwhere Φ(x 1 , t) is the time-dependent electric field of the rf trap\n\nΦ(x, t) = 1 2 (u x x 2 + u y y 2 + u z z 2 ) + 1 2 (v x x 2 + v y y 2 + v z z 2 ) cos ω rf t. ( A6\n\n)\n\nHere, ω rf is the frequency of the time-dependent part of the electric potential, and u k , v k (k = x, y, z) are amplitudes depending on the geometry of the trap [5]. The electric field at every instant of time has to fulfill the Laplace equation: ∆Φ = 0, hence, the coefficients u k , v k are subject to the following conditions:\n\nu x + u y + u z = 0, v x + v y + v z = 0.\n\nBelow we indicate the basic steps of the derivation. Expansion in the basis of Born-Oppenheimer wave functions for electron motion. We start from generating a complete set of electronic wave functions, parametrized by the positions of the atomic core and of the ion\n\nH e Φ n (x e |x 1 , x 2 ) = E n (|x 2 -x 1 |)Φ n (x e |x 1 , x 2 ) (A7)\n\nIn this way the total wave function can be expanded in the basis of Born-Oppenheimer electronic wave functions\n\nΨ(x 1 , x 2 , x e , t) = n c n (x 1 , x 2 , t)Φ n (x e |x 1 , x 2 ). (A8)\n\nSince the basis is complete, the expansion of the function does not involve any approximations.\n\nRetaining in the expansion only the modes coupled by the laser. In the expansion (A8) we keep only two modes coupled by the laser light creating the optical lattice: the electronic ground state Φ g (x e |x 1 , x 2 ) and the electronic excited state Φ e (x e |x 1 , x 2 ). In this way we utilize the Born-Oppenheimer approximation, treating the electron 16 motion in the adiabatic approximation. This assumes that the time scale of the electron dynamics is much faster than the dynamics of the atomic nucleus and of the ion, which is typically fulfilled since the electron is much lighter than the other two particles. The approximation of the two coupled channels can be easily generalized to more, or even infinite number of channels, since the other channels are weakly populated and we treat them within the perturbation theory. Adiabatic elimination of the excited electronic state coupled through the laser: derivation of the optical trap potential. The expansion coefficients c g (x 1 , x 2 , t) and c e (x 1 , x 2 , t) fulfill the following set of coupled equations:\n\ni ∂c g (x 1 , x 2 , t) ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E g (|x 2 -x 1 |) + eΦ(x 2 , t) c g (x 1 , x 2 , t) -d eg E ⊥ (x 1 , t)c e (x 1 , x 2 , t) ( A9\n\n) i ∂c e (x 1 , x 2 , t) ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E e (|x 2 -x 1 |) + eΦ(x 2 , t) c e (x 1 , x 2 , t) -d eg E ⊥ (x 1 , t)c g (x 1 , x 2 , t), ( A10\n\n)\n\nwhere d eg (x 1 , x 2 ) = Φ e |d|Φ\n\ng is the dipole matrix element between the ground and excited electronic states, which in general depends on the position of atom and ion. In the derivation of (A9)-(A10) we have neglected the action of the rf field on the atom core and the electron, putting\n\nΦ k |H rf |Φ k ≈ eΦ(x 2 , t), k = e, g (A11) Φ e |H rf |Φ g ≈ 0. ( A12\n\n)\n\nWe focus on the regime of far-detuned laser: ∆ ≫ Ω L , where ∆ denotes the detuning ∆\n\n(|x 2 -x 1 |) = E e (|x 2 - x 1 |) -E g (|x 2 -x 1 |) -ω L\n\n, and Ω L = -d eg E 0 / is the the Rabi frequency. For such conditions, the excited state is only weakly populated and can be adiabatically eliminated. Additional simplification comes from the fact that the transitions between states g and e, due to the laser light, occur on a time scale much shorter than the motion of atom and ion, and the dynamics of the atom and ion centers of masses can be decoupled from the internal dynamics, in full analogy to the Born-Oppenheimer approximation. Eliminating the excited state, in basically the same manner as in the standard derivation of the AC Stark shift, we obtain the following equation that governs the dynamics of the atom and ion center of masses:\n\ni ∂c g (x 1 , x 2 , t) ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E g (|x 2 -x 1 |) + V opt (x 1 ) + eΦ(x 2 , t) c g (x 1 , x 2 , t), ( A13\n\n) where V opt (x) = - Ω 2 L 4∆ cos(k L x + φ x ) 2 + (x → y, z) ( A14\n\n)\n\nis the effective potential due to the laser field. In the formula (A14) we have neglected the position dependence of the detuning ∆(|x 2x 1 |) and of the dipole matrix element d eg (x 1 , x 2 ), assuming that they are modified only at very short distances between the particles. This can be justified, since in the atomic collisions, the probability of finding the particles at short distances, comparable to the range of chemical binding forces, is typically very small. Time averaging over fast oscillations of the rf field: derivation of the Paul trapping potential. We replace the time-dependent rf field by an effective, adiabatic potential. In this way we neglect the fast micromotion of the ion on the time scale of ω rf [5] . Finally we obtain the following equation that describes the dynamics of the wave function c g (x 1 , x 2 , t) dependent on the atom and ion positions\n\ni ∂c g ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E g (|x 2 -x 1 |) + V opt (x 1 ) V rf (x 2 ) c g (x 1 , x 2 , t). ( A15\n\n)\n\nHere V rf denotes the effective potential\n\nV rf (x) = 1 2 m 2 ω x x 2 + ω y y 2 + ω z z 2 , ( A16\n\n)\n\nwhere ω k = (a k + q 2 k /2) 1/2 ω rf /2, a k = 4eu k /(m 2 ω 2 rf ) and q k = 2ev k /(m 2 ω 2 rf ) for k = x, y, z. The interaction between atom and ion is given by\n\nE g (|x 2 -x 1 |) -\n\nthe ground-state energy of the electron motion, which at large distances behaves as E g (r) ∼ -αe 2 /(2r 4 ).\n\nAPPENDIX B: DETERMINING OF ϕe AND ϕo FROM PSEUDOPOTENTIALS\n\nIn principle, the replacement of the r -4 interaction by the pseudopotential is strictly valid when R * ≪ l ⊥ . Nevertheless, it is possible to try to determine ϕ e and ϕ o for R * ∼ l * using the energy-dependent pseudopotentials. To this end we replace the r -4 interaction with [17, 18] V s (r) = 2π 2 a(k) µ δ(r) ∂ ∂r r (B1) 17 for even scattering, and [19] V p (r) = π 2 a p (k) 3 µ ← ∇ δ(r) → ∇ r ∂ 3 ∂r 3 r 2 , (B2) for odd scattering. Here, a(k) = -tan δ 0 (k)/k is the energy-dependent s-wave scattering length, a p is the pwave scattering length: a p (k) 3 = -tan δ 1 (k)/k 3 , the symbol ← ∇ ( → ∇) denotes the gradient operator that acts to the left (right) of the pseudopotential, and δ 0 (k), δ 1 (k) are the s-and p-wave phase shifts, respectively.\n\nIn 1D the even (ψ e ) and odd (ψ o ) scattering waves have the following asymptotic behavior: ψ e (z, k) ∼ sin(k|z| + δ e (k)), and ψ o (z, k) ∼ z/|z| sin(k|z| + δ o (k)) (|z| → ∞), where δ e (k) and δ o (k) denote even and odd scattering phases, respectively. In analogy to the three-dimensional scattering theory, we define 1D even (a e 1D ) and 1D odd (a o 1D ) scattering lengths: a e,o 1D = lim k→0 -(tan δ e,o (k))/k. From this definition it follows that for k = 0 the even and odd scattering waves behaves as: ψ e (z, k = 0) ∼ |z|-a e 1D and ψ o (z, k = 0) ∼ z -a o 1D z/|z| (|z| → ∞). The latter result applied to Eqs. (10) and (11) leads to a e 1D /R * = -cot ϕ e (B3) a o 1D /R * = -cot ϕ o (B4)\n\nIn the pseudopotential approximation one can solve the quasi-1D scattering problem exactly, and calculate values of the 1D scattering lengths for even [20, 21] and odd [22] waves a e 1D (k) = -\n\nl 2 ⊥ 2a s (E) - l ⊥ 2 ζ 1 2 , 3 2 - E 2 ω ⊥ , ( B5\n\n)\n\na o 1D (k) = l ⊥ 2 l 3 ⊥ 12a p (E) 3 -ζ - 1 2 , 3 2 - E 2 ω ⊥ -1 , ( B6\n\n)\n\nwhere E = ω ⊥ + 2 k 2 /(2µ) and ζ(s, a) denotes the Hurwitz Zeta function: ζ(s, a) = ∞ k=0 (k + a) -s [23] . Finally, to relate the energy-dependent scattering lenghts a s (E) and a p (E) to ϕ, we apply the exact solutions of the Schrödinger equation for r -4 potential, given by the Mathieu functions [7, 8, 9].\n\n[1] N. Schlosser, G. Reymond, I.E. Protsenko and P. Grangier, Nature (London) 411, 1024 (2001). [2] Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Förster, M. Khudaverdyan, D. Meschede, D. Schrader, and A. Rauschenbeutel, Nature (London), 442, 151 (2006). [3] M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39-44 (2002). [4] R. Folman et al., Adv. At. Mol. Opt. Phys. 48 263 (2002); R. Dumke et al., Phys. Rev. Lett. 89, 97903 (2002). [5] D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev.\n\nMod. Phys. 75, 281 (2003). [6] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, Nature 425, 937-940 (2003). [7] E. Vogt, and G.H. Wannier, Phys Rev. 95, 1190 (1954). [8] T.F. O'Malley, L. Spruch, L. Rosenberg, J. Math. Phys. 2, 491 (1961). [9] R.M. Spector, J. Math. Phys. 5, 1185 (1964). [10] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). [11] R. Stock, I. H. Deutsch, and E. L. Bolda Phys. Rev. Lett.\n\n91, 183201 (2003). [12] Z. Idziaszek, T. Calarco, P. Zoller, in preparation. [13] D.J. Tannor Introduction to Quantum Mechanics: A Time-Dependent Perspective, (University Science Books, 2006). [14] L.D. Landau, and E.M. Lifshitz Quantum Mechanics: Non-relativistic Theory, 3rd edition (Butterworth-Heinemann, Oxford, 2003). [15] C. Zener, Proc. R. Soc. London A137, 696 (1932).\n\n[16] A. Auerbach, and S. Kivelson, Nucl. Phys. B 257, 799 (1985). [17] D. Blume, C.H. Greene, Phys. Rev. A 65, 043613 (2002). [18] E.L. Bolda, E. Tiesinga, and P.S. Julienne, Phys. Rev. A 66, 013403 (2002). [19] Z. Idziaszek, and T. Calarco, Phys. Rev. Lett. 96, 013201 (2006). [20] T. Bergeman, M.G. Moore, and M. Olshanii, Phys. Rev. Lett. 91, 163201 (2003). [21] Z. Idziaszek, and T. Calarco, Phys. Rev. A 74, 022712 (2006). [22] B.E. Granger, D. Blume, Phys. Rev. Lett. 92, 133202 (2004). [23] M. Abramowitz and I.E. Stegun Handbook of Mathematical Functions, 10th edition, (Department of Commerce, Washington DC, 1972). [24] For atomic states with a permanent quadrupole moment, the long-range part of the potential comes from the interaction of ion charge with the atom quadrupole moment. [25] 1D solutions (10) and (11) are valid at all distances for k → 0 (E → ω ⊥ ). [26] COM and relative motions can be decoupled in the lowest order when the size of the molecule is much smaller than the harmonic oscillator lengths li and la. [27] At least for the internal states with the total angular momentum J = 1/2."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"Techniques developed in atomic physics during the last two decades allow the preparation of single atoms and ions in the laboratory. Single neutral atoms can be stored and laser cooled in far off-resonance laser traps (FORT) [1, 2] , and arrays of atoms can be prepared in an optical lattice via a Mott insulator phase, where exactly one atom is stored per lattice site [3] . Similar techniques are being pursued in the context of atom chips [4] . Furthermore, single ions and arrays of ions can be stored in Paul and Penning traps, and sideband laser cooling allows to prepare ions in the vibrational ground state of these trapping potentials [5] . The controlled preparation, manipulation and measurement of electronic internal and motional states of laser driven single atoms and ions, combined with the possibility of controlled entanglement of atoms and ions provides the basic ingredients to for investigations of fundamental aspects of quantum mechanics, and applications such as high-precision measurements and quantum information. In quantum information processing entanglement of qubits stored in internal states of single atoms or ions is achieved by controlled interactions between particles, either in the form of switchable qubit-dependent two-particle interactions or via an auxiliary collective mode of the systems, which serves as a quantum data bus. In particular, for neutral atoms cold controlled collisions of atoms stored in mov-able spin-(i.e. qubit-)dependent optical lattices has been proposed as a means to entangle atomic pairs, and has experimentally implemented to generate N -atom cluster states [6] .\\n\\nIn the present work we study the controlled cold collision of a single atom and a single ion, where the atom (ion) is prepared in a given motional state of an atom (ion) trap, and we move the traps to guide the atom and ion wave packet to \\\"collide\\\" for a given time (see Fig. 1 ). Our focus is the development of a quantumdefect formalism for trapped atoms and ions, in a form which is convenient for future applications and extensions, in particular in the context of quantum information processing to swap qubits stored in atoms and ions, and the entanglement of these qubits in a controlled collision. The atom-ion collision is governed by the potential V (r) → -αe 2 /(2r 4 ) (r → ∞), where α is the dipolar polarizability -to be contrasted to a van der Waals potential V (r) → -C 6 /(r 6 ) which represents the collisional interactions between e.g. alkali atoms in their electronic ground state. Below we will calculate the dynamics of the interacting atom-ion system for a given time-dependence of the motion of the trap. This includes the transition to excited trap states after the collision, the possible formation of an excited atom-ion (molecule) complex, and the description of trap induced resonances.\\n\\nTo study the controlled atom-ion collisions we first derive an effective Hamiltonian, describing effective net forces acting on the particles moving in rapidly changing 2 (d) (c) (b) (a) FIG. 1: (Color online) Schematic drawing of a controlled collision between a single atom and a single ion, whose centerof-mass wavepackets are guided by time-dependent atom trap and ion trap, respectively. Labels distinguish different phases of the process: a) initially particles prepared in the motional ground state; b) collision (overlap of the wave packets); c) excitation of the motional states in the traps; d) particles in some excited states after the collision.\\n\\nlaser and rf fields. In our approach, we include in the Hamiltonian only the asymptotic part of the atom-ion potential, whereas the short-range interactions are taken into account by imposing appropriate boundary conditions on the wave function at r → 0. In the regime of cold collisions considered here, the boundary conditions can be expressed in terms of a single quantum-defect parameter, independent of the collisional energy and angular momentum. The quantum-defect approach is dictated by the relatively long-range character of the atom-ion interaction, which exceeds the typical size of the trapping potentials, and, in contrast to the neutral atoms, excludes the applicability of the pseudopotential.\\n\\nOur method to describe the dynamics of the controlled collision is based on the application of the correlation diagrams, i.e. energy spectra as a function of the trap separation. Such correlation diagrams, widely used in quantum chemistry to characterize reactions of diatomic molecules, connect in our case the asymptotic vibrational states with the molecular and vibrational states at zero trap separations. At the intermediate distances the energy curves exhibit avoided or diabatic crossings, depending on the symmetry of eigenstates and of the coupling term in the Hamiltonian. In the atom-ion system the avoided crossing can be attributed to the resonances between molecular and vibrational states, that appear when the energy of a vibrational level, coincide with the energy of a molecular state shifted by the external trapping. The dynamics in the vicinity of such avoided crossings can be accurately described in the framework of the Landau-Zener theory. In this paper we determine the level splitting at the avoided crossing, at different trap separations, and for different symmetries of the molecular states. In this way we can characterize the time scale appropriate for adiabatic or diabatic traversing of a given avoided crossing.\\n\\nIn our paper we consider two different geometries of the trapping potentials: spherically symmetric traps and very elongated cigar-shape traps (quasi-1D traps). The former case requires full three-dimensional treatment, while the latter one can be described using an effective one-dimensional Hamiltonian. In the latter case, the tight transverse confinement effectively renormalizes the one-dimensional quantum-defect parameters. In addition, very elongated cigar-shape traps exhibit another type of resonances, that can be observed at zero-trap separations, but for changing ratio of the transverse confinement to the s-wave scattering length of the atom-ion interaction. Such confinement-induced resonances appear when the energy of the colliding particles coincide with energy of a bound state lifted up by the tight transverse potential.\\n\\nAs already outlined above, the counterpart to the idea of ultracold controlled trap-guided collisions has been implemented in optical lattices, where application of spindependent potentials has allowed to entangle atoms between neighboring sites [6] . Moreover, resonance phenomena in trapped systems, similar to the ones considered here, have been already thoroughly investigated for interactions between neutral atoms. In quasi-1D systems, collisions between neutral atoms exhibit confinementinduced resonances [10] , whereas displacement of the trapping potentials leads to trap-induced resonances [11] .\\n\\nThe paper is organized as follows. In section II we present our model that we use to describe the controlled atom-ion collisions. In particular, in section II A we introduce an effective Hamiltonian of a single trapped atom and ion. Section II B discuss the quantum-defect treatment of the short-range part of the interaction potential. In sections II C and II D we discuss our model for two particular geometries of the trapping potentials, spherically symmetric traps of the same frequency and elongated cigar-shape traps with the same transverse trapping frequency, respectively. We extend the quantum-defect treatment to quasi-1D systems in section II E. Our approach to describe the dynamics of the atom-ion system is discussed in section II F. Section III presents the results of our calculations. We start from discussing the properties of the scattering in quasi-1D systems in the presence of long-range r -4 potential, in particular, the problem of determining of 1D quantum-defect parameters. Sections III B and III C present adiabatic energy curves and adiabatic eigenstates in, respectively, 1D and 3D traps with the same trapping frequencies for the atom and the ion. The dynamics in the vicinity of the avoided crossing is discussed in section III D, where we apply the Landau-Zener theory and semiclassical methods to address this problem. Section III E discuss most complicated case of different trapping frequencies, focusing on the 1D atomion motion. The outlook and final conclusions are presented in section IV. Finally appendix A presents the microscopic derivation of the effective Hamiltonian and appendix B describes the details of the pseudopotential treatment of the scattering in quasi-1D traps.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. BASIC SETUP AND MODEL\",\n \"text\": \"We consider a system consisting of a single atom and single ion, stored in their respective trapping potentials. Single atoms can be trapped in experiments with optical tweezers [1, 2] or optical lattices [3]. Such potentials are created in far-detuned laser fields due to the AC Stark effect. In another experimental technique, on atom chips [4], single atoms can be trapped in microtraps created by electric and magnetic fields around wires and electrodes. On the other hand, single ions can be confined in radiofrequency traps, which use a rapidly oscillating electric field to create an adiabatic trapping potential for the ion charge [5] . Typically, the trapping potentials close to the trap center are with a good approximation harmonic, and in our approach we consider the simple picture of the atom and ion confined in the harmonic potentials with frequencies ω a and ω i , respectively, with d denoting the displacement between the traps (c.f. Fig. 2.a ). We assume that the atom and ion are prepared initially in a given trap state, e.g. the ground state of the potential, and that the atom and ion are brought to collision by overlapping their center-of-mass (COM) wave packets, i.e. we change d(t) as a function of time as outlined in Fig. 1 .\\n\\nIn studying this collision with COM wave packets guided by movable trapping potentials we will study two cases of (i) three-dimensional (3D) dynamics, where both traps are assumed to be spherically symmetric, and (ii) the case of quasi-1D traps, i.e. elongated, cigar-shaped traps which can be described as an effectively 1D collision dynamics. This second case is conceptually simpler, but also provides collisional features specific to 1D situation (e.g. the appearance of 1D confinement-induced resonances, reminiscent of resonances in [10] ). The values of trapping frequencies in atom and ion traps are typically quite different, thus it is natural to consider ω a = ω i . For such condition, however, the COM and relative motion are coupled, which significantly complicates the theoretical description. Apart from this general regime, we consider also the special case of ω a = ω i , when COM and relative motions can be separated. In this case we perform the full diagonalization for the 3D problem, which is numerically very demanding in the general case of ω a = ω i .\\n\\nAt large distances, the interaction between atom and ion is given by a potential V (r) ∼ -αe 2 /(2r 4 ), where α is the dipolar polarizability. This potential originates from the attraction between the ion charge and the electric dipole that it induces on the atom [24] . At short distances the interaction is more complicated; however, it turns out that a detailed knowledge of the form of core region of the potential is not necessary, and that it can be subsumed in our model by introducing a set of (energy independent) quantum-defect parameters.\\n\\nLet us focus now on characteristic scales that can be associated with the atom-ion interaction. For the sake of clarity table I summarizes all the length scales that d R i , R a d R i , R a (c) (b) l i l a i ION d ATOM a (a) FIG. 2: (Color online) Panel (a): Schematic drawing of the considered setup consisting of a trapped atom and ion. Panels (b) and (c): Long and short-range regimes, respectively, in the controlled collisions between a trapped atom and ion.\\n\\nare introduced in the course of the paper. We define its characteristic length as R * ≡ αe 2 µ/ 2 , and characteristic energy as E * = 2 / 2µ(R * ) 2 , where µ is the reduced mass. Other characteristic lengths and energies are given by the ground states of the trapping potentials. We associate to the trapping frequencies the harmonic oscillator lengths for atom (ion): l a = /(m a ω a ) (l i = /(m i ω i )). For optical and rf traps, the trapping frequencies are typically ω a = 2π × 10 -100kHz and ω i = 2π × 0.1 -10MHz. In table II we collected some example values of R * , l i , l a , and E * for a few systems of alkali atoms and alkaline earth ions. One can observe that the range of the polarization potential is comparable or larger than the size of the trapped ground state. Therefore, for the trapped particles the polarization interaction cannot be replaced with the standard contact pseudopotential familiar from s-wave atom-atom scattering in the context, e.g., of Bose-Einstein condensation.\\n\\nIn the interactions between trapped atom and ion, one can distinguish two different regimes depending on the relative distance d between traps: (i 1/6 for ν = i, a is some characteristic distance at which the atom-ion interaction becomes comparable to the trapping potential of the ion and atom, respectively. Because of the weak dependence of R ν on ω ν and m ν , the characteristic distances R i and R a are roughly the same. The two regimes (i) and (ii) correspond to the cases where the motion in the traps is weakly or strongly affected by the atom-ion interactions, respectively. The discussed regimes are illustrated schematically in panels (b) and (c) of Fig. 2 , and they correspond to snap shots in the collision process when the wavepackets of the particles do not overlap / do overlap during the controlled collision. In the regime (i) of large distances, the atom-ion interaction can be treated perturbatively as a distortion of the trapping potential. In this limit the system can be described in terms of two coupled harmonic oscillators, and all the dy-4 Definition Description R * = p αe 2 µ/ 2 characteristic length of the polarization potential li = p /(miωi) harmonic oscillator length of ion trap la = p /(maωa) harmonic oscillator length of atom trap l = p /(µω) harmonic oscillator length for relative degrees of freedom (axial direction in case of quasi 1D traps) l ⊥ = p /(µω ⊥ ) harmonic oscillator length in the transverse direction for relative degrees of freedom Ri = \\\" αe 2 miω 2 i « 1/6 distance at which the polarization potential becomes equal to the ion trappping potential Ra = \\\" αe 2 maω 2 a « 1/6 distance at which the polarization potential becomes equal to the atom trappping potential R rel = \\\" αe 2 µω 2 « 1/6 distance at which the polarization potential becomes equal to the trappping potential for relative degrees of freedom R ⊥ = \\\" αe 2 µω 2 ⊥ « 1/6 distance at which the polarization potential becomes equal to the trapping potential in the transverse direction for relative degrees of freedom R1D = max(R ⊥ , l ⊥ ) boundary of the quasi-1D regime R0 size of the core region of the atom-ion complex TABLE I: Definitions of the length scales used throughout the paper.\\n\\n) d ≫ R i , R a , (ii) d R i ∼ R a , where R ν = αe 2 /m ν ω 2 ν\\n\\nR * (a0) li(a0) la(a0) E * /h (kHz) 40 Ca + + 87 Rb 3989 300 644 4.143 9 Be + + 87 Rb 2179 632 644 46.59 40 Ca + + 23 Na 2081 300 1252 28.56\\n\\nTABLE II: Characteristic distance R * , characteristic energy E * , harmonic oscillator length li for ion trap of ωi = 1MHz, and harmonic oscillator length la for atom trap of ωa = 100kHz, for some combinations of alkali earth ions and alkali atoms.\\n\\nnamics can be solved analytically. We discuss this case in more details in section III F. On the other hand, in the regime (ii) the description is more difficult, since it requires inclusion of the short-range part of the interaction, and full treatment of the long-range r -4 part. In the present paper we focus mainly on the latter case.\\n\\nA. Effective Hamiltonian We adopt the following time-dependent Hamiltonian to describe the system of a single trapped atom and a single trapped ion (c.f. Fig. 2 ):\\n\\nH(t) = ν=i,a p 2 ν 2m ν + 1 2 m ν ω 2 ν (z ν -d ν (t)) 2 + 1 2 m ν ω 2 ⊥ν ρ 2 ν + V (|r i -r a |), ( 1\\n\\n)\\n\\nHere the label i (a) refers to the ion (atom) respectively, p and r are the momentum and position operators, d ν (t) denotes the positions of the atom and ion traps, respectively, that can be controlled in the course of dynamics, and ρ 2 = x 2 + z 2 . We assume that the trapping potentials are axially symmetric and displaced along the axis of symmetry. The trapping frequencies are denoted by ω ν and ω ⊥ν for the axial and transverse directions, respectively. Finally, V (r) denotes the interaction potential between the atom and the ion. At large distances the main contribution to this interaction comes from the polarization of the atomic cloud, V (r) ∼ -αe 2 /(2r 4 ). A microscopic derivation of the Hamiltonian (1) is presented in appendix A. The basic idea of this derivation is based on the application of the Born-Oppenheimer approximation to the motion of the outer shell electrons, and followed by a time averaging over fast time scale of the rf and laser frequencies. In this picture, the Hamiltonian (1) represents effective net interactions felt by particles moving in the rapidly changing time-dependent potentials. The interaction potential V (r) can be identified with the adiabatic Born-Oppenheimer curve of the electronic ground-state, that depends on the difference between the atom and ion COM coordinates.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Quantum-defect theory\",\n \"text\": \"We denote by R 0 the characteristic distance at which V (r) starts to deviate from the asymptotic r -4 law. Typically, R 0 is the size of the core region of the ionatom complex, and is much smaller than all the other length scales in our problem. Therefore, we can describe the short-range part of interaction in the spirit of the quantum-defect theory. For R 0 ≪ R * , the parameters describing the short-range potential become independent of energy and angular momentum, since for r ≪ R * the interaction potential is much larger than typical kinetic energies and heights of the angular-momentum barrier. This feature is a key ingredient of quantum-defect theory, where the complicated short-range dynamics can be summarized in terms of few, energy-independent constants (phase shifts or quantum defects).\\n\\nTo analyze the short-distance behavior of the wave functions, we omit for the moment the trapping potentials in the Hamiltonian (1), and consider only the part describing the scattering of atom from the ion in free space. In this case, the relative and COM degrees of 5 freedom are decoupled, and the relative motion is governed by the Hamiltonian H 0 = p 2 /2µ + V (r). We apply the partial wave expansion to the relative wave function: Ψ rel (r) = l R l (r)Y lm (θ, φ) with l denoting the angular momentum and Y lm (θ, φ) the spherical harmonics. For r ≫ R 0 , we set V (r) = -αe 2 /(2r 4 ), which allows to solve radial Schrödinger equation in terms of the Mathieu functions of the imaginary argument [7, 8, 9] . In this way we obtain the following short-distance behavior of the radial wave functions\\n\\nR l (r, k) ∼ sin (R * /r + ϕ l (k)) , r ≪ R * /k, ( 2\\n\\n)\\n\\nwhere k is the relative momentum, and ϕ l (k) are some short-range phases. As one can easily verify, the asymptotic solution (2) fulfills the radial Schrödinger equation with the energy 2 k 2 /(2µ) and centrifugal barrier 2 l(l+1)/(2µr 2 ) terms neglected. The short-range phases constitute our quantum-defect parameters. For R 0 ≪ R * we can assume that ϕ l (k) are independent of the energy and angular momentum: ϕ l (k) ≡ ϕ, which reduces description of the short-range interaction to the single quantum-defect parameter φ. In the calculations we replace V (r) by its asymptotic r -4 behavior, effectively letting R 0 → 0 and imposing boundary condition stated by (2) with the short-range phase φ.\\n\\nFor k = 0 the solution (2) becomes valid at all distances. Utilizing the fact that zero-energy solution behaves asymptotically as Ψ(r) ∼ 1 -a s /r (r → ∞), we can relate the short-range phase to the s-wave scattering length a s\\n\\na s = -R * cot ϕ ( 3\\n\\n)\\n\\nIn this way the knowledge of the scattering length a s allows us to calculate the quantum defect parameter ϕ.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Symmetric 3D trapping potential and identical trapping frequencies\",\n \"text\": \"In this section we consider 3D dynamics assuming, spherically symmetric trapping potentials ω ν = ω ⊥ν for ν = i, a and identical trapping frequencies for atom and ion: ω i = ω a = ω. For such conditions the relative and CM motions are decoupled.\\n\\nThe COM motion is governed by the Hamiltonian of a harmonic oscillator with mass M = m i + m a and frequency ω, while the Hamiltonian of the relative motion is given by\\n\\nH rel = p 2 2µ + 1 2 µω 2 (r -d) 2 - αe 2 2r 4 , ( 4\\n\\n)\\n\\nwhere p and r are the relative-motion momentum and position operators, respectively, and d = d ad i . In the case of relative motion we define\\n\\nR rel = αe 2 /µω 2 1/6 = (R * ) 1/3 l 2/3 ( 5\\n\\n)\\n\\n0 1 2 3 -1.0 -0.5 0.0 0.5 1.0 R 0 R rel z/R * /R * -5.000 0 5.000 10.00 15.00 20.00 25.00 30.00 d U(r)/ l FIG. 3: (Color online) Contour plot of the total potential U (r) = V (r) + 1 2 µω 2 (rd) 2 for the relative motion in the atom-ion system. The figure shows the case of spherically symmetric trapping potentials with the same trapping frequencies, separated by d = (0, 0, 2R * ), and for R * = 2l (l = p /(µω)).\\n\\nas a characteristic length at which the atom-ion interaction is comparable to the trapping potential for the relative degrees of freedom, where l = /(µω) denotes harmonic oscillator length.\\n\\nFig. 3 illustrates a typical potential for the atom-ion relative motion. In addition this figure depicts different lengths scales characteristic of our problem. In the presented case R * = 2l, and the distance between trap centers is d = 2R * .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D. Quasi-1D trapping potentials\",\n \"text\": \"As a second geometry we consider cigar-shape traps with a transverse trapping frequency much larger than the axial one: ω ⊥ν ≫ ω ν (quasi-1D traps) for ν = i, a.\\n\\nFor energies smaller than the excitation energy in the transverse direction, the motion in the transverse direction is frozen to zero-point oscillations, and the dynamics takes place along the weakly confined direction. Nevertheless, the transverse motion plays an important role at short distances, effectively renormalizing the short-range phase, as we will show in the next section. In this way quasi-1D traps offer the additional possibility of tuning the interactions, similarly to the case of neutral atoms exhibiting confinement-induced resonances [10] .\\n\\nHere, we consider only some particular situation when ω ⊥i = ω ⊥a is the same for atom and ion. This simplifies our description of the renormalization effects, since in this case the transverse COM and relative motions can be separated. We expect that our results are also qualitatively valid in the general case of different transverse trapping frequencies.\\n\\nTo obtain an effective 1D Hamiltonian, describing the evolution of the wave packets along the z axis, we decompose the total wave function of the atom and ion into a series over eigenmodes of the transverse part of the Hamiltonian. For total energies E < 3 ω ⊥ only the transverse ground-state mode contributes to the total wave function at large distances. In addition, condition 6 |z i -z a | ≫ R ⊥ assures that the axial and the transverse motions are decoupled and the wave function can be written as a product of the axial and the transverse components. Here, R ⊥ = αe 2 /µω 2 ⊥ 1/6 is some characteristic distance at which the atom-ion interaction becomes comparable to the transverse trapping potential, and the condition |z i -z a | ≫ R ⊥ describes the regime where the transverse oscillation frequency is weakly modified by the atom-ion interaction. Hence, we have\\n\\nΨ(r i , r a ) → ψ 0 (ρ i , ρ a )Ψ 1D (z i , z a ) (|z i -z a | → ∞) for E < 3 ω ⊥ .\\n\\nHere, ψ 0 is the ground state of the transverse part of the Hamiltonian:\\n\\nψ 0 (ρ i , ρ a ) = e -ω ⊥ (miρ 2 i +maρ 2 a )/2 /π 1/2 , and Ψ 1D (z i , z a )\\n\\ndenotes the axial part of the wave function. Substituting the decomposition over transverse modes into the Schrödinger equation with the Hamiltonian (1), and retaining only the lowest transverse mode we obtain an effective 1D Hamiltonian, that governs the dynamics of Ψ\\n\\n1D (z i , z a ) H 1D = ν=i,a p 2 ν 2m ν + 1 2 m ν ω 2 ν (z ν -d ν ) 2 + V 1D (|z i -z a |). ( 6\\n\\n)\\n\\nHere V 1D (|z|) is the 1D interaction potential obtained by integrating out the transverse degrees of freedom:\\n\\nV 1D (|z i -z a |) = dρ i dρ a |ψ 0 (ρ i , ρ a )| 2 V (|r i -r a |) ( 7\\n\\n)\\n\\nAt sufficiently large distances the effective 1D interaction has similar power dependence as in 3D\\n\\nV 1D (|z|) = - αe 2 2z 4 , |z| ≫ l ⊥ , ( 8\\n\\n)\\n\\nwhere l ⊥ = /(µω ⊥ ). Summarizing, in our 1D calculations we apply the Hamiltonian (6) with the approximation ( 8\\n\\n), that are valid for |z i -z a | ≫ R 1D ≡ max(R ⊥ , l ⊥ ).\\n\\nFinally in the case of equal longitudinal trapping frequencies: ω i = ω a = ω, the relative and the COM degrees of freedom can be separated, and the dynamics is described by the Hamiltonian of the relative motion\\n\\nH rel1D = p 2 2µ + 1 2 µω 2 (z -d) 2 - αe 2 2z 4 , ( 9\\n\\n) where z = z i -z a and p = p i -p a .\\n\\nIn the next Section we shall solve the problem in the different regimes described before, and investigate the peculiar phenomena that arise from the interplay between the trapping and the interaction potentials involving the two particles.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"E. Quantum-defect theory in 1D\",\n \"text\": \"The quantum-defect treatment of the short-range interactions can be extended to the 1D dynamics, described by the Hamiltonian (6). In 1D the asymptotic behavior of the relative wave function at short distances is governed by Ψ e rel (z, k) ∼ |z| sin (R * /|z| + ϕ e (k)) , z ≪ R * /k, (10)\\n\\nΨ o rel (z, k) ∼ z sin (R * /|z| + ϕ o (k)) , z ≪ R * /k, ( 11\\n\\n)\\n\\nwhere labels e and o refer to the even and odd solutions respectively. In our model of 1D dynamics we treat Eqs. (10)-(11) as boundary conditions for |z| → 0. In section III A we show that the phases ϕ e and ϕ o are uniquely determined by l ⊥ and ϕ, and that their values can be calculated by solving the scattering problem in the quasi-1D geometry. This requires that l ⊥ ≫ R 0 , which allows to use the quantum-defect description of the shortrange potential. As we discuss later, in contrast to 3D traps, in quasi-1D traps the values of ϕ are in general different for scattering waves of different symmetry.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"F. Time-dependent problem\",\n \"text\": \"The atom-ion collision by moving the trapping potential d(t) is an intrinsically time-dependent problem which requires the integration of the time dependent Schrödinger equation for the given initial condition to predict the transition probabilities for the possible final states. In our approach to the dynamics we first calculate the correlation diagrams, showing the energy levels as a function of the trap separations, and on the basis of these diagrams we predict the possible scenario of the atom-ion collision. When the motion of the trap is sufficiently adiabatic, the evolution of the system proceeds along one of the energy curves, therefore the basis of adiabatic eigenstates, dependent parametrically on d, is particularly useful in the analysis of the collision process. Of course, the adiabaticity is usually broken in the vicinity of avoided crossings. In such cases, however, one can apply e.g. the Landau-Zener theory to calculate the probability of the adiabatic and diabatic passage through an avoided crossing.\\n\\nIn our approach to the atom-ion collisions we consider that initially the traps are well separated. In this limit, the asymptotic states of (1) are given by products of the harmonic oscillator states in the two traps. In the course of dynamics, the distance d(t) decreases, particle interact for some definite time, and finally they are again separated, and the final state evolves into some superposition of the harmonic oscillator states (see Fig. 1 for a schematic picture). For such a scheme, we are interested in predicting the final state for some particular realization of d(t). We stress that in the intermediate phase, the system may evolve into atom-ion molecular complex, and such possibility is fully accounted for in our model.\\n\\nWe start from the time-dependent Schrödinger equation with the Hamiltonian (1). We expand the time-7 dependent wave function in the basis of the energyordered adiabatic eigenstates Ψ n (x 1 , x 2 |d) of the Hamiltonian (1)\\n\\nH(d)Ψ n (x i , x a |d) = E n (d)Ψ n (x i , x a |d), ( 12\\n\\n)\\n\\nwhere we explicitly point out its dependence on the distance d. Substituting the expansion\\n\\nΨ(x i , x a , t) = n c n (t) exp - i t 0 dτ E n (d(τ )) × Ψ n (x i , x a |d(t)) ( 13\\n\\n)\\n\\ninto the Schrödinger equation, we obtain a set of coupled differential equations that govern the dynamics of the expansion coefficients c n :\\n\\nċn = -ḋ m =n c m (t) exp i t 0 dτ (E n (d(τ )) -E m (d(τ ))) × Ψ n (d)| ∂ ∂d |Ψ m (d) , ( 14\\n\\n)\\n\\nIn the case of fully adiabatic evolution the coefficients c n (t) remain constant, and the evolution of the system proceeds along the adiabatic energy curves. From Eqs. (14) one can derive the condition for the adiabaticity of the transfer process. Adiabaticity requires that ḋ multiplied by the nonadiabatic coupling Ψ n (d)| ∂ ∂d |Ψ m (d) be much smaller than the frequency of the oscillating factor in the exponential of (14) . In this case the oscillating factor effectively cancels out the contribution due to changes of d. The adiabaticity condition can be written [13] . At trap separations where atom and ion can be approximately described by harmonic oscillator states, one can easily estimate the nonadiabatic couplings Ψ n (d)| ∂H ∂d |Ψ m (d) , which gives the following constraint on the adiabatic changes of the trap separation:\\n\\nas ḋ Ψ n (d)| ∂H ∂d |Ψ m (d) ≪ (E n -E m ) 2 ∀ m,n\\n\\nḋ l k ≪ (E n -E m ) 2 2 ω k , k = i, a, ( 15\\n\\n)\\n\\nwhere k stands for i (a) when the ion trap (atom trap) is moved. We stress that the latter condition is valid for transitions between different vibrational states, that may occur during the transfer of the atom or of the ion. In the case of avoided crossings between vibrational and molecular states, the adiabaticity of the transfer is characterized by the condition that can be determined from the Landau-Zener theory.\"\n },\n {\n \"section_type\": \"RESULTS\",\n \"section_title\": \"III. RESULTS\",\n \"text\": \"We turn now to the analysis of the adiabatic eigenenergies and eigenstates as a function of the trap separation d, and of the trapping potential geometry. We start our analysis from the simplest case of quasi-1D traps, where the dynamics takes place effectively in 1D, while the assumptions of equal trapping frequencies allows us to consider COM and relative motions separately. Before discussing this problem, we first study the dependence of the 1D short-range phases ϕ e and ϕ o on l ⊥ , R * and ϕ. We argue that in the considered range of parameters they are practically independent of the kinetic energy of the scattering particles, which is assumed in our quantumdefect-theory approach. The reader not interested in details of the derivation may start at Section III B, where we analyze 1D relative motion eigenenergies and eigenstates assuming some particular values of ϕ e and ϕ o . In Section III C we switch to 3D geometries, analyzing the system of two spherically symmetric traps with the same trapping frequency for atom and ion. Section III D is devoted to the properties of adiabatic energy spectra in the vicinity of an avoided crossing. Using the semiclassical theory we calculate the level separation at the avoided crossing, and then applying Landau-Zener theory we investigate the conditions for the adiabatic and diabatic transfer, depending on the distance between the traps. Finally, in Section III E we address the most complicated case of different trapping frequencies for atom and ion, coupling COM and relative degrees of freedom. Because of the complexity of this problem, we limit our analysis only to 1D dynamics; we argue, however, that the observed behavior should be qualitatively valid also for the 3D system.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Short-range phases in quasi-1D traps\",\n \"text\": \"As a preliminary technical step in preparation for the calculations of adiabatic energy curves that will be reported in the following, the present subsection deals with calculating the short-range phases ϕ e and ϕ o in quasione-dimensional geometries.\\n\\nTo find the 1D short-range phases we solve the Schrödinger equation for the relative part of the Hamiltonian (1), assuming the same transverse trapping frequency for atom and ion: ω ⊥i = ω ⊥a = ω ⊥ , and neglecting axial trapping frequencies, which is valid for\\n\\nz ≪ R i , R a . The latter condition requires R i , R a ≫ R 1D , where R 1D = max(R ⊥ , l ⊥ )\\n\\ndetermines the boundaries of the 1D regime. In this way we obtain the following equation for the relative wave function Ψ(r)\\n\\n- 2 2µ ∇ 2 - αe 2 2r 4 + 1 2 µω 2 ⊥ ρ 2 -E Ψ(r) = 0, ( 16\\n\\n)\\n\\nIn the case of interaction potentials at distances R 1D much larger than typical kinetic energies in the trapping potential:\\n\\n|V (R 1D )| ≫ ω i , ω a ,\\n\\nwe can neglect the energy dependence of the short-range phases, and solve (16) for E = ω ⊥ . At small distances, r ≪ R 1D , the atom-ion interaction dominates over the transverse trapping potential, and the solution of (16) behaves according to (2). At large distances, z ≫ R 1D , the motion in the transverse direction is frozen to its zero-point oscillations and in the 8 -1.0 -0.5 0.0 0.5 1.0 0.00 0.05 -0.2 0.0 0.2 1D regime /R * -8 -6 -4 -2 0 2 1D regime 2l R 1D V(r)/ l = 0.09 R * (z) |z| sin(|z|/R * + e ) (r) sin(r/R * + ) z/R * (z) 3D regime FIG. 4: (Color online) Potential for the atom-ion relative motion (upper panel) and the relative wave function (lower panel) for the quasi-one dimensional system with l ⊥ = 0.09R * and without trapping in the axial direction.\\n\\nasymptotic regime the scattering wave function assumes the form Ψ(r\\n\\n) → ψ 0 (ρ)Ψ 1D (z) (|z| → ∞), where ψ 0 (ρ)\\n\\nis the ground-state wave function of the 2D harmonic oscillator: ψ 0 (ρ) = e -ρ 2 /(2l 2 ⊥ ) /π 1/2 , and Ψ 1D (z) is a linear combination of odd and even waves: 10 ) and (11) respectively [25]. Fig. 4 shows the potential for the relative motion, and the axial profile (ρ = 0) of the relative wave function for l ⊥ = 0.09R * , which corresponds to a system of 40 Ca + and 87 Rb in a trap with ω ⊥ = 2π × 1MHz. In addition, the figure illustrates different length scales present in the quasi-1D problem, and indicates the 1D (|z| ≫ R 1D ) and 3D regimes (r ≪ R 1D ) in the behavior of the wave function.\\n\\nΨ 1D (z) = c e Ψ e (z) + c o Ψ o (z), ( 17\\n\\n) with Ψ e (z), Ψ g (z) given by (\\n\\nFig. 5 shows an example of the dependence of ϕ e and ϕ o on ϕ, for l ⊥ = R * . It compares the results of numerical calculations with predictions based on the pseudopotential method, that is dicsussed in appendix B. We observe that for even waves the agreement is fairly good, while for odd waves the agreement is poorer, which is probably due to the fact that the p-wave energydependent pseudopotential does not work already in the regime of R * ∼ l ⊥ , or contributions of odd partial waves with l > 1 can be important. The pseudopotentials are expected to give accurate predictions for R * ≪ l ⊥ .\\n\\nWhen R * ≫ l ⊥ , the pseudopotential approach is not applicable at all, and one has to resort to numerical calculations. As an example, we present in Fig. 6 values for the short-range phases for l ⊥ = 0.1R * . We note the presence of several resonances in the dependence of ϕ e and ϕ o . This behavior is related to the contribution of several partial waves for R * ≫ l ⊥ , leading to resonances when the energy of a bound state in the combined harmonic and r -4 potential becomes equal to the energy of the scattered wave. To analyze this issue more carefully -0.4 -0.2 0.0 0.2 0.4 -0.8 -0.6 -0.4 -0.2 0.0 0.2 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 0.6 e l = R * numerical energy-dependent pseudopotential / l = R * o numerical energy-dependent pseudopotential / FIG. 5: (Color online) Even and odd short-range phases ϕe and ϕo calculated for l ⊥ = R * . Numerical results (red solid lines) are compared with predictions of the model replacing r -4 with the energy dependent pseudopotential.\\n\\nwe have calculated the energies of some particular bound states in in the combined harmonic and r -4 potential. Fig. 7 shows the relation between the bound-state energy and the short-range phase for few bound states that are responsible for the scattering resonances close to φ = 0 for φ e (cf. Fig. 6 ). For comparison we include the bound states of pure r -4 potential and we label them by the angular momentum l. We note that, due to the limited resolution of our numerical calculations, Fig. 6 does not allow to resolve the single resonances close to φ = 0, however, one can observe the rapid changes of φ e in this region.\\n\\nFor the numerical calculations we have transformed (16) into cylindrical coordinates and we were solving 2D elliptic partial differential equation using a finiteelement method. To fix the short-range phase we impose the boundary conditions on the logarithmic derivative of Ψ(r), ∂ r Ψ(r)/Ψ(r), at r min , where r min = 0.09R * (r min = 0.022R * ) for l ⊥ = R * (l ⊥ = 0.1R * ) . At large distances we impose the Dirichlet boundary condition on the rectangle with boundaries |z| = z max , ρ = ρ max , assuming that the wave function has a Gaussian transverse profile at |z| = z max and vanishes at ρ = ρ max . For l ⊥ = R * we took z max = 6.3R * , ρ max = 3R * , while for l ⊥ = 0.1R * we have used z max = R * , ρ max = 0.4R * .\\n\\nTo determine the values of ϕ e and ϕ o , we fit at large distances (|z| ≫ R 1D ) the symmetric and antisymmetric solutions of (16) for E = ω ⊥ , to the asymptotic formula (17).\\n\\nRelative motion in a 1D system: adiabatic eigenenergies and eigenstates\\n\\nIn this section we consider the 1D motion of an atom and an ion, assuming ω i = ω a . In this case we can focus only on the relative motion described by the Hamiltonian (9), with the boundary conditions at z → 0, stated by (10) and (11) . To find eigenenergies and eigenfunc-9 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 -0.4 -0.2 0.0 0.2 0.4 (b) e (a) o FIG. 6: (Color online) Even short-range phase ϕe (upper panel) and odd short-range phase ϕo (lower panel) calculated numerically for l ⊥ = 0.1R * , versus 3D short-range phase ϕ.\\n\\n-50 -5 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.0 0.5 1.0 l=6 l=8 E l=10 FIG. 7: (Color online) 3D short-range phase ϕ versus energy of bound states in the presence of the transverse confinement with l ⊥ = 0.1R * . The bound states in the combined harmonic and r -4 potentials (solid lines) are compared with the bound states of pure r -4 interaction (dashed lines).\\n\\ntions for arbitrary value of d we diagonalize the Hamiltonian (9) numerically in the basis of its eigenstates for d = 0, that are found by numerical integration of the 1D Schrödinger equation. In the basis we include the lowest 50 odd and even eigenfunctions, which is sufficient to perform the diagonalization for d/R * 2.5. Fig. 8 present the adiabatic energy spectrum for some example parameters: ϕ e = -π/4, ϕ o = π/4 and R * = 3.48l. The latter value corresponds to the system of 40 Ca + and 87 Rb in the trap with ω = 2π × 100kHz, while the particular choice of the short-range phases ϕ e and ϕ o is explained later in this section. Points with labels correspond to the wave functions shown in Fig. 9 presented together with their eigenenergies and potential energy curves.\\n\\nThe three panels shown in Fig. 9 illustrate three different regimes, where the system exhibits qualitatively different behavior. In the first regime, represented by the eigenstate Ψ a and realized at large distances between traps, d ≫ R rel with R rel defined in (5), the main effect of the atom-ion interaction is the distortion of the trapping potential, and in this limit the Hamiltonian can be diagonalized analytically in the model of two coupled harmonic oscillators (see Section III E for more details). The value of the short-range phase is not important for this model. At large separations the eigenstate Ψ a is typically only weakly perturbed with respect to the eigenstate of the harmonic oscillator.\\n\\nIn the second regime, represented by the eigenstates Ψ b and Ψ c , and realized at distances d ∼ R rel , the system exhibits resonances between vibrational (Ψ b ) and molecular (Ψ c ) states, manifesting themselves as avoided crossings in the adiabatic energy spectrum. The resonances appear when the energies of the two eigenstates become equal, and the energy splitting at the avoided crossing is proportional to the tunneling rate through the potential barrier separating the two regions of the potential. In section III D we calculate this splitting in the framework of a semiclassical approximation, and describe the dynamics in the vicinity of the avoided crossing. The value of the short-range phase determines the energy of the molecular states, but it is practically not important for the atom-ion vibrational states. Since the point of the avoided crossing depends on the short-range phases, the controlled collisions can provide some information on the short-range interaction potential.\\n\\nFinally, at distances d ≪ R rel the barrier separating the external trap from the well given by the atom-ion attraction disappears and all eigenstates have some shortrange component behaving at short distances according to Eqs. (10) and (11) . Hence, in this regime all eigenstates depend on the value of the short-range phases. The molecular states with energies below the dissociation threshold (E = 0) are mainly localized in the well of the atom-ion attractive potential, however for the tight traps considered here, they can be strongly affected by the external trapping (e.g. state Ψ e ). The other type of states, with energies E > 0, are analogs of the vibrational states and they are localized mainly at distances where the external trapping potential dominates (e.g. state Ψ d ).\\n\\nSo far we have discussed the properties of the energy spectrum for some particular choice of ϕ e and ϕ o . It turns out that a qualitatively similar behavior can be observed in all the systems with |ϕ e -ϕ o | = π/2. In the general case, however, the adiabatic energy spectrum has a slightly more complicated structure, as it is illustrated in Fig. 10 , showing the eigenenergies for ϕ e = ϕ o = -π/4. We note that by going from d = 0 10 0.0 0.5 1.0 1.5 2.0 2.5 -10 -5 0 5 e d c b o = + / 4 E/ * e = -/ 4 a FIG. 8: (Color online) Energy spectrum of the relative motion for an atom and an ion in 1D traps versus the distance d between traps, calculated for ϕe = -π/4, ϕo = π/4 and R * = 3.48l (see text for details). The dashed curve shows the line 1 2 µω 2 d 2 , giving the approximate shift of the bound state in the trapping potential (see section III C for derivation).\\n\\nto positive d = 0 all the energies split into two branches.\\n\\nTo understand the nature of this splitting in Fig. 11 we present the wave functions for d = 0.1R * , with the corresponding eigenvalues marked with stars in Fig. 10 . We observe that the branches represent the eigenstates localized on the left-hand side and right-hand side of the point z = 0. The left-localized eigenstates correspond to the rising branches, because they are mainly localized in the region of strong atom-ion attraction. We note that this behavior is similar to the properties of eigenstates in a double-well potential, where appropriate wave functions are constructed by taking symmetric and antisymmetric combination of the states in two wells. In the particular case of ϕ e = ϕ o , the symmetric and antisymmetric combination leads to the states localized on the positive and negative z semi axes, respectively.\\n\\nC. Relative motion in 3D for spherical ωi = ωa traps: adiabatic eigenenergies and eigenstates\\n\\nIn this section we extend our analysis to 3D, and consider spherically symmetric traps for the atom and the ion with the same trapping frequencies: ω i = ω a . In this case the COM and relative motion can be separated, and in the following we focus only on the relative motion governed by the Hamiltonian (4).\\n\\nWe have diagonalized the Hamiltonian (4) taking different values of the short-range phase ϕ, and we have observed that the adiabatic energy spectra exhibit qualitatively the same features as in 1D. In the numerical calculations we first calculated the eigenstates for d = 0, by solving the radial Schrödinger equation, for angular momenta l ≤ 75 and for energies E ≤ 75 ω. These states 0 10 20 -4 0 4 -3 -2 -1 0 1 2 3 4 -4 0 4 c e d b d = 2.5 R * e = -/4 o = + /4 E/ a d = 1.2 R * e = -/4 o = + /4 E/ d = 0 e = -/4 o = + /4 E/ * FIG. 9: (Color online) Eigenstates of the relative motion for an atom and an ion in 1D traps at different separations d between the traps, calculated for ϕe = -π/4, ϕo = π/4 and R * = 3.48l. Presented eigenstates correspond to labeled points in Fig. 8 . The horizontal lines present the corresponding eigenenergies, and the thick black line shows the potential energy.\\n\\n0.0 0.5 1.0 1.5 2.0 2.5 -10 -5 0 5 o = -/ 4 E/ * e = -/ 4 FIG. 10: (Color online) Energy spectrum of the relative motion for trapped atom and ion in 1D traps versus distance d between traps, calculated for ϕe = ϕo = -π/4 and R * = 3.48l (see text for details).\\n\\nwere used as a basis in the numerical diagonalization of the Hamiltonian (4). A sample adiabatic energy spectrum is shown in Fig. 12 , presenting the results for ϕ = -π/4 and R * = 3.48l. At d 0 the angular momentum l is a good quantum number and the states have definite angular 11 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 -2 0 2 4 6 8 d = 0.1 R * e = -/4 o = -/4 E/ z/R * FIG. 11: (Color online) Eigenstates of the relative motion for trapped atom and ion in 1D at trap separation d = 0.1R * , calculated for ϕe = ϕo = -π/4 and R * = 3.48l. The horizontal lines present the corresponding eigenenergies, and the thick black line shows the potential energy.\\n\\n0.0 0.5 1.0 1.5 2.0 -2 0 2 4 6 l = 0 l = 1 l = 2 l = 3 l = 4 E/ * = -/4 FIG. 12: (Color online) Energy spectrum of the relative motion for an atom and an ion confined in spherically symmetric traps versus the distance d between traps, calculated for ϕ = -π/4, and R * = 3.48lr (see text for details). The blue dashed line shows the shift of the bound states with d, given approximately by 1 2 µω 2 d 2 .\\n\\nsymmetry, which is depicted by the appropriate symbols in Fig. 12 . For nonzero d, angular momentum is not conserved, hence in the dynamics the state can change its angular symmetry. We note that the energies of the bound states shift to a good approximation according to µω 2 d 2 /2. This behavior can be explained by noting that bound states |Ψ mol are concentrated around r = 0, and\\n\\nΨ mol (d)|H rel (d)|Ψ mol (d) ≈ E b + 1 2 µω 2 d 2 ,\\n\\nwhere E b is the binding energy at d = 0.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D. Avoided crossings: semiclassical analysis\",\n \"text\": \"In the considered setup, level anticrossings reflect resonances between molecular and trap states. A simple picture of such an avoided crossing is shown in Fig. 13 . If one passes an avoided crossing from the direction of the asymptotic trap state, then for adiabatic evolution the system at small distances evolves into a molecular state. On the other hand, for diabatic passage, the particles remain in their traps, and the state basically does not change, apart from the modification due to the smaller trap separation. In addition, for fast changes of the trap positions, the particles can be excited to higher motional states. To describe quantitatively the dynamics in the vicinity of the avoided crossing one can apply the Landau-Zener theory. Assuming that close to the avoided crossing the eigenenergies are linear in d, and that d(t) varies linearly in time, the probability that the crossing is traversed diabatically is given by [14, 15]\\n\\np |1 →|1 ′ = exp -2π | Ψ 1 |H|Ψ 2 | 2 | ḋ||∂E 12 /∂d| , ( 18\\n\\n)\\n\\nwhere the labels |1 , |2 (|1 ′ , |2 ′ ) respectively refer to the vibrational and molecular states before (after) passing the avoided crossing (cf. Fig. 13) , and E\\n\\n12 (d) = E 1 (d) -E 2 (d). For | Ψ 1 |H|Ψ 2 | 2 ≪ | ḋ||∂E 12 /∂d|,\\n\\nthe probability (18) is close to unity, and the avoided crossing is passed diabatically. In the opposite case:\\n\\n| Ψ 1 |H|Ψ 2 | 2 ≫ | ḋ||∂E 12 /∂d|\\n\\n, p is small, and the avoided crossing is traversed adiabatically. The matrix element Ψ 1 |H|Ψ 2 can be related to the energy gap at the avoided crossing, and in this way we obtain the following constraint on the adiabaticity of the transfer close to the avoided crossing:\\n\\nḋ ∂E 1 ∂d - ∂E 2 ∂d ≪ (E 1 (d) -E 2 (d)) 2 ( 19\\n\\n)\\n\\nThe position of the avoided crossings is directly related to the energies of the bound states; thus, a measurement of the final state after controlled collisions, provides technique for spectroscopy of the trapped atom-ion complex. Since the avoided crossings become weaker as the separation between traps increases, this scheme allows to probe only excited molecular states, having sufficiently small binding energies, comparable to the energy scales of the trapping potentials.\\n\\nIn the case when the tunneling barrier is sufficiently large, the energy splitting at the avoided crossings can be estimated using a semiclassical approximation. We first focus on the relative motion in the 1D system, assuming ω i = ω a = ω. Using the standard WKB method, one can derive the following result for the energy splitting ∆E [14]\\n\\n∆E = EΨ mol (x 1 )Ψ vib (x 2 ) v(x 1 )v(x 2 )T e -W/ . ( 20\\n\\n)\\n\\nHere x 1 , x 2 are some arbitrary points located in the classically forbidden region close to the classical turning 12 FIG. 13: (Color online) Schematic drawing of the avoided crossing between vibrational and molecular states in the system of a trapped atom and an ion. An adiabatic change of the distance d between traps induces a transition from the vibrational to the molecular states, while a diabatic process leaves the particles in the vibrational states.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"points\",\n \"text\": \", W (x 1 , x 2 , E) = x2 x1 dx 2m(V (x) -E)\\n\\nis the action along the classical trajectory from x 1 to x 2 , T = ∂W ∂E is the tunneling time, Ψ mol (x 1 ), Ψ vib (x 2 ) denote, respectively, the molecular and vibrational states (cf. Fig. 13 ) with the same eigenvalues E (the resonance case), and v(x) = 2(V (x) -E)/m is the velocity of a particle with energy -E in the inverted potential -V (x). In principle the choice of x 1 and x 2 is arbitrary, but in our calculations we take x 1 and x 2 at fixed distance from the location of the molecular (x = 0) and vibrational states (x = d), close to the classical turning points. In this way at sufficiently large separations Ψ mol (x 1 ), Ψ vib (x 2 ) becomes independent of d.\\n\\nWe turn now to the case of relative motion in 3D traps. Similar to the analysis for 1D systems, the energy splitting at the avoided crossings can be calculated semiclassically by applying the instanton technique. In our calculation we adopt the formulation based on the path decomposition expansion developed by Auerbach and Kivelson [16] . We obtain the following formula describing the level splitting:\\n\\n∆E = EAΨ mol (x 1 )Ψ vib (x 2 ) v(x 1 )v(x 2 )T e -W/ (21)\\n\\nThe meaning of all the quantities is the same as in the 1D case, the only difference is the prefactor A which accounts for the fluctuations around the instanton path (see [16] for details). In the case ω i = ω a the instanton path that minimizes the classical action is simply a straight line connecting the trap centers.\\n\\nFig. 14 compares the semiclassical formulas Eqs. (20) and (21) with the exact energy splitting determined from the numerical energy spectra, as those presented in Fig. 8 . The presented results correspond to the avoided crossings between the vibrational ground state molecular states. In 1D the energy splitting at the avoided crossing does not 0.8 1.0 1.2 1.4 1.6 1.8 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 WKB 1D instanton 3D (s-wave) 1D numerical 3D numerical (s-wave) E/ d/R * R * = 3.48 l\\n\\nFIG. 14: (Color online) Energy splitting at the avoided crossing between vibrational ground-state and the molecular state in 1D and 3D systems. Semiclassical results (solid lines) are compared to the numerical values extracted from the adiabatic energy spectra for different combinations of ϕe and ϕo.\\n\\n3D results present avoided crossings widths for s-wave bound states.\\n\\ndepend on the symmetry of the molecular state, while in 3D it does depend on its angular momentum l, and we present here only the case l = 0. In the first approximation, the splitting depends on the short-range phases only through the critical trap separation at which the resonance occurs. The numerical data are obtained for different combinations of the short-range phases, for which the adiabatic spectra exhibit avoided crossings at different values of d. In our approach, instead of calculating Ψ mol (x 1 ) and Ψ vib (x 2 ), we fix the overall amplitude in (20) by fitting to the single point at the largest trap separation, where we expect the WKB approximation to be most accurate. The semiclassical curves stop at the distances where the potential barrier disappears. We observe that for the same ratio of R * to l the splittings in 1D are larger than in 3D. Fig. 14 shows the prediction of the semiclassical formula Eq. ( 20 ), and the energy splittings calculated numerically for molecular states with different l. We note that separations at the avoided crossing are largest for the spherically symmetric molecular states, with l = 0. The knowledge of the energy splitting ∆E can be used to calculate the probabilities of an adiabatic and diabatic passage of avoided crossings. Assuming that the avoided crossing is traversed at constant rate, we apply the formula (18) 12) , which leads to\\n\\nwith ∂E 1 (d)/∂d ≈ 0 and ∂E 2 (d)/∂d = Ψ mol (d)|∂H rel (d)/∂d|Ψ mol (d) ≈ µω 2 d (c.f. Figs. 8 and\\n\\np |1 →|1 ′ = exp - π 2 (∆E) 2 | ḋ|µω 2 d , ( 22\\n\\n)\\n\\nwhere we use the same notation as in Fig. 13 . Analyzing Figs. 14 and 15 we can now estimate the rates ḋ required 13 1.0 1.2 1.4 1.6 10 -4 10 -3 10 -2 10 -1 s-wave semiclassical s-wave numerical p-wave semiclassical p-wave numerical d-wave semiclassical d-wave numerical E/ d/R * R * = 3.48 l FIG. 15: (Color online) Energy splitting at the avoided crossing between vibrational ground state and molecular states with different angular momenta, for a 3D spherically symmetric trap. Semiclassical results obtained by means of the instanton technique (solid lines) are compared to the numerical values extracted from the adiabatic energy spectra. Semiclassical calculations stop at the distance d = 1.1R * where the potential barrier disappears.\\n\\nfor adiabatic and diabatic transitions. For instance in 3D, for the parameters of Fig. 14 , the diabatic transfer of particles across the avoided crossings up to distances d ≈ R * (∆E 0.1 ω) can be realized by keeping ḋ/R * ≫ 0.001ω . In summary, the analysis carried out here provides the basis for a description of the dynamics of various processes of interaction between an atom and an ion manipulated through external trapping potentials, and it gives a way to estimate quantitatively with simple analytical means the outcome of controlled interaction experiments in the different regimes described in Sect. II.\\n\\nE. Center of mass coupled to the relative motion: 1D analysis for ωi = ωa\\n\\nIn this section we consider the effects of coupling between COM and relative motions in 1D, that appear when the trapping frequencies for atom and ion are not equal.\\n\\nFig. 16 shows the adiabatic levels as a function of trap separation for some example parameters: ω i = 5.5ω a and l a = 0.9R * . This choice corresponds to the interaction of 40 Ca + and 87 Rb in the traps with ω a = 2π × 10kHz and ω i = 2π × 55kHz. To obtain the adiabatic spectrum presented in Fig. 16 , we performed the diagonalization of the Hamiltonian (6) in the product basis of the COM and relative motion eigenstates evaluated at d = 0. In the calculations we consider all the states with a total energy E ≤ 460 ω a , which leads to about 7100 states in the basis.\\n\\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 -5 0 5\\n\\n10 |2 a |0 i |1 a |0 i |0 a |0 i CM |1 a |1 i |0 a |1 i E/ a d/R * = 5.5\\n\\na e = -/ 4 o = + / 4 CM FIG. 16: Energy spectrum for atom and ion confined in harmonic traps with ωi = 5.5ωa, as a function of the distance d. Calculations are performed for ϕe = -π/4, ϕo = π/4 and la = 0.9R * (see text for details).\\n\\nThe arrows on the right-hand side of Fig. 16 indicate the asymptotic states for large separations, which can be labeled by the number of phonons for the atom and ion trap, respectively. In comparison to the case where relative and COM motions are decoupled, we observe the following new features: (i) the molecular spectrum contains states with different numbers of excitations in the COM degree of freedom. This can be observed at d = 0, when the molecular levels are with a good approximation equally separated by ω CM where\\n\\nω 2 CM = (m a ω 2 a + m i ω 2 i )/M . [26];\\n\\n(ii) the avoided crossings between molecular and vibrational states are weaker for the states involving vibrational excitations of the ion. This can be observed by comparing the avoided crossings for the state |0 a |1 i with the avoided crossings for the neighboring states. This behavior can be understood when we notice that for weaker atom trap (for ω a < ω i ), the particles have larger probability to tunnel when the atom is excited.\\n\\nF. Large distances between traps: two coupled oscillators approximation Finally we turn to the limit of large distances: d ≫ R i , R a . Expansion of the interaction up to second-order terms in the distance r leads to\\n\\nH 1D ≃ ν=i,a p 2 ν 2m ν + 1 2 m ν ω 2 ν (z ν -zν ) 2 + 10 αe 2 d 6 (z i -zi )(z a -za ), ( 23\\n\\n)\\n\\nwhere zν denotes the equilibrium position of the particles and we neglect the second-order terms modifying the trapping frequencies. The Hamiltonian (23) describes the 14 system of two coupled harmonic oscillators, which can be written in the form\\n\\nH 1D = ω a a † a + ω i b † b + ω c (a + a † )(b + b † ), ( 24\\n\\n)\\n\\nwhere we have introduced the usual annihilation and creation operators\\n\\na = m i ω i 2 z i + i p i m i ω i ( 25\\n\\n) b = m a ω a 2 z a + i p a m a ω a . ( 26\\n\\n)\\n\\nHere ω c denotes the coupling frequency: ω c = 10E * (R * ) 4 l a l i /d 6 . The validity of the considered model is limited by the assumption that a stable equilibrium position exists, which is fulfilled for R i , R a 0.57d, as it can be easily verified. Typical, maximal values of ω c for atoms in optical potentials and ions in rf traps are of the order of 10kHz. In practice the model ceases to be valid already at weaker conditions, when the terms higher than the second order cannot be neglected. In any case, this model is relevant for an important class of processes involving coherent transfer of quanta between the atom and the ion, of direct application in a quantum information processing context.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"IV. CONCLUSIONS AND OUTLOOK\",\n \"text\": \"In this paper we analyzed in detail the interaction between a single atom and a single ion guided by external trapping potentials. This kind of work is motivated by recently opened experimental possibilities within combined systems -currently being built in several groups worldwide -where magneto-optical traps or optical lattices for neutral atoms coexist with electromagnetic traps for ions.\\n\\nTight confinement of single particles, associated with independent control of the respective confining potentials, allows for exploring different regimes of the twobody dynamics involving one atom and one ion at a time. At large distances, the interaction is dominated by the inverse quartic term arising from the attraction between the ion's charge and the electric dipole induced by it on the atomic electron wave function. A pseudopotential approximation, similar to that familiar from ultracoldatom collision theory, is not applicable in our case, as the characteristic range of the atom-ion molecular potential often exceeds the size of the tight traps experimentally available. To describe the interaction at short distances, smaller than that range, we employ quantum-defect theory, which allows to deal with different geometries from isotropic three-dimensional traps to very elongated quasione-dimensional ones.\\n\\nA good description of adiabatic dynamics, involving processes where the traps are moved toward or across each other at a rate much slower than the trapping frequencies, can be obtained from quasistatic eigenenergy curves, which we calculate for various trapping configurations based on the methods outlined above. A remarkable feature displayed by the system is the presence of resonances between molecular-ion bound states and motional excitations within the trap. These trap-induced resonances are similar in nature to Feshbach resonances driven by external fields, and they could as well be spectroscopically probed in simple experiments where the interaction is controlled via the external guiding potentials.\\n\\nIn addition to the above aspects, an important motivation for the interest in systems of trapped atoms and ions and their trap-induced resonances resides in possible applications to quantum information processing. In this context, one can utilize controlled atom-ion interactions to effect coherent transfer of qubits, thereby creating interfaces between atoms and ions. By storing quantum information in internal atomic states, and performing gate operations with ions, one would combine the advantages of both: (i) long decoherence times for neutral atoms and (ii) short gate-operation times for charged particles due to the relatively strong interactions. In the present work we have focused on the motional degrees of freedom, which can serve as auxiliary degrees of freedom for quantum gates involving internal-state qubits, provided an appropriate coupling mechanism between internal and external degrees of freedom is employed -e.g., sideband excitation via a laser.\\n\\nAnother potential application of our results is cooling of the atomic motion. In typical neutral-atom quantum computation schemes with qubits stored in internal states of atoms trapped in optical-lattice sites, motional excitations constitute a serious source of errors, and the atomic motion needs to be cooled in a state-insensitive manner between computational steps in order to avoid qubit decoherence. Since the long-range part of the atom-ion interaction is not sensitive to the internal state [27] , our setup can be applied for sympathetic cooling of atoms, through the exchange of energy with laser cooled ions [12] .\\n\\nBeside these applications in the context of quantum information processing, our work opens broader perspectives for the study of new interesting collisional physics in a physical situation never explored before. In principle, our scheme allows for production of ultracold trapped atom-ion molecules (molecular complexes), when the trapping potentials are lowered adiabatically at the stage when the particles remain close to each other. In this way our method can be regarded as a way to perform cold chemical reactions, where the final state of the molecule can be well controlled. Beyond sufficiently large separation between traps, however, the survival probability of such molecules in the final state is negligible. Indeed, in the calculations presented here, we have only regarded processes leading to a final state in which the atom and the ion are still separated, and no molecule has been formed as an outcome of the interaction. In other words, the Hamiltonian describing our single-channel model does not include the possibility of 15 transitions to other Born-Oppenheimer curves. While this is a justified assumption for the dynamics considered here, it is in a sense a limitation of our current approach. Investigation of molecular formation in traps beyond this approximation is certainly among the interesting developments that can arise from our work. Its long-term motivation is really to open, beyond the present examples, a new paradigm for cold collision physics, which can be described as the mechanical control of single-particle chemical reactions.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"The authors thank J. Denschlag, P. Schmidt, R. Stock, and A. Simoni for helpful discussions. This work was supported by the Austrian Science Foundation (FWF), the European Union projects OLAQUI (FP6-013501-OLAQUI), CONQUEST (MRTN-CT-2003-505089), the SCALA network (IST-15714), the Institute for Quantum Information GmbH, the EU Marie Curie Outgoing International Fellowship QOQIP, and the National Science Foundation through a grant for the Institute for Theoretical Atomic, Molecular and Optical Physics at Harvard University and Smithsonian Astrophysical Observatory.\\n\\nAPPENDIX A: DERIVATION OF THE EFFECTIVE HAMILTONIAN (1)\\n\\nWhile the Hamiltonian (1) is intuitively obvious, we find it nonetheless worthwhile to summarize the microscopic derivation in an adiabatic approximation, and to discuss its validity. For simplicity we consider an atom with a single outer-shell electron (alkali atom) and an ion with single positive charge +e. In addition we do not consider the internal structure of the ion. The total Hamiltonian can be written as\\n\\nH = p 2 1 2m 1 + p 2 2 2m 2 + H e + H las + H rf . ( A1\\n\\n)\\n\\nHere the labels 1 and 2 correspond, respectively, to atomic nucleus and ion, and H e is the Hamiltonian of the electron, which includes the Coulomb interactions,\\n\\nH e = p 2 e 2m e - e 2 |x e -x 1 | - e 2 |x e -x 2 | + e 2 |x 2 -x 1 | . (A2)\\n\\nHere, for simplicity, we omit the contributions of the core regions for both the atom and ion. In this way our model refers in fact to H + 2 molecule. For alkali atoms and alkaline-earth ions one should treat the complete structure of core regions, however, our approach can be readily generalized to this more complicated case.\\n\\nThe term H las describes the interaction of the atom with a laser beam creating an optical potential, written in the electric dipole representation\\n\\nH las = -dE ⊥ (x 1 , t), ( A3\\n\\n)\\n\\nwhere d = e(x 1x e ) is the dipole moment of the atom, and for simplicity we neglected the influence of the laser on the ion, which has typically a different electronic structure that the atom. In addition we have applied the long-wavelength approximation, neglecting changes of the electric field on the scale of the atom. In the case of the optical lattice the transverse part of the electric field E ⊥ can be assumed to have the form of a standing wave\\n\\nE ⊥ (x, t) = E 0 cos(ω L t) [cos(k L x + φ x ) + (x → y, z)] , (A4)\\n\\nwhere for simplicity we have assumed the same amplitude E 0 , the same wave vector k L , and the same frequency of the laser light ω L for all three laser beams creating the optical lattice potential. The abbreviation (x → y, z) denotes sum of the terms with x replaced by y and z, and ϕ k for k = x, y, z is the phase factor characterizing the standing wave. Finally, H rf is the electric potential creating the rf-trap\\n\\nH rf = eΦ(x 1 , t) + eΦ(x 2 , t) -eΦ(x e , t), ( A5\\n\\n)\\n\\nwhere Φ(x 1 , t) is the time-dependent electric field of the rf trap\\n\\nΦ(x, t) = 1 2 (u x x 2 + u y y 2 + u z z 2 ) + 1 2 (v x x 2 + v y y 2 + v z z 2 ) cos ω rf t. ( A6\\n\\n)\\n\\nHere, ω rf is the frequency of the time-dependent part of the electric potential, and u k , v k (k = x, y, z) are amplitudes depending on the geometry of the trap [5]. The electric field at every instant of time has to fulfill the Laplace equation: ∆Φ = 0, hence, the coefficients u k , v k are subject to the following conditions:\\n\\nu x + u y + u z = 0, v x + v y + v z = 0.\\n\\nBelow we indicate the basic steps of the derivation. Expansion in the basis of Born-Oppenheimer wave functions for electron motion. We start from generating a complete set of electronic wave functions, parametrized by the positions of the atomic core and of the ion\\n\\nH e Φ n (x e |x 1 , x 2 ) = E n (|x 2 -x 1 |)Φ n (x e |x 1 , x 2 ) (A7)\\n\\nIn this way the total wave function can be expanded in the basis of Born-Oppenheimer electronic wave functions\\n\\nΨ(x 1 , x 2 , x e , t) = n c n (x 1 , x 2 , t)Φ n (x e |x 1 , x 2 ). (A8)\\n\\nSince the basis is complete, the expansion of the function does not involve any approximations.\\n\\nRetaining in the expansion only the modes coupled by the laser. In the expansion (A8) we keep only two modes coupled by the laser light creating the optical lattice: the electronic ground state Φ g (x e |x 1 , x 2 ) and the electronic excited state Φ e (x e |x 1 , x 2 ). In this way we utilize the Born-Oppenheimer approximation, treating the electron 16 motion in the adiabatic approximation. This assumes that the time scale of the electron dynamics is much faster than the dynamics of the atomic nucleus and of the ion, which is typically fulfilled since the electron is much lighter than the other two particles. The approximation of the two coupled channels can be easily generalized to more, or even infinite number of channels, since the other channels are weakly populated and we treat them within the perturbation theory. Adiabatic elimination of the excited electronic state coupled through the laser: derivation of the optical trap potential. The expansion coefficients c g (x 1 , x 2 , t) and c e (x 1 , x 2 , t) fulfill the following set of coupled equations:\\n\\ni ∂c g (x 1 , x 2 , t) ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E g (|x 2 -x 1 |) + eΦ(x 2 , t) c g (x 1 , x 2 , t) -d eg E ⊥ (x 1 , t)c e (x 1 , x 2 , t) ( A9\\n\\n) i ∂c e (x 1 , x 2 , t) ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E e (|x 2 -x 1 |) + eΦ(x 2 , t) c e (x 1 , x 2 , t) -d eg E ⊥ (x 1 , t)c g (x 1 , x 2 , t), ( A10\\n\\n)\\n\\nwhere d eg (x 1 , x 2 ) = Φ e |d|Φ\\n\\ng is the dipole matrix element between the ground and excited electronic states, which in general depends on the position of atom and ion. In the derivation of (A9)-(A10) we have neglected the action of the rf field on the atom core and the electron, putting\\n\\nΦ k |H rf |Φ k ≈ eΦ(x 2 , t), k = e, g (A11) Φ e |H rf |Φ g ≈ 0. ( A12\\n\\n)\\n\\nWe focus on the regime of far-detuned laser: ∆ ≫ Ω L , where ∆ denotes the detuning ∆\\n\\n(|x 2 -x 1 |) = E e (|x 2 - x 1 |) -E g (|x 2 -x 1 |) -ω L\\n\\n, and Ω L = -d eg E 0 / is the the Rabi frequency. For such conditions, the excited state is only weakly populated and can be adiabatically eliminated. Additional simplification comes from the fact that the transitions between states g and e, due to the laser light, occur on a time scale much shorter than the motion of atom and ion, and the dynamics of the atom and ion centers of masses can be decoupled from the internal dynamics, in full analogy to the Born-Oppenheimer approximation. Eliminating the excited state, in basically the same manner as in the standard derivation of the AC Stark shift, we obtain the following equation that governs the dynamics of the atom and ion center of masses:\\n\\ni ∂c g (x 1 , x 2 , t) ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E g (|x 2 -x 1 |) + V opt (x 1 ) + eΦ(x 2 , t) c g (x 1 , x 2 , t), ( A13\\n\\n) where V opt (x) = - Ω 2 L 4∆ cos(k L x + φ x ) 2 + (x → y, z) ( A14\\n\\n)\\n\\nis the effective potential due to the laser field. In the formula (A14) we have neglected the position dependence of the detuning ∆(|x 2x 1 |) and of the dipole matrix element d eg (x 1 , x 2 ), assuming that they are modified only at very short distances between the particles. This can be justified, since in the atomic collisions, the probability of finding the particles at short distances, comparable to the range of chemical binding forces, is typically very small. Time averaging over fast oscillations of the rf field: derivation of the Paul trapping potential. We replace the time-dependent rf field by an effective, adiabatic potential. In this way we neglect the fast micromotion of the ion on the time scale of ω rf [5] . Finally we obtain the following equation that describes the dynamics of the wave function c g (x 1 , x 2 , t) dependent on the atom and ion positions\\n\\ni ∂c g ∂t = p 2 1 2m 1 + p 2 2 2m 2 + E g (|x 2 -x 1 |) + V opt (x 1 ) V rf (x 2 ) c g (x 1 , x 2 , t). ( A15\\n\\n)\\n\\nHere V rf denotes the effective potential\\n\\nV rf (x) = 1 2 m 2 ω x x 2 + ω y y 2 + ω z z 2 , ( A16\\n\\n)\\n\\nwhere ω k = (a k + q 2 k /2) 1/2 ω rf /2, a k = 4eu k /(m 2 ω 2 rf ) and q k = 2ev k /(m 2 ω 2 rf ) for k = x, y, z. The interaction between atom and ion is given by\\n\\nE g (|x 2 -x 1 |) -\\n\\nthe ground-state energy of the electron motion, which at large distances behaves as E g (r) ∼ -αe 2 /(2r 4 ).\\n\\nAPPENDIX B: DETERMINING OF ϕe AND ϕo FROM PSEUDOPOTENTIALS\\n\\nIn principle, the replacement of the r -4 interaction by the pseudopotential is strictly valid when R * ≪ l ⊥ . Nevertheless, it is possible to try to determine ϕ e and ϕ o for R * ∼ l * using the energy-dependent pseudopotentials. To this end we replace the r -4 interaction with [17, 18] V s (r) = 2π 2 a(k) µ δ(r) ∂ ∂r r (B1) 17 for even scattering, and [19] V p (r) = π 2 a p (k) 3 µ ← ∇ δ(r) → ∇ r ∂ 3 ∂r 3 r 2 , (B2) for odd scattering. Here, a(k) = -tan δ 0 (k)/k is the energy-dependent s-wave scattering length, a p is the pwave scattering length: a p (k) 3 = -tan δ 1 (k)/k 3 , the symbol ← ∇ ( → ∇) denotes the gradient operator that acts to the left (right) of the pseudopotential, and δ 0 (k), δ 1 (k) are the s-and p-wave phase shifts, respectively.\\n\\nIn 1D the even (ψ e ) and odd (ψ o ) scattering waves have the following asymptotic behavior: ψ e (z, k) ∼ sin(k|z| + δ e (k)), and ψ o (z, k) ∼ z/|z| sin(k|z| + δ o (k)) (|z| → ∞), where δ e (k) and δ o (k) denote even and odd scattering phases, respectively. In analogy to the three-dimensional scattering theory, we define 1D even (a e 1D ) and 1D odd (a o 1D ) scattering lengths: a e,o 1D = lim k→0 -(tan δ e,o (k))/k. From this definition it follows that for k = 0 the even and odd scattering waves behaves as: ψ e (z, k = 0) ∼ |z|-a e 1D and ψ o (z, k = 0) ∼ z -a o 1D z/|z| (|z| → ∞). The latter result applied to Eqs. (10) and (11) leads to a e 1D /R * = -cot ϕ e (B3) a o 1D /R * = -cot ϕ o (B4)\\n\\nIn the pseudopotential approximation one can solve the quasi-1D scattering problem exactly, and calculate values of the 1D scattering lengths for even [20, 21] and odd [22] waves a e 1D (k) = -\\n\\nl 2 ⊥ 2a s (E) - l ⊥ 2 ζ 1 2 , 3 2 - E 2 ω ⊥ , ( B5\\n\\n)\\n\\na o 1D (k) = l ⊥ 2 l 3 ⊥ 12a p (E) 3 -ζ - 1 2 , 3 2 - E 2 ω ⊥ -1 , ( B6\\n\\n)\\n\\nwhere E = ω ⊥ + 2 k 2 /(2µ) and ζ(s, a) denotes the Hurwitz Zeta function: ζ(s, a) = ∞ k=0 (k + a) -s [23] . Finally, to relate the energy-dependent scattering lenghts a s (E) and a p (E) to ϕ, we apply the exact solutions of the Schrödinger equation for r -4 potential, given by the Mathieu functions [7, 8, 9].\\n\\n[1] N. Schlosser, G. Reymond, I.E. Protsenko and P. Grangier, Nature (London) 411, 1024 (2001). [2] Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Förster, M. Khudaverdyan, D. Meschede, D. Schrader, and A. Rauschenbeutel, Nature (London), 442, 151 (2006). [3] M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, Nature 415, 39-44 (2002). [4] R. Folman et al., Adv. At. Mol. Opt. Phys. 48 263 (2002); R. Dumke et al., Phys. Rev. Lett. 89, 97903 (2002). [5] D. Leibfried, R. Blatt, C. Monroe, D. Wineland, Rev.\\n\\nMod. Phys. 75, 281 (2003). [6] O. Mandel, M. Greiner, A. Widera, T. Rom, T. W. Hänsch, and I. Bloch, Nature 425, 937-940 (2003). [7] E. Vogt, and G.H. Wannier, Phys Rev. 95, 1190 (1954). [8] T.F. O'Malley, L. Spruch, L. Rosenberg, J. Math. Phys. 2, 491 (1961). [9] R.M. Spector, J. Math. Phys. 5, 1185 (1964). [10] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). [11] R. Stock, I. H. Deutsch, and E. L. Bolda Phys. Rev. Lett.\\n\\n91, 183201 (2003). [12] Z. Idziaszek, T. Calarco, P. Zoller, in preparation. [13] D.J. Tannor Introduction to Quantum Mechanics: A Time-Dependent Perspective, (University Science Books, 2006). [14] L.D. Landau, and E.M. Lifshitz Quantum Mechanics: Non-relativistic Theory, 3rd edition (Butterworth-Heinemann, Oxford, 2003). [15] C. Zener, Proc. R. Soc. London A137, 696 (1932).\\n\\n[16] A. Auerbach, and S. Kivelson, Nucl. Phys. B 257, 799 (1985). [17] D. Blume, C.H. Greene, Phys. Rev. A 65, 043613 (2002). [18] E.L. Bolda, E. Tiesinga, and P.S. Julienne, Phys. Rev. A 66, 013403 (2002). [19] Z. Idziaszek, and T. Calarco, Phys. Rev. Lett. 96, 013201 (2006). [20] T. Bergeman, M.G. Moore, and M. Olshanii, Phys. Rev. Lett. 91, 163201 (2003). [21] Z. Idziaszek, and T. Calarco, Phys. Rev. A 74, 022712 (2006). [22] B.E. Granger, D. Blume, Phys. Rev. Lett. 92, 133202 (2004). [23] M. Abramowitz and I.E. Stegun Handbook of Mathematical Functions, 10th edition, (Department of Commerce, Washington DC, 1972). [24] For atomic states with a permanent quadrupole moment, the long-range part of the potential comes from the interaction of ion charge with the atom quadrupole moment. [25] 1D solutions (10) and (11) are valid at all distances for k → 0 (E → ω ⊥ ). [26] COM and relative motions can be decoupled in the lowest order when the size of the molecule is much smaller than the harmonic oscillator lengths li and la. [27] At least for the internal states with the total angular momentum J = 1/2.\"\n }\n]","isConversational":false}}},{"rowIdx":182,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1038"},"arxiv_id":{"kind":"string","value":"0704.1038"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1016/j.nuclphysb.2007.07.026"},"content_hash":{"kind":"string","value":"766e7a4219705289485bd116dc5634201dacdf8ed85b8060d4f8ba56d6c22ef5"},"title":{"kind":"string","value":"Counting BPS operators in N=4 SYM"},"abstract":{"kind":"string","value":"The free field partition function for a generic U(N) gauge theory, where the\nfundamental fields transform in the adjoint representation, is analysed in\nterms of symmetric polynomial techniques. It is shown by these means how this\nis related to the cycle polynomial for the symmetric group and how the large N\nresult may be easily recovered. Higher order corrections for finite N are also\ndiscussed in terms of symmetric group characters. For finite N, the partition\nfunction involving a single bosonic fundamental field is recovered and explicit\ncounting of multi-trace quarter BPS operators in free \\N=4 super Yang Mills\ndiscussed, including a general result for large N. The partition function for\nBPS operators in the chiral ring of \\N=4 super Yang Mills is analysed in terms\nof plane partitions. Asymptotic counting of BPS primary operators with\ndiffering R-symmetry charges is discussed in both free \\N=4 super Yang Mills\nand in the chiral ring. Also, general and explicit expressions are derived for\nSU(2) gauge theory partition functions, when the fundamental fields transform\nin the adjoint, for free field theory."},"authors":{"kind":"list like","value":["F.A. Dolan"],"string":"[\n \"F.A. Dolan\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered. Higher order corrections for finite N are also discussed in terms of symmetric group characters. For finite N, the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free N = 4 super Yang Mills discussed, including a general result for large N. The partition function for quarter BPS operators in the chiral ring of N = 4 super Yang Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free N = 4 super Yang Mills and in the chiral ring. Also, general and explicit expressions are derived for SU(2) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory.\n\nFor the past while there has been intense interest in finite N partition functions for Yang Mills theories, especially in super-symmetric ones, particularly with regard to their construction for BPS states and the counting thereof [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] . Much attention has been devoted to this issue for N = 4 super Yang Mills theory, it being by now the archetypal example of a conformal field theory for which we have a dual description in terms of string theory, by means of the AdS/CFT conjecture [16] . An example is in the assiduous efforts that have been made to explain the entropy of certain BPS black holes in AdS 5 ×S 5 [17, 18, 19, 20] in terms of microscopic counting of dual operators in N = 4 super Yang Mills, with gauge group SU (N ), by means of partition functions [1] . While this problem remains unsolved to date, essentially due to the difficulty of defining what is meant by these dual operators for finite N , there are other equally interesting sectors of (super) Yang Mills theories where more progress with counting has been made, examples being in free N = 4 super Yang Mills and in chiral ring sectors involving BPS operators.\n\nFor N = 4 super Yang Mills, the half BPS sector consists of multi-trace operators involving a single bosonic operator, Z. Similarly, the quarter BPS sector consists of multitrace operators involving two bosonic operators Z, Y while the eighth consists of ones involving three bosonic operators Z, Y, X and two fermionic operators λ, λ. (All these operators are here assumed to belong to the Lie algebra of U (N ).)\n\nFor the chiral ring, the commuting/anti-commuting of these operators is at the heart of why we can write very concise and elegant generating functions for the finite N multitrace partition functions [1] . Analysis of these partition functions, in terms of the counting of operators, has become a sophisticated industry where such approaches as the so-called 'Plethystic Program' have provided substantial results [3, 9] . This paper is devoted largely to the issue of partition functions for free field theory and particularly to the counting of gauge invariant multi-trace operators for the case of two bosonic fundamental fields. This is of relevance to the quarter BPS sector of N = 4 super Yang Mills in the free field limit when the operators Z, Y do not commute, in contrast to the chiral ring when they do.\n\nThe partition function for a free, massless quark-gluon gas was computed long ago [21] . This involved taking particle statistics into account using coherent state techniques and then imposing the gauge singlet condition by integrating over the relevant gauge group. With some modifications to the expression thus derived we may write the multi-trace partition function for some generic bosonic/fermionic fundamental fields in terms of an integral over the gauge group, involving the single particle partition function [22, 23] . This is the starting point here.\n\nFor U (N ) gauge theories we may easily write down the integral, though, even for this case, its evaluation is far from simple. One approach (which we adapt here for the SU (2) case) is to rewrite the expression in terms of an N -fold contour integral, whereby it may in principle be evaluated by summing the contributions from poles inside origin centred unit discs, in each of the N complex planes -similar techniques have been used in [10] . Due to the number of poles this becomes unfeasible for higher values of N . Another approach is to use the fact that the complex integral that interests us provides for an inner product for symmetric polynomials -see Macdonald [24] , pp. 363 -372, for a related discussion. Taking this point more seriously reveals an alternative route to evaluating the free field theory partition function which exposes not only the large N case in an almost trivial way, but also how and where this differs from the finite N case. This treatment also reveals an alternative interpretation of the free field partition function at finite N -it is related to a gauge group average of the cycle polynomial for the symmetric permutation group (after a certain identification of 'letters' with gauge group valued variables). This point is not dwelt on further here though makes the connection between the partition function and Polya enumeration explicit. For single trace operators at large N this connection has already been made [22, 25] for N = 4 super Yang Mills, whereby the partition function for single trace operators is related to the cycle polynomial for the cyclic permutation group. 1 Another issue is how to use the expression for the free field partition function to give explicit counting of gauge invariant multi-trace operators in a Yang-Mills theory, with gauge group U (N ). The case for one bosonic fundamental field has been widely discussed and, for finite N , the operators are counted by partition numbers of non-negative integers into at most N parts (which is here denoted by p N (n) -no closed formula for these numbers for arbitrary N, n exist, though they have a 'nice' generating function). Here, this result is re-derived, from a symmetric function perspective, by employing the well known Cauchy-Littlewood formula.\n\nTo proceed further with counting, for the quarter BPS sector of N = 4 super Yang Mills, for instance, character methods prove to be both natural and indispensable. Characters in relation to conformal field theories prove to be very convenient for encoding the allowable representations [15, 29, 30, 31, 32, 33] and for studying related partition functions, [34, 35, 36, 33] . For N = 4 super Yang Mills, it was shown in [33] that, if we are to distinguish among primary operators with differing conformal dimensions, spins and R-symmetry charges, such counting is most easily achieved using reductions of the full N = 4 superconformal characters, in certain limits that isolate corresponding sectors of short/semi-short operators. (One such limit corresponds to the index constructed in [1] .) This point may be easily illustrated for quarter BPS primary operators in N = 4 super Yang Mills, the case of two bosonic fundamental fields here. (See [37] for an explicit construction and counting of quarter BPS operators.) For N = 4 super Yang Mills, the counting of quarter BPS primary operators is complicated by the fact that, if we are to keep track of differing R-symmetry representations, any partition function restricted to this sector must be expanded in terms of U (2) characters (or two-variable Schur polynomials). Denoting some partition function restricted to this sector by Z(t, u), where t, u are letters corresponding to the fields Z, Y , then by expanding\n\nZ(t, u) = ∞ n=0 n m=0 N (n,m) s (n,m) (t, u) , s (n,m) (t, u) = t n+1 u m -t m u n+1 t -u , (1.1)\n\nin terms of two-variable Schur polynomials s (n,m) (t, u), we obtain the numbers N (n,m) of gauge invariant quarter BPS operators belonging to the [m, n-m, m] SU (4) R R-symmetry representation and having conformal dimension n + m (so that they are superconformal primary highest weight states in the corresponding quarter BPS supermultiplets). (The case m = 0 counts gauge invariant half BPS primary operators.)\n\nHere, the free field partition function is thus expanded in terms of Schur polynomials, depending on the same variables as the one particle partition function, the two boson case being a specialisation. This is quite naturally achieved using the Cauchy-Littlewood formula (and, if we include fermions, another formula due to Littlewood). Generally, we may obtain a result that relates the counting numbers to a sum over Kronecker coefficients. These arise naturally in the theory of the symmetric permutation group, though remain somewhat mysterious from a combinatorial perspective.\n\nSpecialising to the two boson case, a recursive procedure is employed here for the counting of multi-trace quarter BPS operators in free field theory at finite N . This issue was given considerable discussion for the large N case in [33] -here the results of [33] are generalised in terms of a generating function that may be employed to count quarter BPS operators for any R-symmetry charges, at large N . Asymptotic counting is also addressed in the latter case for the numbers N (n,m) in (1.1) for large n and fixed m.\n\nTo complete the discussion, counting of quarter BPS operators in the chiral ring of N = 4 super Yang Mills is investigated in terms of expanding over U (2) characters as in (1.1) . An explicit formula is given for the corresponding finite N partition function, with a short combinatorial interpretation given in terms of plane partitions, and specialised to large N . For the latter case, the exponential behaviour of the numbers N (n,m) in (1.1) is found for n and m both comparably large. This behaviour is consistent with a special case addressed in [3, 9] . By way of completion, a similar discussion for an arbitrary number of bosonic fundamental fields in the chiral ring is included. Two appendices are included; the first establishes some notation used for partitions and gives some standard results for the symmetric group and symmetric polynomials, the second gives some tables of numbers of quarter BPS operators in free N = 4 super Yang Mills, with gauge group U (N ), for which explicit formulae are given in the main text. Footnotes contain further details and points of clarification.\n\nWe start from the single particle partition function which is here denoted by f (t) for some variables 2 t = (t 1 , t 2 . . .). The general form of f (t) is\n\nf (t) = i a i t i , (2.1)\n\nwhere each t i is a letter corresponding to a fundamental field and a i are signs, being +1 for a bosonic field, or -1 for a fermionic field.\n\nFor compact gauge Lie group G, the multi-trace partition function is then given by [21] , (see [22, 23] for refinements,)\n\nZ G (t) = G dµ G (g) exp ∞ n=1 1 n f (t n )χ R (g n ) , (2.2)\n\ninvolving the Haar (or G-invariant, or Hurwitz) measure dµ G (g) for g ∈ G (so that\n\nG dµ G (g)F (g) = G dµ G (g)F (gh) = G dµ G (g)F (hg) for all h ∈ G and G dµ G (g) = 1)\n\nand where χ R (g) is the character for the R representation of G, assuming that the fundamental fields transform in identical gauge group representations R.\n\nFor G = U (N ), so that for any matrix U ∈ U (N ) we may write U = V ΘV † , where V is a unitary matrix and Θ = diag.(e iθ 1 , . . . , e iθ N ), 0 ≤ θ i < 2π, and for some F (U ) = F (Θ), independent of V , then we may write\n\nU(N) dµ U(N) (U )F (U ) = 1 (2π) N N ! 2π 0 N j=1 dθ j 1≤k<l≤N |e iθ k -e iθ l | 2 F (Θ) , (2.3)\n\n2 In what follows roman letters are used to denote a collection of variables and, for x = (x 1 , . . . , x i ), y = (y 1 , . . . , y j ) for example, the shorthand x α is used to mean (x 1 α , . . . , x i α ) and z = xy to mean z = (z 11 , . . . , z ij ) where z rs = x r y s . The latter convenient notation has been used by Macdonald [24] .\n\nwhich is, of course, related to the Weyl parametrisation of U (N ). Thus, for such F (U ), the left-hand side of (2.3) simplifies to an integral over the N torus. Of course, as χ R (U ) generally depends on linear combinations of tr(U j ) k tr(U †l ) m for various non-negative integers j, k, l, m then any function of χ R (U ) is an example of such an F (U ).\n\nWe are interested in the case where R = Adj. is the adjoint representation so that for U (N ) we have that χ Adj. (U ) = tr U tr U † (while for SU (N ) then χ Adj. (U ) = tr U tr U † -1). For U (N ) we then find that, using (2.2) with (2.3),\n\nZ U(N) (t) = 1 (2π) N N ! 2π 0 N j=1 dθ j 1≤k<l≤N |e iθ k -e iθ l | 2 exp ∞ n=1 1 n f (t n ) N j,k=1\n\ne in(θ j -θ k ) .\n\n(2.4)\n\nWe may write (2.4) as an N -fold contour integral by first making the variable change z j = e iθ j so that the integrals in (2.4) are around unit circles in each z j complex plane and then we obtain\n\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) exp ∞ n=1 1 n f (t n )p n (z)p n (z -1 ) , (2.5)\n\nwhere ∆(z) = 1≤i<j≤N (z iz j ) is the Vandermonde determinant and p n (z) = N i=1 z i n is a power symmetric polynomial -see appendix A for a brief discussion of symmetric polynomials. This integral may then in principle be evaluated by deforming the contours so as to extract the residues at poles within the discs\n\n|z j | < 1, 1 ≤ j ≤ N .\n\nA crucial observation is that, for some N variable symmetric polynomials g(z), h(z), then\n\ng, h N = h, g N = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 )g(z)h(z -1 ) , (2.6)\n\nacts as an inner product -this is easy to see in terms of Schur polynomials which provide an orthonormal basis for symmetric polynomials. The reader may now wish to peruse appendix A where notation regarding partitions and a short discussion of symmetric polynomials is included.\n\nThe General and Large N Cases for U (N )\n\nFor application of inner products to (2.5) we have that, in terms of power symmetric polynomials p λ (z) for partitions λ,\n\nexp ∞ n=1 1 n f (t n )p n (z)p n (z -1 ) = ∞ n=1 ∞ a n =0 1 n a n a n ! f (t n ) a n p n (z) a n p n (z -1 ) a n , = λ 1 z λ f λ (t)p λ (z)p λ (z -1 ) , (2.7)\n\nwith the definitions of\n\nz λ = ∞ n=1 n a n a n ! , f λ (t) = ∞ n=1 f (t n ) a n , (2.8)\n\nbeing in terms of the frequency representation of λ, (1 a 1 , 2 a 2 , . . .) with n≥1 n a n = |λ|, the weight of the partition λ (note that the frequency representation of λ is simply a convenient re-ordering of the parts of λ).\n\nIn (2.7) the numbers z λ have a standard combinatorial interpretation -for a given permutation σ ∈ S m with a 1 1-cycles, a 2 2-cycles etc., so that n≥1 n a n = m = |λ|, then z λ = n≥1 n a n a n ! is the size of the centraliser Z σ = {τ ; τ ∈ S m , τ στ -1 = σ} of σ ∈ S m . (This may be easily seen as under conjugation of σ by τ then τ can permute the cycles of length n among themselves in a n ! ways and/or render a cyclic rotation on each of the individual cycles in n a n ways.) More details of the symmetric group are to be found in appendix A.\n\nWe may immediately observe that (2.7) represents a sum over cycle polynomials of the symmetric group S m . This is given by, for letters\n\nu 1 , u 2 , . . . u m , 3 C m (u) = a 1 ,...,a m ≥0 δ a 1 +2a 2 +...+ma m ,m m n=1 1 n a n a n ! u n a n = λ⊢m 1 z λ u λ , (2.9)\n\nwhere 'λ ⊢ m' means that λ is any partition of m -see appendix A for notation -and\n\nu λ = m n=1 u n a n , (2.10)\n\nin terms of the frequency representation of λ above. Identifying u n = f (t n )p n (z)p n (z -1 ) then we may rewrite\n\nZ U(N) (t) = ∞ m=0 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 )C m (u) , (2.11)\n\n3 This formula is easy to see from the definition of the cycle polynomial for a subgroup G of S m . This is given by\n\n1 |G| g∈G u 1 j 1 (g) • • • u m j m (g) = 1 |G| K g |K g |u 1 j 1 (g) • • • u m j m (g) ,\n\nwhere j i (g) denotes the number of i cycles in the unique decomposition of g into disjoint cycles the sum of the U (N ) group averages of each of the cycle polynomials C m (u). (Physically, the interpretation is that the cycle index for the symmetric permutation group accounts for particle statistics while integration over the gauge group imposes the gauge singlet condition. For purposes of clarity, the U (N ) case has been focused upon here, though from the form of (2.2) it is easy to see how this generalises for other gauge groups whereby the letters u n = f (t n )χ R (g n ) for the fundamental fields transforming in identical gauge group representations, R.)\n\nDirectly from (2.7), in terms of the inner product (2.6), then\n\nZ U(N) (t) = λ 1 z λ f λ (t) p λ , p λ N = λ 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 , (2.12)\n\nwhere on the right-hand side of (2.12) we have used an expression for the inner product of two power symmetric polynomials expressed in terms of the characters of the symmetric group, given in appendix A. (Here 'ℓ(µ)' means the number of non-zero parts of the partition µ.)\n\nUsing a result of appendix A (essentially orthogonality relations for symmetric group characters), (2.12) may be rewritten as\n\nZ U(N) (t) = λ |λ|≤N f λ (t) + λ |λ|>N 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\n\n(2.13)\n\nIn the large N limit, Z U(N) (t) simplifies considerably as only the first term in (2.13) need be considered. Using the frequency representation of λ then\n\nZ U(∞) (t) = λ f λ (t) = ∞ n=1 ∞ a n =0 f (t n ) a n = ∞ n=1 1 1 -f (t n ) , (2.14)\n\na result which has been obtained using Polya counting methods for single trace operators and saddle point approximations [22, 23] .\n\nHigher order corrections in |λ|, the weight of the partition λ, to (2.13) may be obtained by successive evaluation of\n\nµ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\n\nOne method is to employ the Murnaghan-Nakayama Rule, used to compute χ µ λ using skew hooks and Young diagrams. (A readable account of the Murnaghan-Nakayama Rule may be found in [38] , though of course it is explained in many standard textbooks that discuss the symmetric group.)\n\nFor the case of |λ| = N +1 then we may observe that,\n\nµ⊢N +1 ℓ(µ)≤N χ µ λ 2 = µ⊢N +1 ℓ(µ)≤N +1 χ µ λ 2 -(χ ν λ ) 2 = z λ -(χ ν λ ) 2 , ν = (1 N+1 ) , (2.15)\n\nsince the partition ν = (1 N+1 ) is the only one excluded among those partitions µ of N +1 with ℓ(µ) ≤ N . By applying the Murnaghan-Nakayama Rule we may determine, for\n\nν = (1 L ), (χ ν λ ) 2 = 1 for |λ| = L 0 otherwise , (2.16)\n\nsince χ ν λ in this case is just a sign. (This may be easily seen as there is only one possible way to remove successive skew hooks, which in this case are just column Young diagrams of length λ i , from the (1 L ) column Young diagram to leave one of normal shape, in this case, another column Young diagram.) Thus, using (2.15) with (2.16) in (2.13), we obtain\n\nZ U(N) (t) = λ |λ|≤N +1 f λ (t) - λ |λ|=N +1 1 z λ f λ (t) + λ |λ|>N +1 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\n\n(2.17) By a similar line of argument we may do the same for the case of |λ| = N +2. We have that, for\n\nν 1 = (1 N+2 ) and ν 2 = (2, 1 N+1 ), µ⊢N +2 ℓ(µ)≤N χ µ λ 2 = µ⊢N +2 ℓ(µ)≤N +2 χ µ λ 2 -(χ ν 1 λ ) 2 -(χ ν 2 λ ) 2 = z λ -(χ ν 1 λ ) 2 -(χ ν 2 λ ) 2 . (2.18)\n\nWe may determine, for ν = (2, 1 L ),\n\n(χ ν λ ) 2 = (a 1 -1) 2 for |λ| = n≥1 n a n = L+2 0 otherwise . ( 2\n\n.19) Using (2.16), (2.19) with (2.18) in (2.17) then we obtain\n\nZ U(N) (t) = λ |λ|≤N +2 f λ (t) - λ N +1≤|λ|≤N +2 1 z λ f λ (t) - λ |λ|=N +2 1 z ′ λ f λ (t) + λ |λ|>N +2 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 , (2.20)\n\nwhere, in the frequency representation of λ,\n\n1 z ′ λ = (a 1 -1) 2 z λ = 1 a 1 ! - 1 (a 1 -1)! + 1 (a 1 -2)! ∞ n=2 n a n a n ! .\n\n(2.21)\n\nWe may proceed in this manner to compute explicit higher order corrections though this becomes cumbersome save for the first few cases as shown. (For |λ| > N + 2 the corrections will always involve contributions from (2.16) and (2.19) as well as extra ones coming from\n\nµ⊢|λ| ℓ(µ)≤|λ| χ µ λ 2 - µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .)\n\nThe One Boson Case for U (N )\n\nFor the case of one bosonic fundamental field (applicable to half BPS operators for N = 4 super Yang Mills), we have f (t) = t in (2.5), so that we may write\n\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) N j,k=1 1 1 -tz j z k -1 . (2.22)\n\nTo evaluate this integral we may use the Cauchy-Littlewood formula,\n\nL i=1 M j=1 1 1 -x i y j = λ ℓ(λ)≤min.{L,M } s λ (x 1 , . . . , x L )s λ (y 1 , . . . , y M ) , (2.23)\n\nwhere the sum on the right-hand side is over all partitions λ such that the corresponding Young diagrams have no more than min.{L,\n\nM } rows, ℓ(λ) ≤ min.{L, M }. With x i = tz i , y i = z i -1 , i = 1, . . . , N , in (2.23\n\n), so that s λ (tz 1 , . . . , tz N ) = t |λ| s λ (z), and employing also (2.6 ) and the orthonormality of Schur polynomials, we may easily obtain,\n\nZ U(N) (t) = λ ℓ(λ)≤N t |λ| s λ , s λ N = λ ℓ(λ)≤N t |λ| . (2.24)\n\nBy changing summation variables so that\n\nλ i -λ i+1 = a i , i = 1, . . . , N -1, λ N = a N then we may write Z U(N) (t) = ∞ a 1 ,...,a N =0 t a 1 +2a 2 +...+Na N = P N (t) , (2.25)\n\nwhere 4\n\nP N (t) = N i=1 1 1 -t i . (2.26)\n\nOf course this is nothing other than the generating function for the number p N (n) of partitions of n into no more than N parts since, by definition,\n\nλ ℓ(λ)≤N δ |λ|,n = p N (n) , (2.27)\n\nso that by the above\n\n∞ n=0 p N (n)t n = P N (t) .\n\n(2.28)\n\n4 1/P ∞ (t) = ∞ n=1 (1-t n ) is commonly called the Euler function, denoted by Φ(t). P ∞ (t) = ∞ n=0 p(n)t n\n\nacts as a generating function for the number of unordered partitions of n, p(n). Note that p N (n) = p(n) for n ≤ N , i.e. the number of partitions of n into no more than N parts is the same as the total number of partitions of n so long as n ≤ N .\n\nThis makes explicit the connection between Z U(N) (t) and the partition numbers p N (n).\n\nHere we first consider f (t) = k j=1 t j in (2.1) so that the variables 0 ≤ t i < 1 represent k bosons in the single particle partition function. For such fields transforming in the adjoint representation of SU (2) then (2.2) simplifies significantly. For any U ∈ SU (2) we may write U = V ΘV † , where V is unitary and Θ = diag.(e iθ , e -iθ ), for 0 ≤ θ < 2π, so that for F (U ) = F (θ), then in usual Weyl parametrisation,\n\nSU(2) dµ SU(2) (U )F (U ) = 1 π 2π 0 dθ sin 2 θ F (θ) = 1 4π 4π 0 dθ 1 -cos θ F ( θ 2 ) = 1 2π 2π 0 dθ 1 -cos θ F ( θ 2 ) , (2.29)\n\nwhere F (θ) = F (θ + π) is assumed in writing the last line. In the present case,\n\nF (U ) = F (θ) = n≥1 f (t n )χ Adj. (U n )/n, where χ Adj. (U ) = tr(U )tr(U † ) -1 = e 2iθ + e -2iθ + 1 = 2 cos 2θ + 1, so that Z SU(2) (t) = 1 2π 2π 0 dθ (1 -cos θ) k j=1 1 (1 -t j )(1 -t j e iθ )(1 -t j e -iθ )\n\n.\n\n(2.30)\n\nMaking the variable change z = e iθ , and using that F (θ) = F (-θ) is even, then\n\nZ SU(2) (t) = 1 2πi dz z (1 -z) k j=1 1 (1 -t j )(1 -t j z)(1 -t j z -1 ) , (2.31)\n\nwhere the integral is around the unit circle |z| = 1. The residues in (2.31) may be easily computed since all the relevant (simple) poles in the disc |z| < 1 occur at the points z = t j . Thus\n\nZ SU(2) (t) = k i=1 t i k-1 1 -t i 2 k j=1 j =i 1 (t i -t j )(1 -t i t j )(1 -t j ) . (2.32)\n\nThis partition function has an interesting interpretation from a group theory perspective. We may write,\n\n1 + t (1 -tz)(1 -tz -1 ) = ∞ n=0 χ n (z)t n , (2.33)\n\nwhere\n\nχ j (z) = z j+ 1 2 -z -j-1 2 z 1 2 -z -1 2 , j ∈ 1 2 Z , (2.34)\n\nis an SU (2) character, corresponding to the spin j irreducible representation, R j . Now the integral in (2.31) acts as an SU (2) inner product,\n\nχ j , χ k = 1 2πi dz z (1 -z)χ j (z)χ k (z -1 ) = δ jk , (2.35)\n\nfor j, k ∈ N. Thus, from (2.31) with (2.33) and (2.35),\n\nk j=1 (1 -t j 2 )Z SU(2) (t) = ∞ n 1 ,...,n k =0 χ n 1 • • • χ n k , 1 t 1 n 1 • • • t k n k , (2.36)\n\nacts as a generating function for the number of singlets in the decomposition of the 5 By using that χ n (z) = n j=-n z j we may use the Cauchy residue theorem to compute explicitly that\n\nSU (2) representation R n 1 ⊗ • • • ⊗ R n k .\n\nχ n 1 • • • χ n k , 1 = 2n 1 j 1 =0 • • • 2n k j k =0 (δ j 1 +•••+j k ,n 1 +•••+n k -δ j 1 +•••+j k ,n 1 +•••+n k +1 ) .\n\n(2.37)\n\nIf we modify the one particle partition function to include k bosons and k fermions and hence consider (2.1) in the form f (t, t) = k j=1 t j -k =1 t then we may similarly as above evaluate\n\nZ SU(2) (t, t) = 1 2πi dz z (1 -z) 1≤j≤k 1≤≤ k (1 -t )(1 -t z)(1 -t z -1 ) (1 -t j )(1 -t j z)(1 -t j z -1 ) , (2.38)\n\nwhere the contour is around the unit disc |z| = 1. So long as k > k then (2.38) receives contributions from only those simple poles at z = t j so that for this case we obtain\n\nZ SU(2) (t, t) k> k = k i=1 t i k-k-1 1 -t i 2 1≤≤ k 1≤j≤k,j =i (t i -t )(1 -t i t )(1 -t ) (t i -t j )(1 -t i t j )(1 -t j ) . (2.39)\n\nFor k ≤ k then (2.38) also receives contributions from poles at z = 0. For instance\n\nZ SU(2) (t, t) k= k = k i=1 1 t i (1 -t i 2 ) 1≤j,≤k j =i (t i -t )(1 -t i t )(1 -t ) (t i -t j )(1 -t i t j )(1 -t j ) + 1≤j,≤k t (1 -t ) t j (1 -t j ) , (2.40)\n\n5 Generating functions for products of Lie algebra representations have been considered elsewhere, in [39] for instance. A generating function for the number of singlets in n products of the fundamental times n products of the anti-fundamental representations for SU (N ) was found by Gessel [40] in terms of Toeplitz determinants involving Bessel functions. See also [41] for a nice physics oriented discussion of similar issues. The special case of R 1 2\n\n⊗ • • • ⊗ R 1 2\n\n(2n products of the fundamental) for SU (2), contains a Catalan number, 1 n+1 2n n , of singlet representations.\n\nwhere the last term on the right-hand side of (2.40) comes from the simple pole at z = 0. These formulae should be useful for computing the multi-trace partition functions, for fundamental fields transforming in an SU (2) gauge group, in other sectors of N = 4 super Yang Mills. For instance, after a suitable identification of the variables t j , t with variables in single particle partition functions for semi-short sectors of N = 4 super Yang Mills, described in detail in [33] , then (2.40) should allow for an explicit expression for corresponding multi-trace partition functions. They may also be useful for computing the N = 4 superconformal index of [1] for SU (2) gauge group, or at least for restrictions of it such as described in [33] or [42] .\n\nIn this section, the counting of half and quarter BPS operators for free N = 4 super Yang Mills, when the fundamental fields transform in the adjoint representation of U (N ), is discussed in some detail.\n\nWe may, of course, proceed to count multi-trace half and quarter BPS primary operators directly, in terms of the fundamental fields, Z, for half BPS operators and Z, Y , for quarter BPS operators.\n\n(Z, Y ) forms a U (2) doublet, where U (2) has generators given by a subset of the SU (4) R generators, H i , E i± , 1 ≤ i ≤ 3, where H i are the Cartan sub-algebra generators and E i± are ladder operators satisfying (in the Chevalley-Serre basis)\n\n[H i , E j± ] = ±K ij E j± , with [K ij ] being the usual SU (4) Cartan matrix. The U (2) generators consist of the SU (2) generators H 2 , E 2± , where explicitly [(H 1 , H 2 , H 3 ), E 2± ] = ∓(1, -2, 1)E 2± , along with the generator H 1 +H 2 +H 3 , whose eigenvalues give the conformal dimensions in this case, [H 1 +H 2 +H 3 , (Z, Y )] = (Z, Y ) [33]. Explicitly, we have that [E 2+ , Z] = 0, [E 2-, Z] = Y , [(H 1 , H 2 , H 3 ), Z] = (0, 1, 0)Z, [(H 1 , H 2 , H 3 ), Y ] = (1, -1, 1)Y so that an operator involving n Z's and m Y 's transforms in the [m, n-m, m] SU (4) R R-symmetry representation.\n\nFor k-trace half BPS primary operators transforming in the [0, n, 0] SU (4) R Rsymmetry representation, with conformal dimension n, then in terms of the fundamental field Z a basis is provided by,\n\ntr(Z n 1 ) • • • tr(Z n k ) , k i=1 n i = n . (3.1)\n\nWe have that, due to trace identities for finite N , tr(Z n ) for n > N is expressible in terms of a sum over multi-trace operators of the form (3.1), for k > 1, and thus, a minimal basis for multi-trace half BPS primary operators consists of (3.1) for all 1 ≤ k ≤ n and with every n i ≤ N , ordered so that n 1 ≥ n 2 ≥ . . . ≥ n k ≥ 0, i.e. so that (n 1 , . . . , n k ) is a partition of n where each part n i ≤ N . With this restriction, the number of multi-trace half BPS primary operators for a given n is\n\nN (n) = p N (n) , (3.2)\n\nsince the number p N (n), in (2.27), of partitions of n into ≤ N parts is the same as the number of partitions of n in which each part is ≤ N -see [43] for a simple proof employing generating functions.\n\nFor quarter BPS operators belonging to the [m, n-m, m] SU (4) R R-symmetry representation, a basis for k-trace operators is\n\ntr j Z n 1j Y m 1j • • • tr j Z n kj Y m kj , i,j n ij = n , i,j m ij = m , (3.3)\n\nwhere there is a choice of ordering in each trace. (Note that the m = 0 case corresponds to the half BPS case already considered.) Using the basis provided by (3.3) for all allowable k, then to avoid over-counting of multi-trace quarter BPS operators, the cyclicity of each trace and also trace identities for finite N must be accounted for. Assuming that this is done, let M (n,m) denote the number of elements in this minimal basis for multi-trace operators of the form (3. The number of such U (2) descendants coincides with N (n+m-j,j) , the number of corresponding primary operators. In this way, we obtain that\n\nM (n,m) = N (n,m) + N (n+1,m-1) + . . . + N (n+m-1,1) + N (n+m) , (3.4)\n\nso that N (n,m) = M (n,m) -M (n+1,m-1) may be obtained recursively for each m.\n\nWe may illustrate by counting all multi-trace quarter BPS primary operators in the [1, n-1, 1] R-symmetry representation. In this case a basis for k+1-trace operators is provided by\n\ntr(Z n 1 ) • • • tr(Z n k )tr(Z j Y ) , k i=1 n i = n -j . (3.5)\n\nCyclicity of traces implies that we may arrange Y as shown, to avoid over-counting. U (N ) trace identities imply, similarly as for the half BPS case, that a minimal basis for multitrace operators requires j < N and each n i ≤ N in (3.5) for every 1 ≤ k ≤ n-j, so that (n 1 , . . . , n k ) forms a partition of n-j, with every part ≤ N . Thus, by a similar argument as for the half BPS case, M (n,1) = N-1 j=0 p N (nj). Finally, to ensure that only primary operators are counted then we must subtract off contributions from descendants of half BPS primary operators in the [0, n+1, 0] SU (4) R representation, of which there are p N (n+1). Using (3.4) with (3.2) we then conclude that\n\nN (n,1) = N-1 j=0 p N (n -j) -p N (n+1) , (3.6)\n\ngives the number of multi-trace quarter BPS primary operators in the [1, n-1, 1] Rsymmetry representation.\n\nCounting in this fashion becomes more difficult for greater m and now a procedure is described employing symmetric polynomials to find a generating function for the numbers of multi-trace quarter BPS primary operators in the [m, n-m, m] SU (4) R representation, for m = 0, 1, 2 at finite N and for any n, m at large N . This generating function is subsequently used to provide asymptotic counting for fixed m, large n in the large N limit.\n\nFor k bosonic fundamental fields, we may take f (t) = k j=1 t j in (2.1) so that (2.5) may be written as\n\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) k j=1 N r,s=1 1 1 -t j z r z s -1 . (3.7)\n\nOften it is the case that such partition functions should be expanded in terms of s λ (t), the k variable Schur polynomial labelled by partitions λ. An example is provided by (1.1) for counting multi-trace quarter BPS operators. We may use the Cauchy-Littlewood formula (2.23) to expand in this way, to obtain\n\nZ U(N) (t) = λ ℓ(λ)≤min.{k,N 2 } N λ s λ (t) , (3.8)\n\nwhere\n\nN λ = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s λ (zz -1 ) , (3.9)\n\nwhere zz -1 has components z i z j -1 , 1 ≤ i, j ≤ N .\n\nFrom Macdonald [24] we have that\n\ns λ (xy) = µ,ν⊢|λ| γ λ µν s µ (x)s ν (y) , (3.10)\n\nin terms of Kronecker coefficients,\n\nγ λ µν = 1 |λ|! σ∈S |λ| χ λ (σ)χ µ (σ)χ ν (σ) = ρ⊢|λ| 1 z ρ χ λ ρ χ µ ρ χ ν ρ , (3.11)\n\nbeing a sum over irreducible S |λ| characters evaluated at σ ∈ S |λ| , related to a sum over irreducible S |λ| characters evaluated on the conjugacy classes labelled by the partitions ρ in the second line. Using (2.6) along with the orthonormality property of Schur polynomials we find that\n\nN λ = µ⊢|λ| ℓ(µ)≤N γ λ µµ .\n\n(3.12)\n\nThe situation becomes much more involved if we include also k fermionic fields, so that (2.1) may be written in the form f (t, t) = k j=1 t j -k =1 t , and attempt to expand Z U(N) (t, t) in terms of products of Schur polynomials s λ (t)s µ ( t). Such expansion is required for counting, for instance, for the free field partition function in the eighth BPS sector of N = 4 super Yang Mills. In this case the partition function is expanded, analogous to (1.1), in terms of SU (2|3) characters, which may be expressed in terms of a linear combination of products of two-variable and three-variable Schur polynomials. (See [33] for a discussion of counting for the eighth BPS sector along these lines.) Including fermions, (3.7) becomes modified by\n\nZ U(N) (t, t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) 1≤j≤k 1≤≤ k N r,s=1 1 -t z r z s -1 1 -t j z r z s -1 . (3.13)\n\nTo achieve the expansion, we may use the Cauchy-Littlewood formula (2.23) along with another formula of Littlewood,\n\nL i=1 M j=1 (1 + x i y j ) = λ ℓ(λ)≤L,ℓ( λ)≤M s λ (x 1 , . . . , x L ) s λ(y 1 , . . . , y M ) , (3.14)\n\nwhere λ is the partition conjugate to λ (where the rows and columns of the Young diagram corresponding to λ are interchanged) and where the sum is restricted to those λ whereby the corresponding Young diagrams have at most L rows, ℓ(λ) ≤ L, and M columns, ℓ( λ) ≤ M . We may thus write\n\nZ U(N) (t, t) = λ ℓ(λ)≤min.{k,N 2 } µ ℓ(µ)≤ k,ℓ( μ)≤N 2 N λ,µ s λ (t) s µ ( t) , (3.15)\n\nwhere\n\nN λ,µ = (-1) |µ| (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s λ (zz -1 )s μ(zz -1 ) . (3.16)\n\nObviously these numbers are considerably more involved than those in (3.9). We may of course use (3.10) again to interpret (3.16) in terms of Kronecker coefficients.\n\nThe two bosonic fundamental field case is now focused upon. 6 In particular, the numbers N (n,m) in (1.1) are evaluated using results of the last sub-section.\n\nWe may proceed to evaluate N λ recursively. The simplest case is for N λ = N (n) , whereby introducing a formal variable t then it is clear, by (2.23) with (2.22), (2.25) and (2.26), that ∞ n=0\n\nN (n) t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) N j,k=1 1 1 -tz j z k -1 = P N (t) , (3.17)\n\nso that, by (2.28), N (n) is given by (3.2).\n\nMore generally to evaluate N (n,m) from (3.9) we may use, for y = zz -1 ,\n\ns (m) (y)s (n) (y) = s (n,m) (y) + s (n+1,m-1) (y) + . . . + s (n+m-1,1) (y) + s (n+m) (y) , s (m) (zz -1 ) = µ⊢m ℓ(µ)≤N s µ (z)s µ (z -1 ) , (3.18)\n\n6 The two boson case leads to an interesting generalisation of an identity in [44] involving Littlewood-Richardson coefficients c ν λµ , the coefficients that appear in the decomposition s λ (x)s µ (x) = ν c ν λµ s ν (x). With f (t) in (2.1) given by f (t) = t 1 + t 2 , and expanding appropriately the corresponding integrand in (3.7) using (2.23); then using (2.6), the orthonormality of Schur polynomials and the result (2.14), we obtain (note that\n\nc ν λµ = 0 if |ν| = |λ| + |µ|) Z U(∞) (t 1 , t 2 ) = λ,µ t 1 |λ| t 2 |µ| s λ s µ , s λ s µ ∞ = λ,µ,ν ν⊢|λ|+|µ| t 1 |λ| t 2 |µ| (c ν λµ ) 2 = n≥1 1 1 -t 1 n -t 2 n\n\nwhich reduces to Theorem 4.1 of [44] if we take t 1 = t 2 = t.\n\nwhere the expression in the first line of (3.18) may be easily seen using Young tableaux multiplication rules while (2.23) determines the expression in the second line. From (3.9) with (3.18), we may find a useful generating function, in terms of a formal variable t, for the numbers in (3.4) as follows,\n\nF (m) N (t) = ∞ n=0 M (n,m) t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s (m) (zz -1 ) ∞ n=0 s (n) (zz -1 )t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) µ⊢m ℓ(µ)≤N s µ (z)s µ (z -1 ) N j,k=1 1 1 -tz j z k -1 = µ⊢m ℓ(µ)≤N λ t |λ| s λ s µ , s λ s µ N , (3.19)\n\nso that we may write\n\nN (n,m) = 1 2πi dt t 1 t n F (m) N (t) - 1 t n+1 F (m-1) N (t) , (3.20)\n\nwhich allows for recursive determination of N (n,m) .\n\nApplying this to the case of N λ = N (n,1) we have, from (3.18)\n\ns (1) (zz -1 ) = s (1) (z)s (1) (z -1 ) , (3.21) so that, from (3.19), F (1)\n\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = λ ℓ(λ)≤N t |λ| s (1) s λ , s (1) s λ N . (3.22)\n\nUsing (again, this may be easily seen from Young tableaux multiplication rules)\n\ns (1) (z)s λ (z) = N r=1 s λ+e r (z) , (3.23)\n\nfor {e r ; 1 ≤ r ≤ N, e r • e s = δ rs } being usual orthonormal vectors, we find that\n\nF (1)\n\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = λ ℓ(λ)≤N t |λ| N r,s=1\n\ns λ+e r , s λ+e s N .\n\n(3.24)\n\nNow for any partition λ, s λ+e r , s λ+e s N vanishes unless e r = e s for any r, s and λ r-1 -λ r > 0 for r = 2, . . . , N , due to\n\ns (λ 1 ,...,λ r-1 ,λ r +1,...,λ N ) (z) = 0 for λ r-1 = λ r , r > 1 . (3.25)\n\nChanging summation variables to those in (2.25) then we have, with the definition (2.26), F\n\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = a 1 ,...,a N ≥0 t a 1 +...+Na N + N-1 r=1 a 1 ,...,a N ≥0 a r ≥1 t a 1 +...+Na N = N-1 i=0 t i a 1 ,...,a N ≥0 t a 1 +...+Na N = 1 1 -t P N-1 (t) . (1)\n\n(3.26) Thus, using (2.28), (3.2) with (3.26), 7\n\nN (n,1) = n j=0 p N-1 (j) -N (n+1) = n j=0 p N-1 (j) -p N (n + 1) . (3.27)\n\nFor the case of N λ = N (n,2) we have that, from (3.18),\n\ns (2) (zz -1 ) = s (2) (z)s (2) (z -1 ) + s (1,1) (z)s (1,1) (z -1 ) , (3.28)\n\nso that, from (3.19), we have\n\nF (2) N (t) = λ ℓ(λ)≤N t |λ| s (2) s λ , s (2) s λ N + s (1,1) s λ , s (1,1) s λ N . (3.29) Using s (2) (z)s λ (z) = N r=1 s λ+2e r (z) + 1≤r<s≤N s λ+e r +e s (z) , s (1,1) (z)s λ (z) = 1≤r<s≤N s λ+e r +e s (z) , (3.30)\n\n7 This formula agrees with (3.6) due to\n\nN -1 j=0 p N (n -j) = n j=0 p N -1 (\n\nj) which follows because the corresponding generating functions match,\n\n∞ n=0 N -1 j=0 p N (n -j)t n = (1 + t + . . . + t N -1 )P N (t) = 1 1 -t P N -1 (t) = ∞ n=0 n j=0 p N -1 (j)t n .\n\nalong with (3.25) and\n\ns (λ 1 ,...,λ r-1 ,λ r +2,...,λ N ) (z) = -s (λ 1 ,...,λ r +1,λ r-1 +1,...,λ N ) (z) , (3.31)\n\nfor the cases where λ r-1 = λ r , we may obtain, with the definition (2.26), F\n\nN (t) = 1-t N+1 (1-t)(1-t 2 ) P N-1 (t) + 1 (1-t)(1-t 2 ) P N-2 (t) , (2)\n\nwhere the first contribution comes from λ t |λ| s (2) s λ , s (2) s λ N while the second comes from λ t |λ| s (1,1) s λ , s (1,1) s λ N . Since the partition number p k (-n) = 0 for n = 1, 2, . . . we may write, using (2.28), 8\n\n1 (1 -t)(1 -t 2 ) P k (t) = ∞ n,i,j=0 p k (n -i -2j)t n . (3.33)\n\nThus, from (3.32) with (3.27),\n\nN (n,2) = - n+1 j=0 p N-1 (j) + ∞ i,j=0 (p N-2 (n-i-2j) + p N-1 (n-i-2j)) if n ≤ N , ∞ i,j=0 (p N-2 (n-i-2j) + p N-1 (n-i-2j) -p N-1 (N +1-n-i-2j)) if n ≥ N +1. (3.34)\n\nTables of the numbers (3.2), (3.27) and (3.34) are given in appendix B for some few cases of n, N . Notice from these tables that the numbers N (n,m) below the diagonal line N ≥ n + m for a given n are the same for all N . This is a general feature that derives from values of N (n,m) for N ≥ n + m, which numbers may be obtained from a corresponding generating function that is now constructed.\n\nUsing these techniques, we may provide a consistency check of (3.17), (3.26), (3.32) along with a general result for N (n,m) for high enough values of N , N ≥ m + n. This employs the orthogonality property of power symmetric polynomials p λ (z) (in the large N 8 This is a special case of the following: for any f (n), n ∈ Z, that satisfies f (-n) = 0, n = 1, 2, . . ., then we may (at least formally) write\n\nP k (t) ∞ n=0 f (n)t n = ∞ n,i 1 ,...,i k =0 f (n -i 1 -2i 2 -. . . -ki k )t n . limit) along with s (n) (z) = λ⊢n 1 z λ p λ (z) = ∞ i 1 ,...,i n =0 1 i 1 !i 2 ! • • • i n ! δ i 1 +2i 2 +...+ni n ,n p 1 (z) i 1 1 2 p 2 (z) i 2 • • • 1 n p n (z) i n . (3.35)\n\nUsing the trivial identity p λ (xy) = p λ (x)p λ (y) then from (2.6), (3.19) with (3.35) we have\n\nF (m) ∞ (t) = ∞ n=0 λ⊢n µ⊢m 1 z λ z µ p λ p µ , p λ p µ ∞ t n = ∞ n=0 λ⊢n µ⊢m 1 z λ z µ p ν , p ν ∞ t n = ∞ n=0 λ⊢n µ⊢m z ν z λ z µ t n , (3.36)\n\nwhere for (1 a 1 , 2 a 2 , . . .) being the frequency representation of λ and (1 b 1 , 2 b 2 , . . .) being that of µ then ν has frequency representation (1 a 1 +a 2 , 2 a 2 +b 2 , . . .) so that |ν| = n + m. This agrees with F\n\nN (t) in a series expansion up to O(t N-m ) (since the last equation in (3.36) is also valid for finite N so long as |ν| = n + m ≤ N , by a result of appendix A). Now, since\n\nz ν z λ z µ = ∞ j=1 (a j +b j )! a j !b j ! , (3.37)\n\nwe obtain from (3.36) that,\n\nF (m) ∞ (t) = ∞ n=0 ∞ a 1 ,...,a n =0 ∞ b 1 ,...,b m =0 δ a 1 +•••+na n ,n δ b 1 +•••+mb m ,m m j=1 (a j + b j )! a j !b j ! t n = ∞ b 1 ,...,b m =0 δ b 1 +•••+mb m ,m m j=1 ∞ a j =0 (a j + b j )! a j !b j ! t ja j j>m 1 1 -t j = ∞ b 1 ,...,b m =0 δ b 1 +•••+mb m ,m m j=1 1 (1 -t j ) b j +1 j>m 1 1 -t j . (3.38)\n\nFor the first few cases we have that, with P N (t) as defined in (2.26),\n\nF (m) ∞ (t) = P ∞ (t) for m = 0 1 1-t P ∞ (t) for m = 1 2 (1-t)(1-t 2 ) P ∞ (t) for m = 2 , (3.39)\n\nwhose series expansion agrees with (3.17), (3.26), (3.32) up to O(t N-m ) for, respectively, m = 0, 1, 2. We may use (3.20) with (3.36) to determine N (n,m) exactly for N ≥ n + m.\n\nAsymptotic Counting of Quarter BPS Operators at Large N Asymptotic counting for the one boson case in the large N limit, for which, with P N (t) as defined in (2.26), with p(n) being the total number of (unordered) partitions of n,\n\nZ U(∞) (t) = P ∞ (t) = ∞ n=1 p(n)t n , (3.40)\n\nis the multi-trace partition function, entails finding an asymptotic value for the partition number p(n) for 'large' n. This may be achieved by performing a saddle point approxima-\n\ntion of p(n) = 1 2πi dtP ∞ (t)t -n-1 .\n\nThe function P ∞ (t) has a 'large' singularity at t = 1, but in addition has singularities at all other roots of unity -see [45] on the validity of ignoring these contributions asymptotically. This method was used by Hardy and Ramanujan to find their celebrated formula, here given in a less detailed form as,\n\np(n) ∼ 1 4n √ 3 exp π 2 3 n , (3.41)\n\nwhich was improved by Rademacher to give p(n) exactly. Their method relied crucially on the modular properties of P ∞ (t).\n\nFocusing now on the two bosonic fundamental field case for which, in the large N limit, (3.20) with (3.38) gives exact counting, at issue is first finding asymptotic values for the numbers\n\nQ(n, m, b) = Q(n, m, b 1 , . . . , b m ), with constraint equation m j=1 jb j = m, defined by m j=1 1 (1 -t j ) b j +1 j>m 1 1 -t j = 1 + ∞ n=1 Q(n, m, b)t n . (3.42)\n\nHaving found these we may then attempt to find the dominant contribution to (3.20) with (3.38) for large N . In order to give asymptotic values for Q(n, m, b) we may follow [46] and apply a formula due to Meinardus which gives a general result for the generating function\n\n∞ n=1 (1 -t n ) -a n = 1 + ∞ n=1 r(n)t n . (3.43)\n\nA detailed version of Meinardus' theorem may be found in [46] but for purposes of brevity we may note that it implies that, as n → ∞,\n\nr(n) ∼ C n κ exp AΓ(α + 1)ζ(α + 1)n α 1/(α+1) (α + 1)/α , (3.44)\n\nwhere ζ(s) = ∞ j=1 j -s is the Riemann zeta function and the constants C, κ, α, A are determined by the auxiliary Dirichlet series,\n\nD(s) = ∞ j=1 a j j s , (3.45)\n\nwhich must converge for Re(s) > α, a positive real number, and possess an analytic continuation in the region Re(s) ≥ c, -1 < c < 0, such that, in this region, D(s) is analytic except at a simple pole at s = α where it has residue A. In terms of α, A then\n\nC = 1 2π(1+α) AΓ(α + 1)ζ(α + 1) (1-2D(0))/2(α+1) exp D ′ (0) , κ = (D(0) -1 -1 2 α)/(α + 1) . (3.46)\n\nApplying Meinardus' theorem to the case of (3.42), clearly we have\n\nD(s) = m j=1 b j j s + ζ(s) , (3.47)\n\nso that, assuming m = m j=1 jb j is fixed, D(s) has a simple pole at s = α = 1 where it has residue A = 1. Using\n\nD(0) = -1 2 + m j=1 b j , exp D ′ (0) = 1 √ 2π m j=1 1 j b j , (3.48)\n\nthen, from (3.44) with (3.46), we may easily determine that, as n → ∞,\n\nQ(n, m, b) ∼ 1 4n √ 3 √ 6n π m j=1 b j m j=1 1 j b j exp π 2 3 n . (3.49)\n\nThis reduces to (3.41) when b j = 0, 1 ≤ j ≤ m, whereby Q(n, 0, . . . , 0) = p(n). Using (3.20) for (3.38) with (3.42) and (3.49) then, as n → ∞,\n\nN (n,m) = b 1 ,...,b m ≥0 m j=1 jb j =m Q(n, m, b) - b 1 ,...,b m-1 ≥0 m-1 j=1 jb j =m-1 Q(n+1, m-1, b) ∼ 1 4n √ 3 √ 6n π m exp π 2 3 n , (3.50)\n\nsince Q(n, m, m, 0, . . . , 0), for b 1 = m, b j = 0, j > 1, dominates over all other terms in (3.50) . This gives asymptotic values for the numbers in (1.1), for counting quarter BPS operators, transforming in [m, n-m, m] SU (4) R representations, in the large N limit of free N = 4 super Yang Mills, as previously described.\n\nFor the purposes of counting operators in the chiral ring of N = 4 super Yang Mills, we denote corresponding multi-trace partition functions by C U(N) (t).\n\nThe generating function for C U(N) (t) for the case of one bosonic fundamental field has been written in the form [1, 3, 9 ]\n\nC(ν, t) = ∞ n=0 1 1 -νt n = ∞ N=0 ν N C U(N) (t) , (4.1)\n\nso that ν acts as a chemical potential for the rank of the gauge group U (N ). The equivalence C U(N) (t) = Z U(N) (t) = P N (t), with Z U(N) (t) as in (2.25) , is actually a special case of the q-Binomial theorem. Writing -see [43] for notation -\n\n(a; q) k = (1 -a)(1 -aq) • • • (1 -aq k-1 ) , (4.2)\n\nthen the q-Binomial theorem is, for\n\n|x|, |q| < 1, ∞ k=0 (a; q) k (q; q) k x k = (ax; q) ∞ (x; q) ∞ . (4.3)\n\n(Identifying ν = x and q = t and setting a = 0 in (4.3), so that 1/(ν; t) ∞ = C(ν, t) above and 1/(t; t) N = P N (t) in (2.26), then C U(N) (t) = Z U(N) (t) = P N (t) straightforwardly. This special case of the q-Binomial theorem is due to Euler.)\n\nFor the two boson case, so that the single particle partition function is given by f (t, u) = t + u for some t, u, then the generating function for the finite N chiral ring partition function C U(N) (t, u) is given by [1, 3, 9]\n\nC(ν, t, u) = ∞ n,m=0 1 1 -νt n u m = ∞ N=0 ν N C U(N) (t, u) . (4.4)\n\nThis function is more difficult to analyse in terms of counting though has been investigated by Stanley [47] in relation to partitions -there it has been dubbed the 'double Eulerian' generating function. Through use of the Cauchy-Littlewood formula, then we may expand C U(N) (t, u) in terms of partitions of N as,\n\nC U(N) (t, u) = λ⊢N h λ (t)h λ (u) , (4.5)\n\nwhere h λ (t) = s λ (1, t, t 2 , . . .) , (4.6) so that using an identity for Schur polynomials to be found in [24, 47] then\n\nC U(N) (t, u) = λ⊢N 1≤i<j≤N (1 -t λ i -λ j +j-i )(1 -u λ i -λ j +j-i ) N i=1 (t; t) λ i +N -i (u; u) λ i +N -i (tu) N i=1 (i-\n\n1)λ i . (4.7) (4.5) with (4.6) has a natural interpretation in terms of plane partitions in that, for π being all column-strict plane partitions of shape λ, |π| = i,j π ij , 9 h λ (t) = s λ (1, t, t 2 , . . .) = π t |π| . (4.8) Obviously, (4.5) with (4.8) generalise for other chiral ring sectors. (For a different connection between the 'double Eulerian' generating function and major indices of permutations see [47], p. 385.) As an illustration of (4.5) with (4.8), we may consider the case N = 2 whereby λ = (2, 0), (1, 1) gives the two possible partitions of 2. For λ = (2, 0) (corresponding to a Young diagram with a single row of two boxes) π 11 ≥ π 12 ≥ 0 gives all column-strict plane partitions of shape (2, 0), while for λ = (1, 1) (corresponding to a Young diagram with a single column of two boxes) then π 11 > π 21 ≥ 0 gives all columnstrict plane partitions of shape (1, 1). Thus, h (2,0) (t) = π 11 ,π 12 ≥0 π 11 ≥π 12 t π 11 +π 12 = 1 (1t)(1t 2 ) , h (1,1) (t) = π 11 ,π 21 ≥0 π 11 >π 21 t π 11 +π 21 = t (1t)(1t 2 ) ,\n\nso that, from (4.5) for N = 2,\n\nC U(2) (t, u) = h (2,0) (t)h (2,0) (u) +h (1,1) (t)h (1,1) (u) = 1 + ut (1 -t)(1 -t 2 )(1 -u)(1 -u 2 ) , (4.10)\n\nwhich is the correct result as may be verified by extracting the ν 2 coefficient in an expansion of (4.4) up to O(ν 2 ).\n\n9 See [47] for a detailed description of plane partitions. Briefly, a column-strict plane partition of shape λ is an array π = (π ij ) of non-negative integers with finitely many non-zero entries, that is arranged in a Young tableaux with shape λ -see appendix A -such that the numbers π ij are weakly decreasing along each row, π ij ≥ π i j+1 ≥ 0, and strictly decreasing down each column, π ij > π i+1 j ≥ 0. The sum of the parts of π is given by |π| = i,j π ij . (Note that in contrast to the definition in [47] , here we are allowing π ij = 0, for some i, j, to be a part of the plane partition π with shape λ.)\n\nIn the large N limit,\n\nC U(∞) (t, u) = n 1 ,n 2 ≥0 n 1 +n 2 >0 1 1 -t n 1 u n 2 , (4.11)\n\nupon which attention is shortly focused.\n\nFor the numbers N (n,m) → N (n,m) counting quarter BPS primary operators for the chiral ring of N = 4 super Yang Mills, belonging to [m, nm, m] SU (4) R R-symmetry representations, as in (1.1), we have foot_1\n\nN (n,m) = 1 8π 2 dt du C U(N) (t, u) s (n,m) (t -1 , u -1 ) (t -1 -u -1 ) 2 . (4.12)\n\nThese may be more conveniently evaluated in terms of the numbers in (3.4) M (n,m) → M (n,m) , counting all chiral ring quarter BPS operators in the [m, n-m, m] SU (4) R representation, given by\n\nM (n,m) = 1 (2πi) 2 dt du C U(N) (t, u) t -n-1 u -m-1 , (4.13)\n\nso that N (n,m) = M (n,m) -M (n+1,m-1) . Defining P λ (n) to be the number of column-strict plane partitions π of shape λ so that |π| = i,j π ij = n, then, from (4.5) with (4.8) and (4.13), M (n,m) = λ⊢N P λ (n)P λ (m). Thus,\n\nN (n,m) = λ⊢N (P λ (n)P λ (m) -P λ (n+1)P λ (m-1)) , (4.14)\n\ncounts chiral ring quarter BPS primary operators in SU (4) R representations [m, nm, m] for any n, m at finite N .\n\nAsymptotic Counting for Chiral Ring BPS Operators at Large N For asymptotic counting of operators in the chiral ring of N = 4 super Yang Mills at large N , a relatively crude method is employed here which nevertheless captures the exponential behaviour of counting numbers of interest. This method is based on saddle point approximations of functions near a dominant singularity -see [45] for a useful summary. (Often for physical applications in thermodynamics, e.g. for entropy formulae, we are interested only in the exponential behaviour of such numbers anyhow.)\n\nTo illustrate, we consider the one boson case in the large N limit again. We first find a convenient 'approximating function' as follows,\n\nP ∞ (t) = ∞ n=1 1 1 -t n = exp - ∞ n=1 ln(1 -t n ) ∼ exp - ∞ 0 ds ln(1 -t s ) = exp - π 2 6 ln t , (4.15)\n\nwhich has an 'easier' singularity structure. (The approximation in the second step may be justified by the Euler-Maclaurin formula for approximating sums by integrals.) Using (4.15) then for large enough n,\n\np(n) = 1 2πi dt P ∞ (t)t -n-1 ∼ 1 2πi dt e g(t) , g(t) = - π 2 6 ln t -n ln t . (4.16)\n\nWe may approximate the latter integral for large n by noting that the dominant contribution is at the saddle point t ′ = e -π/ √ 6n ∼ 1 for which (4.18) Thus, ln p(n) ∼ π 2 3 n ,\n\ng(t ′ ) = π 2 3 n , g ′ (t ′ ) = 0 , g ′′ (t ′ ) = 2 π √ 6n 3 e π √ 2/3n = α , (4.17\n\nwhich captures the correct behaviour of ln p(n) for large n, according to (3.41).\n\nWe may proceed analogously for the quarter BPS chiral ring multi-trace partition function at large N , (4.11), which we approximate by\n\nC U(∞) (t, u) = n 1 ,n 2 ≥0 n 1 +n 2 >0 1 1 -t n 1 u n 2 = exp - n 1 ,n 2 ≥0 n 1 +n 2 >0 ln(1 -t n 1 u n 2 ) ∼ exp - ∞ 0 ∞ 0 dv dw ln(1 -t v u w ) = exp ζ(3) ln t ln u . (4.20)\n\nIn this case we have, from (4.13),\n\nM (n,m) ∼ 1 (2πi) 2\n\ndt du e g(t,u) , (\n\nwhere\n\ng(t, u) = ζ(3) ln t ln u -n ln t -m ln u , (4.22)\n\nfor n, m large. The dominant contribution to the integral, for n, m both large and of the same order, occurs about the point (t ′ , u ′ ) ∼ (1, 1) where\n\nt ′ = e -(ζ(3)mn -2 ) 1/3 , u ′ = e -(ζ(3)nm -2 ) 1/3 , (4.23)\n\nfor which\n\ng(t ′ , u ′ ) = 3 3 ζ(3)nm , ∂ ∂t g(t, u) (t ′ ,u ′ ) = ∂ ∂u g(t, u) (t ′ ,u ′ ) = 0 , ∂ 2 ∂t 2 g(t, u) (t ′ ,u ′ ) = 2(ζ(3) -1 n 5 m -1 ) 1 3 e 2(ζ(3)mn -2 ) 1/3 = α , ∂ 2 ∂u 2 g(t, u) (t ′ ,u ′ ) = 2(ζ(3) -1 m 5 n -1 ) 1 3 e 2(ζ(3)nm -2 ) 1/3 = β , ∂ 2 ∂t∂u g(t, u) (t ′ ,u ′ ) = (ζ(3) -1 n 2 m 2 ) 1 3 e (ζ(3)mn -2 ) 1/3 +(ζ(3)nm -2 ) 1/3 = γ . (4.24)\n\nSo long as m, n are both large and of the same order, the saddle point approximation is justified and we obtain\n\nM (n,m) ∼ e 3 3 √ ζ(3)nm 1 4π 2 ∞ -∞ ∞ -∞ dv dw e -1 2 (αv 2 +βw 2 +2γvw) = h(α, β, γ)e 3 3 √ ζ(3)nm , (4.25) where h(α, β, γ) = 1 2π αβ -γ 2 = 1 2π √ 3 (ζ(3)m -2 n -2 ) 1 3 e -(ζ(3)mn -2 ) 1/3 -(ζ(3)nm -2 ) 1/3 . (4.26)\n\nWe thus have that for n, m both comparably large,\n\nln M (n,m) ∼ 3 3 ζ(3)nm , (4.27) so that ln N (n,m) = ln M (n,m) -M (n+1,m-1) ∼ ln M (n,m) ∼ 3 3 ζ(3)nm . (4.28)\n\nIt is difficult to check the consistency of this result given the dearth of literature on these types of multi-variable generating functions and their asymptotic behaviour, however, we may consider the simpler function, also considered in [9] ,\n\nC U(∞) (t, t) = ∞ n=1 1 (1 -t n ) n+1 = ∞ r=0 E(r) t r , (4.29)\n\nwhere, in terms of the counting numbers N (n,m) , from (1.1), By considering (4.29) directly, we may employ Meinardus' theorem (described in the third section) to find the behaviour of ln E(r) as r → ∞. Note, however that Meinardus' theorem may not be applied directly to (4.29) since the corresponding auxiliary Dirichlet series (3.45), with a j = j+1, has two simple poles. To overcome this difficulty we split (4.29) into a product of two functions, both separately amenable to application of Meinardus' theorem. One is the reciprocal of the Euler function, P ∞ (t) in (4.15) . The other, the MacMahon function, is given by\n\nC U(∞) (t, t) = ∞ n=0 n m=0 (n-m+1) N (n,m) t n+m ⇒ E(r) = [ 1 2 r] m=0 (r-2m+1) N (r-m,m) . ( 4\n\nM (t) = ∞ n=1 1 (1 -t n ) n = ∞ r=0 q(r) t r , (4.32)\n\nand has been considered in a similar context as here in [3] . 11 Writing\n\nC U(∞) (t, t) = P ∞ (t)M (t) , (4.33)\n\nwith P ∞ (t) as in (4.15), so that, using (4.29),\n\nE(r) = r 1 ,r 2 ≥0 r 1 +r 2 =r p(r 1 )q(r 2 ) , (4.34)\n\nwe may find the asymptotic behaviour of ln E(r), as r → ∞, by extracting the dominant contribution from (4.34) using the asymptotic behaviour of p(r), q(r). The auxiliary Dirichlet series for M (t) in (4.32) is, from (3.45) with a j = j,\n\nD(s) = ζ(s -1) , (4.35)\n\n11\n\nThe relation of M (t) in (4.32) to plane partitions is given a description in [46] . Briefly, q(r) gives the number of ordinary plane partitions π, so that π ij ≥ π i+1 j > 0, π ij ≥ π i j+1 > 0, with |π| = i,j π ij = r. p(r) < q(r) as ordinary partitions λ are a special case of plane partitions.\n\nIn fact, the formula for ln q(r) found here is a special case of a more exact asymptotic formula first found by Wright [48] for the number of plane partitions q(r) of the number r.\n\nat the saddle point (t ′ 1 , . . . , t ′ k-1 ) ∼ (1, . . . , 1), where (ln t ′ 1 , . . . , ln t ′ k-1 ) = -k ζ(k) m 1 • • • m k-1 (1/m 1 , . . . , 1/m k-1 ) , (4.41) so that ∂ ∂t j g(t 1 , . . . , t k-1 ) (t ′ 1 ,...,t ′ k-1 ) = 0 , j = 1, . . . , k-1 . (4.42) (4.39) is consistent with a result implied by Meinardus' theorem. The function, 13 C U(∞) (t, . . . , t) ∼ j=1 m j = r, for which the solution is m j = m M = r/(k -1) (relaxing the constraint that m j be non-negative integers, which is irrelevant asymptotically), so that ln M (m M ,...,m M ) ∼ ln c(k, r). This is applicable to counting multi-trace operators in the eighth BPS chiral ring sector for N = 4 super Yang Mills with fundamental fields Z, Y, X involving m 1 Z's, m 2 Y 's, m 3 X's. Expanding the corresponding partition function (4.37), with k = 4, in terms of Schur polynomials s (m 1 ,m 2 ,m 3 ) (t), m 1 ≥ m 2 ≥ m 3 ≥ 0, similar to (1.1), the expansion coefficients N (m 1 ,m 2 ,m 3 ) count spinless multi-trace primary operators transforming in the [m 2 + m 3 , m 1m 2 , m 2m 3 ] SU (4) R R-symmetry representation, with conformal dimensions m 1 +m 2 +m 3 [33] . Just as in (4.28), asymptotically ln N (m 1 ,m 2 ,m 3 ) ∼ ln M (m 1 ,m 2 ,m 3 ) . This counting, however, ignores contributions of the fermionic fields λ, λ, which it may be important to include in order to give correct counting of eighth BPS chiral ring operators. 13 This may be easily seen from (4.37), as the number of solutions to k-1 j=1 m j = n, where m j are non-negative integers, is the binomial number n+k-2 k-2 which, to leading order in large n, behaves like n k-2 /(k-2)!. More properly, we should split the product C U(∞) (t, . . . , t) into pieces separately amenable to Meinardus' theorem, as for the prior case for k = 3, however, just as for that case, the numbers c(k, r) dominate, and so other contributions are ignored here.\n\nThere are some obvious questions not answered by this work. The first is whether or not the approach in the second section using symmetric polynomials can give insight into thermodynamics at finite N , such as for the Hagedorn transition, for example. While it gives the large N expression (2.14) in an elementary way, its wider applicability or usefulness to such questions is unclear. The approach is undoubtedly useful for finding exact expressions for counting numbers (as in (3.2), (3.27) and (3.34) for quarter BPS operators) and (3.9), (3.16 ) may be useful for analysing counting for more complicated sectors of N = 4 super Yang Mills, with gauge group U (N ).\n\nThe second question is how the arguments employing symmetric polynomial techniques here may be extended to other gauge groups, the most pertinent being perhaps SU (N ). Arguments here employing (2.6) and the orthonormality property of Schur polynomials should remain largely unaffected for SU (N ). Exact values for counting numbers obtained here should require some modification for SU (N ), though asymptotic values may be unchanged.\n\nThe third question concerns asymptotic values for counting numbers and how these may be improved. The asymptotic counting formulae given in such papers as [3, 9] for chiral ring sectors are special cases of formulae such as those of Hardy and Ramanujan, Meinardus, etc., all of which derive from single variable generating functions. It is hoped that the expressions (3.50), (4.28), (4.39), given here for asymptotic counting of BPS operators, that distinguishes between differing R-symmetry charges, represents a serious attempt at going beyond consideration of single variable generating functions. 14 Improving upon these formulae will require more sophisticated techniques, perhaps along the lines used to find those of Hardy and Ramanujan or Meinardus and employing any modular properties of the multi-variable functions involved. This issue may also be important for microscopic counting for Black Holes, as the BPS solutions found thus far, for N = 4 superconformal symmetry, depend on special values of R-symmetry charges [17, 18, 19, 20] -see [50] for a related detailed discussion.\n\nThus far, the elegant results for finite N partition functions for chiral ring sectors have been interpreted from a largely geometric perspective -it may be interesting to investigate more how such results are related to the theory of random matrices and/or symmetric polynomials."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered. Higher order corrections for finite N are also discussed in terms of symmetric group characters. For finite N, the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free N = 4 super Yang Mills discussed, including a general result for large N. The partition function for quarter BPS operators in the chiral ring of N = 4 super Yang Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free N = 4 super Yang Mills and in the chiral ring. Also, general and explicit expressions are derived for SU(2) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory."},{"section_type":"BACKGROUND","section_title":"Introduction","text":"For the past while there has been intense interest in finite N partition functions for Yang Mills theories, especially in super-symmetric ones, particularly with regard to their construction for BPS states and the counting thereof [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] . Much attention has been devoted to this issue for N = 4 super Yang Mills theory, it being by now the archetypal example of a conformal field theory for which we have a dual description in terms of string theory, by means of the AdS/CFT conjecture [16] . An example is in the assiduous efforts that have been made to explain the entropy of certain BPS black holes in AdS 5 ×S 5 [17, 18, 19, 20] in terms of microscopic counting of dual operators in N = 4 super Yang Mills, with gauge group SU (N ), by means of partition functions [1] . While this problem remains unsolved to date, essentially due to the difficulty of defining what is meant by these dual operators for finite N , there are other equally interesting sectors of (super) Yang Mills theories where more progress with counting has been made, examples being in free N = 4 super Yang Mills and in chiral ring sectors involving BPS operators.\n\nFor N = 4 super Yang Mills, the half BPS sector consists of multi-trace operators involving a single bosonic operator, Z. Similarly, the quarter BPS sector consists of multitrace operators involving two bosonic operators Z, Y while the eighth consists of ones involving three bosonic operators Z, Y, X and two fermionic operators λ, λ. (All these operators are here assumed to belong to the Lie algebra of U (N ).)\n\nFor the chiral ring, the commuting/anti-commuting of these operators is at the heart of why we can write very concise and elegant generating functions for the finite N multitrace partition functions [1] . Analysis of these partition functions, in terms of the counting of operators, has become a sophisticated industry where such approaches as the so-called 'Plethystic Program' have provided substantial results [3, 9] . This paper is devoted largely to the issue of partition functions for free field theory and particularly to the counting of gauge invariant multi-trace operators for the case of two bosonic fundamental fields. This is of relevance to the quarter BPS sector of N = 4 super Yang Mills in the free field limit when the operators Z, Y do not commute, in contrast to the chiral ring when they do.\n\nThe partition function for a free, massless quark-gluon gas was computed long ago [21] . This involved taking particle statistics into account using coherent state techniques and then imposing the gauge singlet condition by integrating over the relevant gauge group. With some modifications to the expression thus derived we may write the multi-trace partition function for some generic bosonic/fermionic fundamental fields in terms of an integral over the gauge group, involving the single particle partition function [22, 23] . This is the starting point here.\n\nFor U (N ) gauge theories we may easily write down the integral, though, even for this case, its evaluation is far from simple. One approach (which we adapt here for the SU (2) case) is to rewrite the expression in terms of an N -fold contour integral, whereby it may in principle be evaluated by summing the contributions from poles inside origin centred unit discs, in each of the N complex planes -similar techniques have been used in [10] . Due to the number of poles this becomes unfeasible for higher values of N . Another approach is to use the fact that the complex integral that interests us provides for an inner product for symmetric polynomials -see Macdonald [24] , pp. 363 -372, for a related discussion. Taking this point more seriously reveals an alternative route to evaluating the free field theory partition function which exposes not only the large N case in an almost trivial way, but also how and where this differs from the finite N case. This treatment also reveals an alternative interpretation of the free field partition function at finite N -it is related to a gauge group average of the cycle polynomial for the symmetric permutation group (after a certain identification of 'letters' with gauge group valued variables). This point is not dwelt on further here though makes the connection between the partition function and Polya enumeration explicit. For single trace operators at large N this connection has already been made [22, 25] for N = 4 super Yang Mills, whereby the partition function for single trace operators is related to the cycle polynomial for the cyclic permutation group. 1 Another issue is how to use the expression for the free field partition function to give explicit counting of gauge invariant multi-trace operators in a Yang-Mills theory, with gauge group U (N ). The case for one bosonic fundamental field has been widely discussed and, for finite N , the operators are counted by partition numbers of non-negative integers into at most N parts (which is here denoted by p N (n) -no closed formula for these numbers for arbitrary N, n exist, though they have a 'nice' generating function). Here, this result is re-derived, from a symmetric function perspective, by employing the well known Cauchy-Littlewood formula.\n\nTo proceed further with counting, for the quarter BPS sector of N = 4 super Yang Mills, for instance, character methods prove to be both natural and indispensable. Characters in relation to conformal field theories prove to be very convenient for encoding the allowable representations [15, 29, 30, 31, 32, 33] and for studying related partition functions, [34, 35, 36, 33] . For N = 4 super Yang Mills, it was shown in [33] that, if we are to distinguish among primary operators with differing conformal dimensions, spins and R-symmetry charges, such counting is most easily achieved using reductions of the full N = 4 superconformal characters, in certain limits that isolate corresponding sectors of short/semi-short operators. (One such limit corresponds to the index constructed in [1] .) This point may be easily illustrated for quarter BPS primary operators in N = 4 super Yang Mills, the case of two bosonic fundamental fields here. (See [37] for an explicit construction and counting of quarter BPS operators.) For N = 4 super Yang Mills, the counting of quarter BPS primary operators is complicated by the fact that, if we are to keep track of differing R-symmetry representations, any partition function restricted to this sector must be expanded in terms of U (2) characters (or two-variable Schur polynomials). Denoting some partition function restricted to this sector by Z(t, u), where t, u are letters corresponding to the fields Z, Y , then by expanding\n\nZ(t, u) = ∞ n=0 n m=0 N (n,m) s (n,m) (t, u) , s (n,m) (t, u) = t n+1 u m -t m u n+1 t -u , (1.1)\n\nin terms of two-variable Schur polynomials s (n,m) (t, u), we obtain the numbers N (n,m) of gauge invariant quarter BPS operators belonging to the [m, n-m, m] SU (4) R R-symmetry representation and having conformal dimension n + m (so that they are superconformal primary highest weight states in the corresponding quarter BPS supermultiplets). (The case m = 0 counts gauge invariant half BPS primary operators.)\n\nHere, the free field partition function is thus expanded in terms of Schur polynomials, depending on the same variables as the one particle partition function, the two boson case being a specialisation. This is quite naturally achieved using the Cauchy-Littlewood formula (and, if we include fermions, another formula due to Littlewood). Generally, we may obtain a result that relates the counting numbers to a sum over Kronecker coefficients. These arise naturally in the theory of the symmetric permutation group, though remain somewhat mysterious from a combinatorial perspective.\n\nSpecialising to the two boson case, a recursive procedure is employed here for the counting of multi-trace quarter BPS operators in free field theory at finite N . This issue was given considerable discussion for the large N case in [33] -here the results of [33] are generalised in terms of a generating function that may be employed to count quarter BPS operators for any R-symmetry charges, at large N . Asymptotic counting is also addressed in the latter case for the numbers N (n,m) in (1.1) for large n and fixed m.\n\nTo complete the discussion, counting of quarter BPS operators in the chiral ring of N = 4 super Yang Mills is investigated in terms of expanding over U (2) characters as in (1.1) . An explicit formula is given for the corresponding finite N partition function, with a short combinatorial interpretation given in terms of plane partitions, and specialised to large N . For the latter case, the exponential behaviour of the numbers N (n,m) in (1.1) is found for n and m both comparably large. This behaviour is consistent with a special case addressed in [3, 9] . By way of completion, a similar discussion for an arbitrary number of bosonic fundamental fields in the chiral ring is included. Two appendices are included; the first establishes some notation used for partitions and gives some standard results for the symmetric group and symmetric polynomials, the second gives some tables of numbers of quarter BPS operators in free N = 4 super Yang Mills, with gauge group U (N ), for which explicit formulae are given in the main text. Footnotes contain further details and points of clarification."},{"section_type":"OTHER","section_title":"Free Field Partition Functions","text":"We start from the single particle partition function which is here denoted by f (t) for some variables 2 t = (t 1 , t 2 . . .). The general form of f (t) is\n\nf (t) = i a i t i , (2.1)\n\nwhere each t i is a letter corresponding to a fundamental field and a i are signs, being +1 for a bosonic field, or -1 for a fermionic field.\n\nFor compact gauge Lie group G, the multi-trace partition function is then given by [21] , (see [22, 23] for refinements,)\n\nZ G (t) = G dµ G (g) exp ∞ n=1 1 n f (t n )χ R (g n ) , (2.2)\n\ninvolving the Haar (or G-invariant, or Hurwitz) measure dµ G (g) for g ∈ G (so that\n\nG dµ G (g)F (g) = G dµ G (g)F (gh) = G dµ G (g)F (hg) for all h ∈ G and G dµ G (g) = 1)\n\nand where χ R (g) is the character for the R representation of G, assuming that the fundamental fields transform in identical gauge group representations R.\n\nFor G = U (N ), so that for any matrix U ∈ U (N ) we may write U = V ΘV † , where V is a unitary matrix and Θ = diag.(e iθ 1 , . . . , e iθ N ), 0 ≤ θ i < 2π, and for some F (U ) = F (Θ), independent of V , then we may write\n\nU(N) dµ U(N) (U )F (U ) = 1 (2π) N N ! 2π 0 N j=1 dθ j 1≤k<l≤N |e iθ k -e iθ l | 2 F (Θ) , (2.3)\n\n2 In what follows roman letters are used to denote a collection of variables and, for x = (x 1 , . . . , x i ), y = (y 1 , . . . , y j ) for example, the shorthand x α is used to mean (x 1 α , . . . , x i α ) and z = xy to mean z = (z 11 , . . . , z ij ) where z rs = x r y s . The latter convenient notation has been used by Macdonald [24] .\n\nwhich is, of course, related to the Weyl parametrisation of U (N ). Thus, for such F (U ), the left-hand side of (2.3) simplifies to an integral over the N torus. Of course, as χ R (U ) generally depends on linear combinations of tr(U j ) k tr(U †l ) m for various non-negative integers j, k, l, m then any function of χ R (U ) is an example of such an F (U ).\n\nWe are interested in the case where R = Adj. is the adjoint representation so that for U (N ) we have that χ Adj. (U ) = tr U tr U † (while for SU (N ) then χ Adj. (U ) = tr U tr U † -1). For U (N ) we then find that, using (2.2) with (2.3),\n\nZ U(N) (t) = 1 (2π) N N ! 2π 0 N j=1 dθ j 1≤k<l≤N |e iθ k -e iθ l | 2 exp ∞ n=1 1 n f (t n ) N j,k=1\n\ne in(θ j -θ k ) .\n\n(2.4)\n\nWe may write (2.4) as an N -fold contour integral by first making the variable change z j = e iθ j so that the integrals in (2.4) are around unit circles in each z j complex plane and then we obtain\n\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) exp ∞ n=1 1 n f (t n )p n (z)p n (z -1 ) , (2.5)\n\nwhere ∆(z) = 1≤i<j≤N (z iz j ) is the Vandermonde determinant and p n (z) = N i=1 z i n is a power symmetric polynomial -see appendix A for a brief discussion of symmetric polynomials. This integral may then in principle be evaluated by deforming the contours so as to extract the residues at poles within the discs\n\n|z j | < 1, 1 ≤ j ≤ N .\n\nA crucial observation is that, for some N variable symmetric polynomials g(z), h(z), then\n\ng, h N = h, g N = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 )g(z)h(z -1 ) , (2.6)\n\nacts as an inner product -this is easy to see in terms of Schur polynomials which provide an orthonormal basis for symmetric polynomials. The reader may now wish to peruse appendix A where notation regarding partitions and a short discussion of symmetric polynomials is included.\n\nThe General and Large N Cases for U (N )\n\nFor application of inner products to (2.5) we have that, in terms of power symmetric polynomials p λ (z) for partitions λ,\n\nexp ∞ n=1 1 n f (t n )p n (z)p n (z -1 ) = ∞ n=1 ∞ a n =0 1 n a n a n ! f (t n ) a n p n (z) a n p n (z -1 ) a n , = λ 1 z λ f λ (t)p λ (z)p λ (z -1 ) , (2.7)\n\nwith the definitions of\n\nz λ = ∞ n=1 n a n a n ! , f λ (t) = ∞ n=1 f (t n ) a n , (2.8)\n\nbeing in terms of the frequency representation of λ, (1 a 1 , 2 a 2 , . . .) with n≥1 n a n = |λ|, the weight of the partition λ (note that the frequency representation of λ is simply a convenient re-ordering of the parts of λ).\n\nIn (2.7) the numbers z λ have a standard combinatorial interpretation -for a given permutation σ ∈ S m with a 1 1-cycles, a 2 2-cycles etc., so that n≥1 n a n = m = |λ|, then z λ = n≥1 n a n a n ! is the size of the centraliser Z σ = {τ ; τ ∈ S m , τ στ -1 = σ} of σ ∈ S m . (This may be easily seen as under conjugation of σ by τ then τ can permute the cycles of length n among themselves in a n ! ways and/or render a cyclic rotation on each of the individual cycles in n a n ways.) More details of the symmetric group are to be found in appendix A.\n\nWe may immediately observe that (2.7) represents a sum over cycle polynomials of the symmetric group S m . This is given by, for letters\n\nu 1 , u 2 , . . . u m , 3 C m (u) = a 1 ,...,a m ≥0 δ a 1 +2a 2 +...+ma m ,m m n=1 1 n a n a n ! u n a n = λ⊢m 1 z λ u λ , (2.9)\n\nwhere 'λ ⊢ m' means that λ is any partition of m -see appendix A for notation -and\n\nu λ = m n=1 u n a n , (2.10)\n\nin terms of the frequency representation of λ above. Identifying u n = f (t n )p n (z)p n (z -1 ) then we may rewrite\n\nZ U(N) (t) = ∞ m=0 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 )C m (u) , (2.11)\n\n3 This formula is easy to see from the definition of the cycle polynomial for a subgroup G of S m . This is given by\n\n1 |G| g∈G u 1 j 1 (g) • • • u m j m (g) = 1 |G| K g |K g |u 1 j 1 (g) • • • u m j m (g) ,\n\nwhere j i (g) denotes the number of i cycles in the unique decomposition of g into disjoint cycles the sum of the U (N ) group averages of each of the cycle polynomials C m (u). (Physically, the interpretation is that the cycle index for the symmetric permutation group accounts for particle statistics while integration over the gauge group imposes the gauge singlet condition. For purposes of clarity, the U (N ) case has been focused upon here, though from the form of (2.2) it is easy to see how this generalises for other gauge groups whereby the letters u n = f (t n )χ R (g n ) for the fundamental fields transforming in identical gauge group representations, R.)\n\nDirectly from (2.7), in terms of the inner product (2.6), then\n\nZ U(N) (t) = λ 1 z λ f λ (t) p λ , p λ N = λ 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 , (2.12)\n\nwhere on the right-hand side of (2.12) we have used an expression for the inner product of two power symmetric polynomials expressed in terms of the characters of the symmetric group, given in appendix A. (Here 'ℓ(µ)' means the number of non-zero parts of the partition µ.)\n\nUsing a result of appendix A (essentially orthogonality relations for symmetric group characters), (2.12) may be rewritten as\n\nZ U(N) (t) = λ |λ|≤N f λ (t) + λ |λ|>N 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\n\n(2.13)\n\nIn the large N limit, Z U(N) (t) simplifies considerably as only the first term in (2.13) need be considered. Using the frequency representation of λ then\n\nZ U(∞) (t) = λ f λ (t) = ∞ n=1 ∞ a n =0 f (t n ) a n = ∞ n=1 1 1 -f (t n ) , (2.14)\n\na result which has been obtained using Polya counting methods for single trace operators and saddle point approximations [22, 23] .\n\nHigher order corrections in |λ|, the weight of the partition λ, to (2.13) may be obtained by successive evaluation of\n\nµ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\n\nOne method is to employ the Murnaghan-Nakayama Rule, used to compute χ µ λ using skew hooks and Young diagrams. (A readable account of the Murnaghan-Nakayama Rule may be found in [38] , though of course it is explained in many standard textbooks that discuss the symmetric group.)\n\nFor the case of |λ| = N +1 then we may observe that,\n\nµ⊢N +1 ℓ(µ)≤N χ µ λ 2 = µ⊢N +1 ℓ(µ)≤N +1 χ µ λ 2 -(χ ν λ ) 2 = z λ -(χ ν λ ) 2 , ν = (1 N+1 ) , (2.15)\n\nsince the partition ν = (1 N+1 ) is the only one excluded among those partitions µ of N +1 with ℓ(µ) ≤ N . By applying the Murnaghan-Nakayama Rule we may determine, for\n\nν = (1 L ), (χ ν λ ) 2 = 1 for |λ| = L 0 otherwise , (2.16)\n\nsince χ ν λ in this case is just a sign. (This may be easily seen as there is only one possible way to remove successive skew hooks, which in this case are just column Young diagrams of length λ i , from the (1 L ) column Young diagram to leave one of normal shape, in this case, another column Young diagram.) Thus, using (2.15) with (2.16) in (2.13), we obtain\n\nZ U(N) (t) = λ |λ|≤N +1 f λ (t) - λ |λ|=N +1 1 z λ f λ (t) + λ |λ|>N +1 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\n\n(2.17) By a similar line of argument we may do the same for the case of |λ| = N +2. We have that, for\n\nν 1 = (1 N+2 ) and ν 2 = (2, 1 N+1 ), µ⊢N +2 ℓ(µ)≤N χ µ λ 2 = µ⊢N +2 ℓ(µ)≤N +2 χ µ λ 2 -(χ ν 1 λ ) 2 -(χ ν 2 λ ) 2 = z λ -(χ ν 1 λ ) 2 -(χ ν 2 λ ) 2 . (2.18)\n\nWe may determine, for ν = (2, 1 L ),\n\n(χ ν λ ) 2 = (a 1 -1) 2 for |λ| = n≥1 n a n = L+2 0 otherwise . ( 2\n\n.19) Using (2.16), (2.19) with (2.18) in (2.17) then we obtain\n\nZ U(N) (t) = λ |λ|≤N +2 f λ (t) - λ N +1≤|λ|≤N +2 1 z λ f λ (t) - λ |λ|=N +2 1 z ′ λ f λ (t) + λ |λ|>N +2 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 , (2.20)\n\nwhere, in the frequency representation of λ,\n\n1 z ′ λ = (a 1 -1) 2 z λ = 1 a 1 ! - 1 (a 1 -1)! + 1 (a 1 -2)! ∞ n=2 n a n a n ! .\n\n(2.21)\n\nWe may proceed in this manner to compute explicit higher order corrections though this becomes cumbersome save for the first few cases as shown. (For |λ| > N + 2 the corrections will always involve contributions from (2.16) and (2.19) as well as extra ones coming from\n\nµ⊢|λ| ℓ(µ)≤|λ| χ µ λ 2 - µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .)\n\nThe One Boson Case for U (N )\n\nFor the case of one bosonic fundamental field (applicable to half BPS operators for N = 4 super Yang Mills), we have f (t) = t in (2.5), so that we may write\n\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) N j,k=1 1 1 -tz j z k -1 . (2.22)\n\nTo evaluate this integral we may use the Cauchy-Littlewood formula,\n\nL i=1 M j=1 1 1 -x i y j = λ ℓ(λ)≤min.{L,M } s λ (x 1 , . . . , x L )s λ (y 1 , . . . , y M ) , (2.23)\n\nwhere the sum on the right-hand side is over all partitions λ such that the corresponding Young diagrams have no more than min.{L,\n\nM } rows, ℓ(λ) ≤ min.{L, M }. With x i = tz i , y i = z i -1 , i = 1, . . . , N , in (2.23\n\n), so that s λ (tz 1 , . . . , tz N ) = t |λ| s λ (z), and employing also (2.6 ) and the orthonormality of Schur polynomials, we may easily obtain,\n\nZ U(N) (t) = λ ℓ(λ)≤N t |λ| s λ , s λ N = λ ℓ(λ)≤N t |λ| . (2.24)\n\nBy changing summation variables so that\n\nλ i -λ i+1 = a i , i = 1, . . . , N -1, λ N = a N then we may write Z U(N) (t) = ∞ a 1 ,...,a N =0 t a 1 +2a 2 +...+Na N = P N (t) , (2.25)\n\nwhere 4\n\nP N (t) = N i=1 1 1 -t i . (2.26)\n\nOf course this is nothing other than the generating function for the number p N (n) of partitions of n into no more than N parts since, by definition,\n\nλ ℓ(λ)≤N δ |λ|,n = p N (n) , (2.27)\n\nso that by the above\n\n∞ n=0 p N (n)t n = P N (t) .\n\n(2.28)\n\n4 1/P ∞ (t) = ∞ n=1 (1-t n ) is commonly called the Euler function, denoted by Φ(t). P ∞ (t) = ∞ n=0 p(n)t n\n\nacts as a generating function for the number of unordered partitions of n, p(n). Note that p N (n) = p(n) for n ≤ N , i.e. the number of partitions of n into no more than N parts is the same as the total number of partitions of n so long as n ≤ N .\n\nThis makes explicit the connection between Z U(N) (t) and the partition numbers p N (n)."},{"section_type":"OTHER","section_title":"The SU (2) Gauge Group Case","text":"Here we first consider f (t) = k j=1 t j in (2.1) so that the variables 0 ≤ t i < 1 represent k bosons in the single particle partition function. For such fields transforming in the adjoint representation of SU (2) then (2.2) simplifies significantly. For any U ∈ SU (2) we may write U = V ΘV † , where V is unitary and Θ = diag.(e iθ , e -iθ ), for 0 ≤ θ < 2π, so that for F (U ) = F (θ), then in usual Weyl parametrisation,\n\nSU(2) dµ SU(2) (U )F (U ) = 1 π 2π 0 dθ sin 2 θ F (θ) = 1 4π 4π 0 dθ 1 -cos θ F ( θ 2 ) = 1 2π 2π 0 dθ 1 -cos θ F ( θ 2 ) , (2.29)\n\nwhere F (θ) = F (θ + π) is assumed in writing the last line. In the present case,\n\nF (U ) = F (θ) = n≥1 f (t n )χ Adj. (U n )/n, where χ Adj. (U ) = tr(U )tr(U † ) -1 = e 2iθ + e -2iθ + 1 = 2 cos 2θ + 1, so that Z SU(2) (t) = 1 2π 2π 0 dθ (1 -cos θ) k j=1 1 (1 -t j )(1 -t j e iθ )(1 -t j e -iθ )\n\n.\n\n(2.30)\n\nMaking the variable change z = e iθ , and using that F (θ) = F (-θ) is even, then\n\nZ SU(2) (t) = 1 2πi dz z (1 -z) k j=1 1 (1 -t j )(1 -t j z)(1 -t j z -1 ) , (2.31)\n\nwhere the integral is around the unit circle |z| = 1. The residues in (2.31) may be easily computed since all the relevant (simple) poles in the disc |z| < 1 occur at the points z = t j . Thus\n\nZ SU(2) (t) = k i=1 t i k-1 1 -t i 2 k j=1 j =i 1 (t i -t j )(1 -t i t j )(1 -t j ) . (2.32)\n\nThis partition function has an interesting interpretation from a group theory perspective. We may write,\n\n1 + t (1 -tz)(1 -tz -1 ) = ∞ n=0 χ n (z)t n , (2.33)\n\nwhere\n\nχ j (z) = z j+ 1 2 -z -j-1 2 z 1 2 -z -1 2 , j ∈ 1 2 Z , (2.34)\n\nis an SU (2) character, corresponding to the spin j irreducible representation, R j . Now the integral in (2.31) acts as an SU (2) inner product,\n\nχ j , χ k = 1 2πi dz z (1 -z)χ j (z)χ k (z -1 ) = δ jk , (2.35)\n\nfor j, k ∈ N. Thus, from (2.31) with (2.33) and (2.35),\n\nk j=1 (1 -t j 2 )Z SU(2) (t) = ∞ n 1 ,...,n k =0 χ n 1 • • • χ n k , 1 t 1 n 1 • • • t k n k , (2.36)\n\nacts as a generating function for the number of singlets in the decomposition of the 5 By using that χ n (z) = n j=-n z j we may use the Cauchy residue theorem to compute explicitly that\n\nSU (2) representation R n 1 ⊗ • • • ⊗ R n k .\n\nχ n 1 • • • χ n k , 1 = 2n 1 j 1 =0 • • • 2n k j k =0 (δ j 1 +•••+j k ,n 1 +•••+n k -δ j 1 +•••+j k ,n 1 +•••+n k +1 ) .\n\n(2.37)\n\nIf we modify the one particle partition function to include k bosons and k fermions and hence consider (2.1) in the form f (t, t) = k j=1 t j -k =1 t then we may similarly as above evaluate\n\nZ SU(2) (t, t) = 1 2πi dz z (1 -z) 1≤j≤k 1≤≤ k (1 -t )(1 -t z)(1 -t z -1 ) (1 -t j )(1 -t j z)(1 -t j z -1 ) , (2.38)\n\nwhere the contour is around the unit disc |z| = 1. So long as k > k then (2.38) receives contributions from only those simple poles at z = t j so that for this case we obtain\n\nZ SU(2) (t, t) k> k = k i=1 t i k-k-1 1 -t i 2 1≤≤ k 1≤j≤k,j =i (t i -t )(1 -t i t )(1 -t ) (t i -t j )(1 -t i t j )(1 -t j ) . (2.39)\n\nFor k ≤ k then (2.38) also receives contributions from poles at z = 0. For instance\n\nZ SU(2) (t, t) k= k = k i=1 1 t i (1 -t i 2 ) 1≤j,≤k j =i (t i -t )(1 -t i t )(1 -t ) (t i -t j )(1 -t i t j )(1 -t j ) + 1≤j,≤k t (1 -t ) t j (1 -t j ) , (2.40)\n\n5 Generating functions for products of Lie algebra representations have been considered elsewhere, in [39] for instance. A generating function for the number of singlets in n products of the fundamental times n products of the anti-fundamental representations for SU (N ) was found by Gessel [40] in terms of Toeplitz determinants involving Bessel functions. See also [41] for a nice physics oriented discussion of similar issues. The special case of R 1 2\n\n⊗ • • • ⊗ R 1 2\n\n(2n products of the fundamental) for SU (2), contains a Catalan number, 1 n+1 2n n , of singlet representations.\n\nwhere the last term on the right-hand side of (2.40) comes from the simple pole at z = 0. These formulae should be useful for computing the multi-trace partition functions, for fundamental fields transforming in an SU (2) gauge group, in other sectors of N = 4 super Yang Mills. For instance, after a suitable identification of the variables t j , t with variables in single particle partition functions for semi-short sectors of N = 4 super Yang Mills, described in detail in [33] , then (2.40) should allow for an explicit expression for corresponding multi-trace partition functions. They may also be useful for computing the N = 4 superconformal index of [1] for SU (2) gauge group, or at least for restrictions of it such as described in [33] or [42] ."},{"section_type":"OTHER","section_title":"Counting Operators in Free N = 4 Super Yang Mills","text":"In this section, the counting of half and quarter BPS operators for free N = 4 super Yang Mills, when the fundamental fields transform in the adjoint representation of U (N ), is discussed in some detail."},{"section_type":"OTHER","section_title":"Counting Operators Directly","text":"We may, of course, proceed to count multi-trace half and quarter BPS primary operators directly, in terms of the fundamental fields, Z, for half BPS operators and Z, Y , for quarter BPS operators.\n\n(Z, Y ) forms a U (2) doublet, where U (2) has generators given by a subset of the SU (4) R generators, H i , E i± , 1 ≤ i ≤ 3, where H i are the Cartan sub-algebra generators and E i± are ladder operators satisfying (in the Chevalley-Serre basis)\n\n[H i , E j± ] = ±K ij E j± , with [K ij ] being the usual SU (4) Cartan matrix. The U (2) generators consist of the SU (2) generators H 2 , E 2± , where explicitly [(H 1 , H 2 , H 3 ), E 2± ] = ∓(1, -2, 1)E 2± , along with the generator H 1 +H 2 +H 3 , whose eigenvalues give the conformal dimensions in this case, [H 1 +H 2 +H 3 , (Z, Y )] = (Z, Y ) [33]. Explicitly, we have that [E 2+ , Z] = 0, [E 2-, Z] = Y , [(H 1 , H 2 , H 3 ), Z] = (0, 1, 0)Z, [(H 1 , H 2 , H 3 ), Y ] = (1, -1, 1)Y so that an operator involving n Z's and m Y 's transforms in the [m, n-m, m] SU (4) R R-symmetry representation.\n\nFor k-trace half BPS primary operators transforming in the [0, n, 0] SU (4) R Rsymmetry representation, with conformal dimension n, then in terms of the fundamental field Z a basis is provided by,\n\ntr(Z n 1 ) • • • tr(Z n k ) , k i=1 n i = n . (3.1)\n\nWe have that, due to trace identities for finite N , tr(Z n ) for n > N is expressible in terms of a sum over multi-trace operators of the form (3.1), for k > 1, and thus, a minimal basis for multi-trace half BPS primary operators consists of (3.1) for all 1 ≤ k ≤ n and with every n i ≤ N , ordered so that n 1 ≥ n 2 ≥ . . . ≥ n k ≥ 0, i.e. so that (n 1 , . . . , n k ) is a partition of n where each part n i ≤ N . With this restriction, the number of multi-trace half BPS primary operators for a given n is\n\nN (n) = p N (n) , (3.2)\n\nsince the number p N (n), in (2.27), of partitions of n into ≤ N parts is the same as the number of partitions of n in which each part is ≤ N -see [43] for a simple proof employing generating functions.\n\nFor quarter BPS operators belonging to the [m, n-m, m] SU (4) R R-symmetry representation, a basis for k-trace operators is\n\ntr j Z n 1j Y m 1j • • • tr j Z n kj Y m kj , i,j n ij = n , i,j m ij = m , (3.3)\n\nwhere there is a choice of ordering in each trace. (Note that the m = 0 case corresponds to the half BPS case already considered.) Using the basis provided by (3.3) for all allowable k, then to avoid over-counting of multi-trace quarter BPS operators, the cyclicity of each trace and also trace identities for finite N must be accounted for. Assuming that this is done, let M (n,m) denote the number of elements in this minimal basis for multi-trace operators of the form (3. The number of such U (2) descendants coincides with N (n+m-j,j) , the number of corresponding primary operators. In this way, we obtain that\n\nM (n,m) = N (n,m) + N (n+1,m-1) + . . . + N (n+m-1,1) + N (n+m) , (3.4)\n\nso that N (n,m) = M (n,m) -M (n+1,m-1) may be obtained recursively for each m.\n\nWe may illustrate by counting all multi-trace quarter BPS primary operators in the [1, n-1, 1] R-symmetry representation. In this case a basis for k+1-trace operators is provided by\n\ntr(Z n 1 ) • • • tr(Z n k )tr(Z j Y ) , k i=1 n i = n -j . (3.5)\n\nCyclicity of traces implies that we may arrange Y as shown, to avoid over-counting. U (N ) trace identities imply, similarly as for the half BPS case, that a minimal basis for multitrace operators requires j < N and each n i ≤ N in (3.5) for every 1 ≤ k ≤ n-j, so that (n 1 , . . . , n k ) forms a partition of n-j, with every part ≤ N . Thus, by a similar argument as for the half BPS case, M (n,1) = N-1 j=0 p N (nj). Finally, to ensure that only primary operators are counted then we must subtract off contributions from descendants of half BPS primary operators in the [0, n+1, 0] SU (4) R representation, of which there are p N (n+1). Using (3.4) with (3.2) we then conclude that\n\nN (n,1) = N-1 j=0 p N (n -j) -p N (n+1) , (3.6)\n\ngives the number of multi-trace quarter BPS primary operators in the [1, n-1, 1] Rsymmetry representation.\n\nCounting in this fashion becomes more difficult for greater m and now a procedure is described employing symmetric polynomials to find a generating function for the numbers of multi-trace quarter BPS primary operators in the [m, n-m, m] SU (4) R representation, for m = 0, 1, 2 at finite N and for any n, m at large N . This generating function is subsequently used to provide asymptotic counting for fixed m, large n in the large N limit."},{"section_type":"OTHER","section_title":"Counting Operators via Expansion of Partition Functions in Schur Polynomials","text":"For k bosonic fundamental fields, we may take f (t) = k j=1 t j in (2.1) so that (2.5) may be written as\n\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) k j=1 N r,s=1 1 1 -t j z r z s -1 . (3.7)\n\nOften it is the case that such partition functions should be expanded in terms of s λ (t), the k variable Schur polynomial labelled by partitions λ. An example is provided by (1.1) for counting multi-trace quarter BPS operators. We may use the Cauchy-Littlewood formula (2.23) to expand in this way, to obtain\n\nZ U(N) (t) = λ ℓ(λ)≤min.{k,N 2 } N λ s λ (t) , (3.8)\n\nwhere\n\nN λ = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s λ (zz -1 ) , (3.9)\n\nwhere zz -1 has components z i z j -1 , 1 ≤ i, j ≤ N .\n\nFrom Macdonald [24] we have that\n\ns λ (xy) = µ,ν⊢|λ| γ λ µν s µ (x)s ν (y) , (3.10)\n\nin terms of Kronecker coefficients,\n\nγ λ µν = 1 |λ|! σ∈S |λ| χ λ (σ)χ µ (σ)χ ν (σ) = ρ⊢|λ| 1 z ρ χ λ ρ χ µ ρ χ ν ρ , (3.11)\n\nbeing a sum over irreducible S |λ| characters evaluated at σ ∈ S |λ| , related to a sum over irreducible S |λ| characters evaluated on the conjugacy classes labelled by the partitions ρ in the second line. Using (2.6) along with the orthonormality property of Schur polynomials we find that\n\nN λ = µ⊢|λ| ℓ(µ)≤N γ λ µµ .\n\n(3.12)\n\nThe situation becomes much more involved if we include also k fermionic fields, so that (2.1) may be written in the form f (t, t) = k j=1 t j -k =1 t , and attempt to expand Z U(N) (t, t) in terms of products of Schur polynomials s λ (t)s µ ( t). Such expansion is required for counting, for instance, for the free field partition function in the eighth BPS sector of N = 4 super Yang Mills. In this case the partition function is expanded, analogous to (1.1), in terms of SU (2|3) characters, which may be expressed in terms of a linear combination of products of two-variable and three-variable Schur polynomials. (See [33] for a discussion of counting for the eighth BPS sector along these lines.) Including fermions, (3.7) becomes modified by\n\nZ U(N) (t, t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) 1≤j≤k 1≤≤ k N r,s=1 1 -t z r z s -1 1 -t j z r z s -1 . (3.13)\n\nTo achieve the expansion, we may use the Cauchy-Littlewood formula (2.23) along with another formula of Littlewood,\n\nL i=1 M j=1 (1 + x i y j ) = λ ℓ(λ)≤L,ℓ( λ)≤M s λ (x 1 , . . . , x L ) s λ(y 1 , . . . , y M ) , (3.14)\n\nwhere λ is the partition conjugate to λ (where the rows and columns of the Young diagram corresponding to λ are interchanged) and where the sum is restricted to those λ whereby the corresponding Young diagrams have at most L rows, ℓ(λ) ≤ L, and M columns, ℓ( λ) ≤ M . We may thus write\n\nZ U(N) (t, t) = λ ℓ(λ)≤min.{k,N 2 } µ ℓ(µ)≤ k,ℓ( μ)≤N 2 N λ,µ s λ (t) s µ ( t) , (3.15)\n\nwhere\n\nN λ,µ = (-1) |µ| (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s λ (zz -1 )s μ(zz -1 ) . (3.16)\n\nObviously these numbers are considerably more involved than those in (3.9). We may of course use (3.10) again to interpret (3.16) in terms of Kronecker coefficients."},{"section_type":"METHOD","section_title":"Counting Quarter BPS Operators by Symmetric Polynomial Methods","text":"The two bosonic fundamental field case is now focused upon. 6 In particular, the numbers N (n,m) in (1.1) are evaluated using results of the last sub-section.\n\nWe may proceed to evaluate N λ recursively. The simplest case is for N λ = N (n) , whereby introducing a formal variable t then it is clear, by (2.23) with (2.22), (2.25) and (2.26), that ∞ n=0\n\nN (n) t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) N j,k=1 1 1 -tz j z k -1 = P N (t) , (3.17)\n\nso that, by (2.28), N (n) is given by (3.2).\n\nMore generally to evaluate N (n,m) from (3.9) we may use, for y = zz -1 ,\n\ns (m) (y)s (n) (y) = s (n,m) (y) + s (n+1,m-1) (y) + . . . + s (n+m-1,1) (y) + s (n+m) (y) , s (m) (zz -1 ) = µ⊢m ℓ(µ)≤N s µ (z)s µ (z -1 ) , (3.18)\n\n6 The two boson case leads to an interesting generalisation of an identity in [44] involving Littlewood-Richardson coefficients c ν λµ , the coefficients that appear in the decomposition s λ (x)s µ (x) = ν c ν λµ s ν (x). With f (t) in (2.1) given by f (t) = t 1 + t 2 , and expanding appropriately the corresponding integrand in (3.7) using (2.23); then using (2.6), the orthonormality of Schur polynomials and the result (2.14), we obtain (note that\n\nc ν λµ = 0 if |ν| = |λ| + |µ|) Z U(∞) (t 1 , t 2 ) = λ,µ t 1 |λ| t 2 |µ| s λ s µ , s λ s µ ∞ = λ,µ,ν ν⊢|λ|+|µ| t 1 |λ| t 2 |µ| (c ν λµ ) 2 = n≥1 1 1 -t 1 n -t 2 n\n\nwhich reduces to Theorem 4.1 of [44] if we take t 1 = t 2 = t.\n\nwhere the expression in the first line of (3.18) may be easily seen using Young tableaux multiplication rules while (2.23) determines the expression in the second line. From (3.9) with (3.18), we may find a useful generating function, in terms of a formal variable t, for the numbers in (3.4) as follows,\n\nF (m) N (t) = ∞ n=0 M (n,m) t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s (m) (zz -1 ) ∞ n=0 s (n) (zz -1 )t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) µ⊢m ℓ(µ)≤N s µ (z)s µ (z -1 ) N j,k=1 1 1 -tz j z k -1 = µ⊢m ℓ(µ)≤N λ t |λ| s λ s µ , s λ s µ N , (3.19)\n\nso that we may write\n\nN (n,m) = 1 2πi dt t 1 t n F (m) N (t) - 1 t n+1 F (m-1) N (t) , (3.20)\n\nwhich allows for recursive determination of N (n,m) .\n\nApplying this to the case of N λ = N (n,1) we have, from (3.18)\n\ns (1) (zz -1 ) = s (1) (z)s (1) (z -1 ) , (3.21) so that, from (3.19), F (1)\n\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = λ ℓ(λ)≤N t |λ| s (1) s λ , s (1) s λ N . (3.22)\n\nUsing (again, this may be easily seen from Young tableaux multiplication rules)\n\ns (1) (z)s λ (z) = N r=1 s λ+e r (z) , (3.23)\n\nfor {e r ; 1 ≤ r ≤ N, e r • e s = δ rs } being usual orthonormal vectors, we find that\n\nF (1)\n\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = λ ℓ(λ)≤N t |λ| N r,s=1\n\ns λ+e r , s λ+e s N .\n\n(3.24)\n\nNow for any partition λ, s λ+e r , s λ+e s N vanishes unless e r = e s for any r, s and λ r-1 -λ r > 0 for r = 2, . . . , N , due to\n\ns (λ 1 ,...,λ r-1 ,λ r +1,...,λ N ) (z) = 0 for λ r-1 = λ r , r > 1 . (3.25)\n\nChanging summation variables to those in (2.25) then we have, with the definition (2.26), F\n\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = a 1 ,...,a N ≥0 t a 1 +...+Na N + N-1 r=1 a 1 ,...,a N ≥0 a r ≥1 t a 1 +...+Na N = N-1 i=0 t i a 1 ,...,a N ≥0 t a 1 +...+Na N = 1 1 -t P N-1 (t) . (1)\n\n(3.26) Thus, using (2.28), (3.2) with (3.26), 7\n\nN (n,1) = n j=0 p N-1 (j) -N (n+1) = n j=0 p N-1 (j) -p N (n + 1) . (3.27)\n\nFor the case of N λ = N (n,2) we have that, from (3.18),\n\ns (2) (zz -1 ) = s (2) (z)s (2) (z -1 ) + s (1,1) (z)s (1,1) (z -1 ) , (3.28)\n\nso that, from (3.19), we have\n\nF (2) N (t) = λ ℓ(λ)≤N t |λ| s (2) s λ , s (2) s λ N + s (1,1) s λ , s (1,1) s λ N . (3.29) Using s (2) (z)s λ (z) = N r=1 s λ+2e r (z) + 1≤r<s≤N s λ+e r +e s (z) , s (1,1) (z)s λ (z) = 1≤r<s≤N s λ+e r +e s (z) , (3.30)\n\n7 This formula agrees with (3.6) due to\n\nN -1 j=0 p N (n -j) = n j=0 p N -1 (\n\nj) which follows because the corresponding generating functions match,\n\n∞ n=0 N -1 j=0 p N (n -j)t n = (1 + t + . . . + t N -1 )P N (t) = 1 1 -t P N -1 (t) = ∞ n=0 n j=0 p N -1 (j)t n .\n\nalong with (3.25) and\n\ns (λ 1 ,...,λ r-1 ,λ r +2,...,λ N ) (z) = -s (λ 1 ,...,λ r +1,λ r-1 +1,...,λ N ) (z) , (3.31)\n\nfor the cases where λ r-1 = λ r , we may obtain, with the definition (2.26), F\n\nN (t) = 1-t N+1 (1-t)(1-t 2 ) P N-1 (t) + 1 (1-t)(1-t 2 ) P N-2 (t) , (2)\n\nwhere the first contribution comes from λ t |λ| s (2) s λ , s (2) s λ N while the second comes from λ t |λ| s (1,1) s λ , s (1,1) s λ N . Since the partition number p k (-n) = 0 for n = 1, 2, . . . we may write, using (2.28), 8\n\n1 (1 -t)(1 -t 2 ) P k (t) = ∞ n,i,j=0 p k (n -i -2j)t n . (3.33)\n\nThus, from (3.32) with (3.27),\n\nN (n,2) = - n+1 j=0 p N-1 (j) + ∞ i,j=0 (p N-2 (n-i-2j) + p N-1 (n-i-2j)) if n ≤ N , ∞ i,j=0 (p N-2 (n-i-2j) + p N-1 (n-i-2j) -p N-1 (N +1-n-i-2j)) if n ≥ N +1. (3.34)\n\nTables of the numbers (3.2), (3.27) and (3.34) are given in appendix B for some few cases of n, N . Notice from these tables that the numbers N (n,m) below the diagonal line N ≥ n + m for a given n are the same for all N . This is a general feature that derives from values of N (n,m) for N ≥ n + m, which numbers may be obtained from a corresponding generating function that is now constructed.\n\nUsing these techniques, we may provide a consistency check of (3.17), (3.26), (3.32) along with a general result for N (n,m) for high enough values of N , N ≥ m + n. This employs the orthogonality property of power symmetric polynomials p λ (z) (in the large N 8 This is a special case of the following: for any f (n), n ∈ Z, that satisfies f (-n) = 0, n = 1, 2, . . ., then we may (at least formally) write\n\nP k (t) ∞ n=0 f (n)t n = ∞ n,i 1 ,...,i k =0 f (n -i 1 -2i 2 -. . . -ki k )t n . limit) along with s (n) (z) = λ⊢n 1 z λ p λ (z) = ∞ i 1 ,...,i n =0 1 i 1 !i 2 ! • • • i n ! δ i 1 +2i 2 +...+ni n ,n p 1 (z) i 1 1 2 p 2 (z) i 2 • • • 1 n p n (z) i n . (3.35)\n\nUsing the trivial identity p λ (xy) = p λ (x)p λ (y) then from (2.6), (3.19) with (3.35) we have\n\nF (m) ∞ (t) = ∞ n=0 λ⊢n µ⊢m 1 z λ z µ p λ p µ , p λ p µ ∞ t n = ∞ n=0 λ⊢n µ⊢m 1 z λ z µ p ν , p ν ∞ t n = ∞ n=0 λ⊢n µ⊢m z ν z λ z µ t n , (3.36)\n\nwhere for (1 a 1 , 2 a 2 , . . .) being the frequency representation of λ and (1 b 1 , 2 b 2 , . . .) being that of µ then ν has frequency representation (1 a 1 +a 2 , 2 a 2 +b 2 , . . .) so that |ν| = n + m. This agrees with F\n\nN (t) in a series expansion up to O(t N-m ) (since the last equation in (3.36) is also valid for finite N so long as |ν| = n + m ≤ N , by a result of appendix A). Now, since\n\nz ν z λ z µ = ∞ j=1 (a j +b j )! a j !b j ! , (3.37)\n\nwe obtain from (3.36) that,\n\nF (m) ∞ (t) = ∞ n=0 ∞ a 1 ,...,a n =0 ∞ b 1 ,...,b m =0 δ a 1 +•••+na n ,n δ b 1 +•••+mb m ,m m j=1 (a j + b j )! a j !b j ! t n = ∞ b 1 ,...,b m =0 δ b 1 +•••+mb m ,m m j=1 ∞ a j =0 (a j + b j )! a j !b j ! t ja j j>m 1 1 -t j = ∞ b 1 ,...,b m =0 δ b 1 +•••+mb m ,m m j=1 1 (1 -t j ) b j +1 j>m 1 1 -t j . (3.38)\n\nFor the first few cases we have that, with P N (t) as defined in (2.26),\n\nF (m) ∞ (t) = P ∞ (t) for m = 0 1 1-t P ∞ (t) for m = 1 2 (1-t)(1-t 2 ) P ∞ (t) for m = 2 , (3.39)\n\nwhose series expansion agrees with (3.17), (3.26), (3.32) up to O(t N-m ) for, respectively, m = 0, 1, 2. We may use (3.20) with (3.36) to determine N (n,m) exactly for N ≥ n + m.\n\nAsymptotic Counting of Quarter BPS Operators at Large N Asymptotic counting for the one boson case in the large N limit, for which, with P N (t) as defined in (2.26), with p(n) being the total number of (unordered) partitions of n,\n\nZ U(∞) (t) = P ∞ (t) = ∞ n=1 p(n)t n , (3.40)\n\nis the multi-trace partition function, entails finding an asymptotic value for the partition number p(n) for 'large' n. This may be achieved by performing a saddle point approxima-\n\ntion of p(n) = 1 2πi dtP ∞ (t)t -n-1 .\n\nThe function P ∞ (t) has a 'large' singularity at t = 1, but in addition has singularities at all other roots of unity -see [45] on the validity of ignoring these contributions asymptotically. This method was used by Hardy and Ramanujan to find their celebrated formula, here given in a less detailed form as,\n\np(n) ∼ 1 4n √ 3 exp π 2 3 n , (3.41)\n\nwhich was improved by Rademacher to give p(n) exactly. Their method relied crucially on the modular properties of P ∞ (t).\n\nFocusing now on the two bosonic fundamental field case for which, in the large N limit, (3.20) with (3.38) gives exact counting, at issue is first finding asymptotic values for the numbers\n\nQ(n, m, b) = Q(n, m, b 1 , . . . , b m ), with constraint equation m j=1 jb j = m, defined by m j=1 1 (1 -t j ) b j +1 j>m 1 1 -t j = 1 + ∞ n=1 Q(n, m, b)t n . (3.42)\n\nHaving found these we may then attempt to find the dominant contribution to (3.20) with (3.38) for large N . In order to give asymptotic values for Q(n, m, b) we may follow [46] and apply a formula due to Meinardus which gives a general result for the generating function\n\n∞ n=1 (1 -t n ) -a n = 1 + ∞ n=1 r(n)t n . (3.43)\n\nA detailed version of Meinardus' theorem may be found in [46] but for purposes of brevity we may note that it implies that, as n → ∞,\n\nr(n) ∼ C n κ exp AΓ(α + 1)ζ(α + 1)n α 1/(α+1) (α + 1)/α , (3.44)\n\nwhere ζ(s) = ∞ j=1 j -s is the Riemann zeta function and the constants C, κ, α, A are determined by the auxiliary Dirichlet series,\n\nD(s) = ∞ j=1 a j j s , (3.45)\n\nwhich must converge for Re(s) > α, a positive real number, and possess an analytic continuation in the region Re(s) ≥ c, -1 < c < 0, such that, in this region, D(s) is analytic except at a simple pole at s = α where it has residue A. In terms of α, A then\n\nC = 1 2π(1+α) AΓ(α + 1)ζ(α + 1) (1-2D(0))/2(α+1) exp D ′ (0) , κ = (D(0) -1 -1 2 α)/(α + 1) . (3.46)\n\nApplying Meinardus' theorem to the case of (3.42), clearly we have\n\nD(s) = m j=1 b j j s + ζ(s) , (3.47)\n\nso that, assuming m = m j=1 jb j is fixed, D(s) has a simple pole at s = α = 1 where it has residue A = 1. Using\n\nD(0) = -1 2 + m j=1 b j , exp D ′ (0) = 1 √ 2π m j=1 1 j b j , (3.48)\n\nthen, from (3.44) with (3.46), we may easily determine that, as n → ∞,\n\nQ(n, m, b) ∼ 1 4n √ 3 √ 6n π m j=1 b j m j=1 1 j b j exp π 2 3 n . (3.49)\n\nThis reduces to (3.41) when b j = 0, 1 ≤ j ≤ m, whereby Q(n, 0, . . . , 0) = p(n). Using (3.20) for (3.38) with (3.42) and (3.49) then, as n → ∞,\n\nN (n,m) = b 1 ,...,b m ≥0 m j=1 jb j =m Q(n, m, b) - b 1 ,...,b m-1 ≥0 m-1 j=1 jb j =m-1 Q(n+1, m-1, b) ∼ 1 4n √ 3 √ 6n π m exp π 2 3 n , (3.50)\n\nsince Q(n, m, m, 0, . . . , 0), for b 1 = m, b j = 0, j > 1, dominates over all other terms in (3.50) . This gives asymptotic values for the numbers in (1.1), for counting quarter BPS operators, transforming in [m, n-m, m] SU (4) R representations, in the large N limit of free N = 4 super Yang Mills, as previously described."},{"section_type":"OTHER","section_title":"Counting Operators in the Chiral Ring of N = 4 Super Yang Mills","text":"For the purposes of counting operators in the chiral ring of N = 4 super Yang Mills, we denote corresponding multi-trace partition functions by C U(N) (t).\n\nThe generating function for C U(N) (t) for the case of one bosonic fundamental field has been written in the form [1, 3, 9 ]\n\nC(ν, t) = ∞ n=0 1 1 -νt n = ∞ N=0 ν N C U(N) (t) , (4.1)\n\nso that ν acts as a chemical potential for the rank of the gauge group U (N ). The equivalence C U(N) (t) = Z U(N) (t) = P N (t), with Z U(N) (t) as in (2.25) , is actually a special case of the q-Binomial theorem. Writing -see [43] for notation -\n\n(a; q) k = (1 -a)(1 -aq) • • • (1 -aq k-1 ) , (4.2)\n\nthen the q-Binomial theorem is, for\n\n|x|, |q| < 1, ∞ k=0 (a; q) k (q; q) k x k = (ax; q) ∞ (x; q) ∞ . (4.3)\n\n(Identifying ν = x and q = t and setting a = 0 in (4.3), so that 1/(ν; t) ∞ = C(ν, t) above and 1/(t; t) N = P N (t) in (2.26), then C U(N) (t) = Z U(N) (t) = P N (t) straightforwardly. This special case of the q-Binomial theorem is due to Euler.)\n\nFor the two boson case, so that the single particle partition function is given by f (t, u) = t + u for some t, u, then the generating function for the finite N chiral ring partition function C U(N) (t, u) is given by [1, 3, 9]\n\nC(ν, t, u) = ∞ n,m=0 1 1 -νt n u m = ∞ N=0 ν N C U(N) (t, u) . (4.4)\n\nThis function is more difficult to analyse in terms of counting though has been investigated by Stanley [47] in relation to partitions -there it has been dubbed the 'double Eulerian' generating function. Through use of the Cauchy-Littlewood formula, then we may expand C U(N) (t, u) in terms of partitions of N as,\n\nC U(N) (t, u) = λ⊢N h λ (t)h λ (u) , (4.5)\n\nwhere h λ (t) = s λ (1, t, t 2 , . . .) , (4.6) so that using an identity for Schur polynomials to be found in [24, 47] then\n\nC U(N) (t, u) = λ⊢N 1≤i<j≤N (1 -t λ i -λ j +j-i )(1 -u λ i -λ j +j-i ) N i=1 (t; t) λ i +N -i (u; u) λ i +N -i (tu) N i=1 (i-\n\n1)λ i . (4.7) (4.5) with (4.6) has a natural interpretation in terms of plane partitions in that, for π being all column-strict plane partitions of shape λ, |π| = i,j π ij , 9 h λ (t) = s λ (1, t, t 2 , . . .) = π t |π| . (4.8) Obviously, (4.5) with (4.8) generalise for other chiral ring sectors. (For a different connection between the 'double Eulerian' generating function and major indices of permutations see [47], p. 385.) As an illustration of (4.5) with (4.8), we may consider the case N = 2 whereby λ = (2, 0), (1, 1) gives the two possible partitions of 2. For λ = (2, 0) (corresponding to a Young diagram with a single row of two boxes) π 11 ≥ π 12 ≥ 0 gives all column-strict plane partitions of shape (2, 0), while for λ = (1, 1) (corresponding to a Young diagram with a single column of two boxes) then π 11 > π 21 ≥ 0 gives all columnstrict plane partitions of shape (1, 1). Thus, h (2,0) (t) = π 11 ,π 12 ≥0 π 11 ≥π 12 t π 11 +π 12 = 1 (1t)(1t 2 ) , h (1,1) (t) = π 11 ,π 21 ≥0 π 11 >π 21 t π 11 +π 21 = t (1t)(1t 2 ) ,\n\nso that, from (4.5) for N = 2,\n\nC U(2) (t, u) = h (2,0) (t)h (2,0) (u) +h (1,1) (t)h (1,1) (u) = 1 + ut (1 -t)(1 -t 2 )(1 -u)(1 -u 2 ) , (4.10)\n\nwhich is the correct result as may be verified by extracting the ν 2 coefficient in an expansion of (4.4) up to O(ν 2 ).\n\n9 See [47] for a detailed description of plane partitions. Briefly, a column-strict plane partition of shape λ is an array π = (π ij ) of non-negative integers with finitely many non-zero entries, that is arranged in a Young tableaux with shape λ -see appendix A -such that the numbers π ij are weakly decreasing along each row, π ij ≥ π i j+1 ≥ 0, and strictly decreasing down each column, π ij > π i+1 j ≥ 0. The sum of the parts of π is given by |π| = i,j π ij . (Note that in contrast to the definition in [47] , here we are allowing π ij = 0, for some i, j, to be a part of the plane partition π with shape λ.)\n\nIn the large N limit,\n\nC U(∞) (t, u) = n 1 ,n 2 ≥0 n 1 +n 2 >0 1 1 -t n 1 u n 2 , (4.11)\n\nupon which attention is shortly focused.\n\nFor the numbers N (n,m) → N (n,m) counting quarter BPS primary operators for the chiral ring of N = 4 super Yang Mills, belonging to [m, nm, m] SU (4) R R-symmetry representations, as in (1.1), we have foot_1\n\nN (n,m) = 1 8π 2 dt du C U(N) (t, u) s (n,m) (t -1 , u -1 ) (t -1 -u -1 ) 2 . (4.12)\n\nThese may be more conveniently evaluated in terms of the numbers in (3.4) M (n,m) → M (n,m) , counting all chiral ring quarter BPS operators in the [m, n-m, m] SU (4) R representation, given by\n\nM (n,m) = 1 (2πi) 2 dt du C U(N) (t, u) t -n-1 u -m-1 , (4.13)\n\nso that N (n,m) = M (n,m) -M (n+1,m-1) . Defining P λ (n) to be the number of column-strict plane partitions π of shape λ so that |π| = i,j π ij = n, then, from (4.5) with (4.8) and (4.13), M (n,m) = λ⊢N P λ (n)P λ (m). Thus,\n\nN (n,m) = λ⊢N (P λ (n)P λ (m) -P λ (n+1)P λ (m-1)) , (4.14)\n\ncounts chiral ring quarter BPS primary operators in SU (4) R representations [m, nm, m] for any n, m at finite N .\n\nAsymptotic Counting for Chiral Ring BPS Operators at Large N For asymptotic counting of operators in the chiral ring of N = 4 super Yang Mills at large N , a relatively crude method is employed here which nevertheless captures the exponential behaviour of counting numbers of interest. This method is based on saddle point approximations of functions near a dominant singularity -see [45] for a useful summary. (Often for physical applications in thermodynamics, e.g. for entropy formulae, we are interested only in the exponential behaviour of such numbers anyhow.)\n\nTo illustrate, we consider the one boson case in the large N limit again. We first find a convenient 'approximating function' as follows,\n\nP ∞ (t) = ∞ n=1 1 1 -t n = exp - ∞ n=1 ln(1 -t n ) ∼ exp - ∞ 0 ds ln(1 -t s ) = exp - π 2 6 ln t , (4.15)\n\nwhich has an 'easier' singularity structure. (The approximation in the second step may be justified by the Euler-Maclaurin formula for approximating sums by integrals.) Using (4.15) then for large enough n,\n\np(n) = 1 2πi dt P ∞ (t)t -n-1 ∼ 1 2πi dt e g(t) , g(t) = - π 2 6 ln t -n ln t . (4.16)\n\nWe may approximate the latter integral for large n by noting that the dominant contribution is at the saddle point t ′ = e -π/ √ 6n ∼ 1 for which (4.18) Thus, ln p(n) ∼ π 2 3 n ,\n\ng(t ′ ) = π 2 3 n , g ′ (t ′ ) = 0 , g ′′ (t ′ ) = 2 π √ 6n 3 e π √ 2/3n = α , (4.17\n\nwhich captures the correct behaviour of ln p(n) for large n, according to (3.41).\n\nWe may proceed analogously for the quarter BPS chiral ring multi-trace partition function at large N , (4.11), which we approximate by\n\nC U(∞) (t, u) = n 1 ,n 2 ≥0 n 1 +n 2 >0 1 1 -t n 1 u n 2 = exp - n 1 ,n 2 ≥0 n 1 +n 2 >0 ln(1 -t n 1 u n 2 ) ∼ exp - ∞ 0 ∞ 0 dv dw ln(1 -t v u w ) = exp ζ(3) ln t ln u . (4.20)\n\nIn this case we have, from (4.13),\n\nM (n,m) ∼ 1 (2πi) 2\n\ndt du e g(t,u) , (\n\nwhere\n\ng(t, u) = ζ(3) ln t ln u -n ln t -m ln u , (4.22)\n\nfor n, m large. The dominant contribution to the integral, for n, m both large and of the same order, occurs about the point (t ′ , u ′ ) ∼ (1, 1) where\n\nt ′ = e -(ζ(3)mn -2 ) 1/3 , u ′ = e -(ζ(3)nm -2 ) 1/3 , (4.23)\n\nfor which\n\ng(t ′ , u ′ ) = 3 3 ζ(3)nm , ∂ ∂t g(t, u) (t ′ ,u ′ ) = ∂ ∂u g(t, u) (t ′ ,u ′ ) = 0 , ∂ 2 ∂t 2 g(t, u) (t ′ ,u ′ ) = 2(ζ(3) -1 n 5 m -1 ) 1 3 e 2(ζ(3)mn -2 ) 1/3 = α , ∂ 2 ∂u 2 g(t, u) (t ′ ,u ′ ) = 2(ζ(3) -1 m 5 n -1 ) 1 3 e 2(ζ(3)nm -2 ) 1/3 = β , ∂ 2 ∂t∂u g(t, u) (t ′ ,u ′ ) = (ζ(3) -1 n 2 m 2 ) 1 3 e (ζ(3)mn -2 ) 1/3 +(ζ(3)nm -2 ) 1/3 = γ . (4.24)\n\nSo long as m, n are both large and of the same order, the saddle point approximation is justified and we obtain\n\nM (n,m) ∼ e 3 3 √ ζ(3)nm 1 4π 2 ∞ -∞ ∞ -∞ dv dw e -1 2 (αv 2 +βw 2 +2γvw) = h(α, β, γ)e 3 3 √ ζ(3)nm , (4.25) where h(α, β, γ) = 1 2π αβ -γ 2 = 1 2π √ 3 (ζ(3)m -2 n -2 ) 1 3 e -(ζ(3)mn -2 ) 1/3 -(ζ(3)nm -2 ) 1/3 . (4.26)\n\nWe thus have that for n, m both comparably large,\n\nln M (n,m) ∼ 3 3 ζ(3)nm , (4.27) so that ln N (n,m) = ln M (n,m) -M (n+1,m-1) ∼ ln M (n,m) ∼ 3 3 ζ(3)nm . (4.28)\n\nIt is difficult to check the consistency of this result given the dearth of literature on these types of multi-variable generating functions and their asymptotic behaviour, however, we may consider the simpler function, also considered in [9] ,\n\nC U(∞) (t, t) = ∞ n=1 1 (1 -t n ) n+1 = ∞ r=0 E(r) t r , (4.29)\n\nwhere, in terms of the counting numbers N (n,m) , from (1.1), By considering (4.29) directly, we may employ Meinardus' theorem (described in the third section) to find the behaviour of ln E(r) as r → ∞. Note, however that Meinardus' theorem may not be applied directly to (4.29) since the corresponding auxiliary Dirichlet series (3.45), with a j = j+1, has two simple poles. To overcome this difficulty we split (4.29) into a product of two functions, both separately amenable to application of Meinardus' theorem. One is the reciprocal of the Euler function, P ∞ (t) in (4.15) . The other, the MacMahon function, is given by\n\nC U(∞) (t, t) = ∞ n=0 n m=0 (n-m+1) N (n,m) t n+m ⇒ E(r) = [ 1 2 r] m=0 (r-2m+1) N (r-m,m) . ( 4\n\nM (t) = ∞ n=1 1 (1 -t n ) n = ∞ r=0 q(r) t r , (4.32)\n\nand has been considered in a similar context as here in [3] . 11 Writing\n\nC U(∞) (t, t) = P ∞ (t)M (t) , (4.33)\n\nwith P ∞ (t) as in (4.15), so that, using (4.29),\n\nE(r) = r 1 ,r 2 ≥0 r 1 +r 2 =r p(r 1 )q(r 2 ) , (4.34)\n\nwe may find the asymptotic behaviour of ln E(r), as r → ∞, by extracting the dominant contribution from (4.34) using the asymptotic behaviour of p(r), q(r). The auxiliary Dirichlet series for M (t) in (4.32) is, from (3.45) with a j = j,\n\nD(s) = ζ(s -1) , (4.35)\n\n11\n\nThe relation of M (t) in (4.32) to plane partitions is given a description in [46] . Briefly, q(r) gives the number of ordinary plane partitions π, so that π ij ≥ π i+1 j > 0, π ij ≥ π i j+1 > 0, with |π| = i,j π ij = r. p(r) < q(r) as ordinary partitions λ are a special case of plane partitions.\n\nIn fact, the formula for ln q(r) found here is a special case of a more exact asymptotic formula first found by Wright [48] for the number of plane partitions q(r) of the number r.\n\nat the saddle point (t ′ 1 , . . . , t ′ k-1 ) ∼ (1, . . . , 1), where (ln t ′ 1 , . . . , ln t ′ k-1 ) = -k ζ(k) m 1 • • • m k-1 (1/m 1 , . . . , 1/m k-1 ) , (4.41) so that ∂ ∂t j g(t 1 , . . . , t k-1 ) (t ′ 1 ,...,t ′ k-1 ) = 0 , j = 1, . . . , k-1 . (4.42) (4.39) is consistent with a result implied by Meinardus' theorem. The function, 13 C U(∞) (t, . . . , t) ∼ j=1 m j = r, for which the solution is m j = m M = r/(k -1) (relaxing the constraint that m j be non-negative integers, which is irrelevant asymptotically), so that ln M (m M ,...,m M ) ∼ ln c(k, r). This is applicable to counting multi-trace operators in the eighth BPS chiral ring sector for N = 4 super Yang Mills with fundamental fields Z, Y, X involving m 1 Z's, m 2 Y 's, m 3 X's. Expanding the corresponding partition function (4.37), with k = 4, in terms of Schur polynomials s (m 1 ,m 2 ,m 3 ) (t), m 1 ≥ m 2 ≥ m 3 ≥ 0, similar to (1.1), the expansion coefficients N (m 1 ,m 2 ,m 3 ) count spinless multi-trace primary operators transforming in the [m 2 + m 3 , m 1m 2 , m 2m 3 ] SU (4) R R-symmetry representation, with conformal dimensions m 1 +m 2 +m 3 [33] . Just as in (4.28), asymptotically ln N (m 1 ,m 2 ,m 3 ) ∼ ln M (m 1 ,m 2 ,m 3 ) . This counting, however, ignores contributions of the fermionic fields λ, λ, which it may be important to include in order to give correct counting of eighth BPS chiral ring operators. 13 This may be easily seen from (4.37), as the number of solutions to k-1 j=1 m j = n, where m j are non-negative integers, is the binomial number n+k-2 k-2 which, to leading order in large n, behaves like n k-2 /(k-2)!. More properly, we should split the product C U(∞) (t, . . . , t) into pieces separately amenable to Meinardus' theorem, as for the prior case for k = 3, however, just as for that case, the numbers c(k, r) dominate, and so other contributions are ignored here."},{"section_type":"CONCLUSION","section_title":"Conclusions","text":"There are some obvious questions not answered by this work. The first is whether or not the approach in the second section using symmetric polynomials can give insight into thermodynamics at finite N , such as for the Hagedorn transition, for example. While it gives the large N expression (2.14) in an elementary way, its wider applicability or usefulness to such questions is unclear. The approach is undoubtedly useful for finding exact expressions for counting numbers (as in (3.2), (3.27) and (3.34) for quarter BPS operators) and (3.9), (3.16 ) may be useful for analysing counting for more complicated sectors of N = 4 super Yang Mills, with gauge group U (N ).\n\nThe second question is how the arguments employing symmetric polynomial techniques here may be extended to other gauge groups, the most pertinent being perhaps SU (N ). Arguments here employing (2.6) and the orthonormality property of Schur polynomials should remain largely unaffected for SU (N ). Exact values for counting numbers obtained here should require some modification for SU (N ), though asymptotic values may be unchanged.\n\nThe third question concerns asymptotic values for counting numbers and how these may be improved. The asymptotic counting formulae given in such papers as [3, 9] for chiral ring sectors are special cases of formulae such as those of Hardy and Ramanujan, Meinardus, etc., all of which derive from single variable generating functions. It is hoped that the expressions (3.50), (4.28), (4.39), given here for asymptotic counting of BPS operators, that distinguishes between differing R-symmetry charges, represents a serious attempt at going beyond consideration of single variable generating functions. 14 Improving upon these formulae will require more sophisticated techniques, perhaps along the lines used to find those of Hardy and Ramanujan or Meinardus and employing any modular properties of the multi-variable functions involved. This issue may also be important for microscopic counting for Black Holes, as the BPS solutions found thus far, for N = 4 superconformal symmetry, depend on special values of R-symmetry charges [17, 18, 19, 20] -see [50] for a related detailed discussion.\n\nThus far, the elegant results for finite N partition functions for chiral ring sectors have been interpreted from a largely geometric perspective -it may be interesting to investigate more how such results are related to the theory of random matrices and/or symmetric polynomials."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered. Higher order corrections for finite N are also discussed in terms of symmetric group characters. For finite N, the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free N = 4 super Yang Mills discussed, including a general result for large N. The partition function for quarter BPS operators in the chiral ring of N = 4 super Yang Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free N = 4 super Yang Mills and in the chiral ring. Also, general and explicit expressions are derived for SU(2) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"For the past while there has been intense interest in finite N partition functions for Yang Mills theories, especially in super-symmetric ones, particularly with regard to their construction for BPS states and the counting thereof [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15] . Much attention has been devoted to this issue for N = 4 super Yang Mills theory, it being by now the archetypal example of a conformal field theory for which we have a dual description in terms of string theory, by means of the AdS/CFT conjecture [16] . An example is in the assiduous efforts that have been made to explain the entropy of certain BPS black holes in AdS 5 ×S 5 [17, 18, 19, 20] in terms of microscopic counting of dual operators in N = 4 super Yang Mills, with gauge group SU (N ), by means of partition functions [1] . While this problem remains unsolved to date, essentially due to the difficulty of defining what is meant by these dual operators for finite N , there are other equally interesting sectors of (super) Yang Mills theories where more progress with counting has been made, examples being in free N = 4 super Yang Mills and in chiral ring sectors involving BPS operators.\\n\\nFor N = 4 super Yang Mills, the half BPS sector consists of multi-trace operators involving a single bosonic operator, Z. Similarly, the quarter BPS sector consists of multitrace operators involving two bosonic operators Z, Y while the eighth consists of ones involving three bosonic operators Z, Y, X and two fermionic operators λ, λ. (All these operators are here assumed to belong to the Lie algebra of U (N ).)\\n\\nFor the chiral ring, the commuting/anti-commuting of these operators is at the heart of why we can write very concise and elegant generating functions for the finite N multitrace partition functions [1] . Analysis of these partition functions, in terms of the counting of operators, has become a sophisticated industry where such approaches as the so-called 'Plethystic Program' have provided substantial results [3, 9] . This paper is devoted largely to the issue of partition functions for free field theory and particularly to the counting of gauge invariant multi-trace operators for the case of two bosonic fundamental fields. This is of relevance to the quarter BPS sector of N = 4 super Yang Mills in the free field limit when the operators Z, Y do not commute, in contrast to the chiral ring when they do.\\n\\nThe partition function for a free, massless quark-gluon gas was computed long ago [21] . This involved taking particle statistics into account using coherent state techniques and then imposing the gauge singlet condition by integrating over the relevant gauge group. With some modifications to the expression thus derived we may write the multi-trace partition function for some generic bosonic/fermionic fundamental fields in terms of an integral over the gauge group, involving the single particle partition function [22, 23] . This is the starting point here.\\n\\nFor U (N ) gauge theories we may easily write down the integral, though, even for this case, its evaluation is far from simple. One approach (which we adapt here for the SU (2) case) is to rewrite the expression in terms of an N -fold contour integral, whereby it may in principle be evaluated by summing the contributions from poles inside origin centred unit discs, in each of the N complex planes -similar techniques have been used in [10] . Due to the number of poles this becomes unfeasible for higher values of N . Another approach is to use the fact that the complex integral that interests us provides for an inner product for symmetric polynomials -see Macdonald [24] , pp. 363 -372, for a related discussion. Taking this point more seriously reveals an alternative route to evaluating the free field theory partition function which exposes not only the large N case in an almost trivial way, but also how and where this differs from the finite N case. This treatment also reveals an alternative interpretation of the free field partition function at finite N -it is related to a gauge group average of the cycle polynomial for the symmetric permutation group (after a certain identification of 'letters' with gauge group valued variables). This point is not dwelt on further here though makes the connection between the partition function and Polya enumeration explicit. For single trace operators at large N this connection has already been made [22, 25] for N = 4 super Yang Mills, whereby the partition function for single trace operators is related to the cycle polynomial for the cyclic permutation group. 1 Another issue is how to use the expression for the free field partition function to give explicit counting of gauge invariant multi-trace operators in a Yang-Mills theory, with gauge group U (N ). The case for one bosonic fundamental field has been widely discussed and, for finite N , the operators are counted by partition numbers of non-negative integers into at most N parts (which is here denoted by p N (n) -no closed formula for these numbers for arbitrary N, n exist, though they have a 'nice' generating function). Here, this result is re-derived, from a symmetric function perspective, by employing the well known Cauchy-Littlewood formula.\\n\\nTo proceed further with counting, for the quarter BPS sector of N = 4 super Yang Mills, for instance, character methods prove to be both natural and indispensable. Characters in relation to conformal field theories prove to be very convenient for encoding the allowable representations [15, 29, 30, 31, 32, 33] and for studying related partition functions, [34, 35, 36, 33] . For N = 4 super Yang Mills, it was shown in [33] that, if we are to distinguish among primary operators with differing conformal dimensions, spins and R-symmetry charges, such counting is most easily achieved using reductions of the full N = 4 superconformal characters, in certain limits that isolate corresponding sectors of short/semi-short operators. (One such limit corresponds to the index constructed in [1] .) This point may be easily illustrated for quarter BPS primary operators in N = 4 super Yang Mills, the case of two bosonic fundamental fields here. (See [37] for an explicit construction and counting of quarter BPS operators.) For N = 4 super Yang Mills, the counting of quarter BPS primary operators is complicated by the fact that, if we are to keep track of differing R-symmetry representations, any partition function restricted to this sector must be expanded in terms of U (2) characters (or two-variable Schur polynomials). Denoting some partition function restricted to this sector by Z(t, u), where t, u are letters corresponding to the fields Z, Y , then by expanding\\n\\nZ(t, u) = ∞ n=0 n m=0 N (n,m) s (n,m) (t, u) , s (n,m) (t, u) = t n+1 u m -t m u n+1 t -u , (1.1)\\n\\nin terms of two-variable Schur polynomials s (n,m) (t, u), we obtain the numbers N (n,m) of gauge invariant quarter BPS operators belonging to the [m, n-m, m] SU (4) R R-symmetry representation and having conformal dimension n + m (so that they are superconformal primary highest weight states in the corresponding quarter BPS supermultiplets). (The case m = 0 counts gauge invariant half BPS primary operators.)\\n\\nHere, the free field partition function is thus expanded in terms of Schur polynomials, depending on the same variables as the one particle partition function, the two boson case being a specialisation. This is quite naturally achieved using the Cauchy-Littlewood formula (and, if we include fermions, another formula due to Littlewood). Generally, we may obtain a result that relates the counting numbers to a sum over Kronecker coefficients. These arise naturally in the theory of the symmetric permutation group, though remain somewhat mysterious from a combinatorial perspective.\\n\\nSpecialising to the two boson case, a recursive procedure is employed here for the counting of multi-trace quarter BPS operators in free field theory at finite N . This issue was given considerable discussion for the large N case in [33] -here the results of [33] are generalised in terms of a generating function that may be employed to count quarter BPS operators for any R-symmetry charges, at large N . Asymptotic counting is also addressed in the latter case for the numbers N (n,m) in (1.1) for large n and fixed m.\\n\\nTo complete the discussion, counting of quarter BPS operators in the chiral ring of N = 4 super Yang Mills is investigated in terms of expanding over U (2) characters as in (1.1) . An explicit formula is given for the corresponding finite N partition function, with a short combinatorial interpretation given in terms of plane partitions, and specialised to large N . For the latter case, the exponential behaviour of the numbers N (n,m) in (1.1) is found for n and m both comparably large. This behaviour is consistent with a special case addressed in [3, 9] . By way of completion, a similar discussion for an arbitrary number of bosonic fundamental fields in the chiral ring is included. Two appendices are included; the first establishes some notation used for partitions and gives some standard results for the symmetric group and symmetric polynomials, the second gives some tables of numbers of quarter BPS operators in free N = 4 super Yang Mills, with gauge group U (N ), for which explicit formulae are given in the main text. Footnotes contain further details and points of clarification.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Free Field Partition Functions\",\n \"text\": \"We start from the single particle partition function which is here denoted by f (t) for some variables 2 t = (t 1 , t 2 . . .). The general form of f (t) is\\n\\nf (t) = i a i t i , (2.1)\\n\\nwhere each t i is a letter corresponding to a fundamental field and a i are signs, being +1 for a bosonic field, or -1 for a fermionic field.\\n\\nFor compact gauge Lie group G, the multi-trace partition function is then given by [21] , (see [22, 23] for refinements,)\\n\\nZ G (t) = G dµ G (g) exp ∞ n=1 1 n f (t n )χ R (g n ) , (2.2)\\n\\ninvolving the Haar (or G-invariant, or Hurwitz) measure dµ G (g) for g ∈ G (so that\\n\\nG dµ G (g)F (g) = G dµ G (g)F (gh) = G dµ G (g)F (hg) for all h ∈ G and G dµ G (g) = 1)\\n\\nand where χ R (g) is the character for the R representation of G, assuming that the fundamental fields transform in identical gauge group representations R.\\n\\nFor G = U (N ), so that for any matrix U ∈ U (N ) we may write U = V ΘV † , where V is a unitary matrix and Θ = diag.(e iθ 1 , . . . , e iθ N ), 0 ≤ θ i < 2π, and for some F (U ) = F (Θ), independent of V , then we may write\\n\\nU(N) dµ U(N) (U )F (U ) = 1 (2π) N N ! 2π 0 N j=1 dθ j 1≤k<l≤N |e iθ k -e iθ l | 2 F (Θ) , (2.3)\\n\\n2 In what follows roman letters are used to denote a collection of variables and, for x = (x 1 , . . . , x i ), y = (y 1 , . . . , y j ) for example, the shorthand x α is used to mean (x 1 α , . . . , x i α ) and z = xy to mean z = (z 11 , . . . , z ij ) where z rs = x r y s . The latter convenient notation has been used by Macdonald [24] .\\n\\nwhich is, of course, related to the Weyl parametrisation of U (N ). Thus, for such F (U ), the left-hand side of (2.3) simplifies to an integral over the N torus. Of course, as χ R (U ) generally depends on linear combinations of tr(U j ) k tr(U †l ) m for various non-negative integers j, k, l, m then any function of χ R (U ) is an example of such an F (U ).\\n\\nWe are interested in the case where R = Adj. is the adjoint representation so that for U (N ) we have that χ Adj. (U ) = tr U tr U † (while for SU (N ) then χ Adj. (U ) = tr U tr U † -1). For U (N ) we then find that, using (2.2) with (2.3),\\n\\nZ U(N) (t) = 1 (2π) N N ! 2π 0 N j=1 dθ j 1≤k<l≤N |e iθ k -e iθ l | 2 exp ∞ n=1 1 n f (t n ) N j,k=1\\n\\ne in(θ j -θ k ) .\\n\\n(2.4)\\n\\nWe may write (2.4) as an N -fold contour integral by first making the variable change z j = e iθ j so that the integrals in (2.4) are around unit circles in each z j complex plane and then we obtain\\n\\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) exp ∞ n=1 1 n f (t n )p n (z)p n (z -1 ) , (2.5)\\n\\nwhere ∆(z) = 1≤i<j≤N (z iz j ) is the Vandermonde determinant and p n (z) = N i=1 z i n is a power symmetric polynomial -see appendix A for a brief discussion of symmetric polynomials. This integral may then in principle be evaluated by deforming the contours so as to extract the residues at poles within the discs\\n\\n|z j | < 1, 1 ≤ j ≤ N .\\n\\nA crucial observation is that, for some N variable symmetric polynomials g(z), h(z), then\\n\\ng, h N = h, g N = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 )g(z)h(z -1 ) , (2.6)\\n\\nacts as an inner product -this is easy to see in terms of Schur polynomials which provide an orthonormal basis for symmetric polynomials. The reader may now wish to peruse appendix A where notation regarding partitions and a short discussion of symmetric polynomials is included.\\n\\nThe General and Large N Cases for U (N )\\n\\nFor application of inner products to (2.5) we have that, in terms of power symmetric polynomials p λ (z) for partitions λ,\\n\\nexp ∞ n=1 1 n f (t n )p n (z)p n (z -1 ) = ∞ n=1 ∞ a n =0 1 n a n a n ! f (t n ) a n p n (z) a n p n (z -1 ) a n , = λ 1 z λ f λ (t)p λ (z)p λ (z -1 ) , (2.7)\\n\\nwith the definitions of\\n\\nz λ = ∞ n=1 n a n a n ! , f λ (t) = ∞ n=1 f (t n ) a n , (2.8)\\n\\nbeing in terms of the frequency representation of λ, (1 a 1 , 2 a 2 , . . .) with n≥1 n a n = |λ|, the weight of the partition λ (note that the frequency representation of λ is simply a convenient re-ordering of the parts of λ).\\n\\nIn (2.7) the numbers z λ have a standard combinatorial interpretation -for a given permutation σ ∈ S m with a 1 1-cycles, a 2 2-cycles etc., so that n≥1 n a n = m = |λ|, then z λ = n≥1 n a n a n ! is the size of the centraliser Z σ = {τ ; τ ∈ S m , τ στ -1 = σ} of σ ∈ S m . (This may be easily seen as under conjugation of σ by τ then τ can permute the cycles of length n among themselves in a n ! ways and/or render a cyclic rotation on each of the individual cycles in n a n ways.) More details of the symmetric group are to be found in appendix A.\\n\\nWe may immediately observe that (2.7) represents a sum over cycle polynomials of the symmetric group S m . This is given by, for letters\\n\\nu 1 , u 2 , . . . u m , 3 C m (u) = a 1 ,...,a m ≥0 δ a 1 +2a 2 +...+ma m ,m m n=1 1 n a n a n ! u n a n = λ⊢m 1 z λ u λ , (2.9)\\n\\nwhere 'λ ⊢ m' means that λ is any partition of m -see appendix A for notation -and\\n\\nu λ = m n=1 u n a n , (2.10)\\n\\nin terms of the frequency representation of λ above. Identifying u n = f (t n )p n (z)p n (z -1 ) then we may rewrite\\n\\nZ U(N) (t) = ∞ m=0 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 )C m (u) , (2.11)\\n\\n3 This formula is easy to see from the definition of the cycle polynomial for a subgroup G of S m . This is given by\\n\\n1 |G| g∈G u 1 j 1 (g) • • • u m j m (g) = 1 |G| K g |K g |u 1 j 1 (g) • • • u m j m (g) ,\\n\\nwhere j i (g) denotes the number of i cycles in the unique decomposition of g into disjoint cycles the sum of the U (N ) group averages of each of the cycle polynomials C m (u). (Physically, the interpretation is that the cycle index for the symmetric permutation group accounts for particle statistics while integration over the gauge group imposes the gauge singlet condition. For purposes of clarity, the U (N ) case has been focused upon here, though from the form of (2.2) it is easy to see how this generalises for other gauge groups whereby the letters u n = f (t n )χ R (g n ) for the fundamental fields transforming in identical gauge group representations, R.)\\n\\nDirectly from (2.7), in terms of the inner product (2.6), then\\n\\nZ U(N) (t) = λ 1 z λ f λ (t) p λ , p λ N = λ 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 , (2.12)\\n\\nwhere on the right-hand side of (2.12) we have used an expression for the inner product of two power symmetric polynomials expressed in terms of the characters of the symmetric group, given in appendix A. (Here 'ℓ(µ)' means the number of non-zero parts of the partition µ.)\\n\\nUsing a result of appendix A (essentially orthogonality relations for symmetric group characters), (2.12) may be rewritten as\\n\\nZ U(N) (t) = λ |λ|≤N f λ (t) + λ |λ|>N 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\\n\\n(2.13)\\n\\nIn the large N limit, Z U(N) (t) simplifies considerably as only the first term in (2.13) need be considered. Using the frequency representation of λ then\\n\\nZ U(∞) (t) = λ f λ (t) = ∞ n=1 ∞ a n =0 f (t n ) a n = ∞ n=1 1 1 -f (t n ) , (2.14)\\n\\na result which has been obtained using Polya counting methods for single trace operators and saddle point approximations [22, 23] .\\n\\nHigher order corrections in |λ|, the weight of the partition λ, to (2.13) may be obtained by successive evaluation of\\n\\nµ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\\n\\nOne method is to employ the Murnaghan-Nakayama Rule, used to compute χ µ λ using skew hooks and Young diagrams. (A readable account of the Murnaghan-Nakayama Rule may be found in [38] , though of course it is explained in many standard textbooks that discuss the symmetric group.)\\n\\nFor the case of |λ| = N +1 then we may observe that,\\n\\nµ⊢N +1 ℓ(µ)≤N χ µ λ 2 = µ⊢N +1 ℓ(µ)≤N +1 χ µ λ 2 -(χ ν λ ) 2 = z λ -(χ ν λ ) 2 , ν = (1 N+1 ) , (2.15)\\n\\nsince the partition ν = (1 N+1 ) is the only one excluded among those partitions µ of N +1 with ℓ(µ) ≤ N . By applying the Murnaghan-Nakayama Rule we may determine, for\\n\\nν = (1 L ), (χ ν λ ) 2 = 1 for |λ| = L 0 otherwise , (2.16)\\n\\nsince χ ν λ in this case is just a sign. (This may be easily seen as there is only one possible way to remove successive skew hooks, which in this case are just column Young diagrams of length λ i , from the (1 L ) column Young diagram to leave one of normal shape, in this case, another column Young diagram.) Thus, using (2.15) with (2.16) in (2.13), we obtain\\n\\nZ U(N) (t) = λ |λ|≤N +1 f λ (t) - λ |λ|=N +1 1 z λ f λ (t) + λ |λ|>N +1 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .\\n\\n(2.17) By a similar line of argument we may do the same for the case of |λ| = N +2. We have that, for\\n\\nν 1 = (1 N+2 ) and ν 2 = (2, 1 N+1 ), µ⊢N +2 ℓ(µ)≤N χ µ λ 2 = µ⊢N +2 ℓ(µ)≤N +2 χ µ λ 2 -(χ ν 1 λ ) 2 -(χ ν 2 λ ) 2 = z λ -(χ ν 1 λ ) 2 -(χ ν 2 λ ) 2 . (2.18)\\n\\nWe may determine, for ν = (2, 1 L ),\\n\\n(χ ν λ ) 2 = (a 1 -1) 2 for |λ| = n≥1 n a n = L+2 0 otherwise . ( 2\\n\\n.19) Using (2.16), (2.19) with (2.18) in (2.17) then we obtain\\n\\nZ U(N) (t) = λ |λ|≤N +2 f λ (t) - λ N +1≤|λ|≤N +2 1 z λ f λ (t) - λ |λ|=N +2 1 z ′ λ f λ (t) + λ |λ|>N +2 1 z λ f λ (t) µ⊢|λ| ℓ(µ)≤N χ µ λ 2 , (2.20)\\n\\nwhere, in the frequency representation of λ,\\n\\n1 z ′ λ = (a 1 -1) 2 z λ = 1 a 1 ! - 1 (a 1 -1)! + 1 (a 1 -2)! ∞ n=2 n a n a n ! .\\n\\n(2.21)\\n\\nWe may proceed in this manner to compute explicit higher order corrections though this becomes cumbersome save for the first few cases as shown. (For |λ| > N + 2 the corrections will always involve contributions from (2.16) and (2.19) as well as extra ones coming from\\n\\nµ⊢|λ| ℓ(µ)≤|λ| χ µ λ 2 - µ⊢|λ| ℓ(µ)≤N χ µ λ 2 .)\\n\\nThe One Boson Case for U (N )\\n\\nFor the case of one bosonic fundamental field (applicable to half BPS operators for N = 4 super Yang Mills), we have f (t) = t in (2.5), so that we may write\\n\\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) N j,k=1 1 1 -tz j z k -1 . (2.22)\\n\\nTo evaluate this integral we may use the Cauchy-Littlewood formula,\\n\\nL i=1 M j=1 1 1 -x i y j = λ ℓ(λ)≤min.{L,M } s λ (x 1 , . . . , x L )s λ (y 1 , . . . , y M ) , (2.23)\\n\\nwhere the sum on the right-hand side is over all partitions λ such that the corresponding Young diagrams have no more than min.{L,\\n\\nM } rows, ℓ(λ) ≤ min.{L, M }. With x i = tz i , y i = z i -1 , i = 1, . . . , N , in (2.23\\n\\n), so that s λ (tz 1 , . . . , tz N ) = t |λ| s λ (z), and employing also (2.6 ) and the orthonormality of Schur polynomials, we may easily obtain,\\n\\nZ U(N) (t) = λ ℓ(λ)≤N t |λ| s λ , s λ N = λ ℓ(λ)≤N t |λ| . (2.24)\\n\\nBy changing summation variables so that\\n\\nλ i -λ i+1 = a i , i = 1, . . . , N -1, λ N = a N then we may write Z U(N) (t) = ∞ a 1 ,...,a N =0 t a 1 +2a 2 +...+Na N = P N (t) , (2.25)\\n\\nwhere 4\\n\\nP N (t) = N i=1 1 1 -t i . (2.26)\\n\\nOf course this is nothing other than the generating function for the number p N (n) of partitions of n into no more than N parts since, by definition,\\n\\nλ ℓ(λ)≤N δ |λ|,n = p N (n) , (2.27)\\n\\nso that by the above\\n\\n∞ n=0 p N (n)t n = P N (t) .\\n\\n(2.28)\\n\\n4 1/P ∞ (t) = ∞ n=1 (1-t n ) is commonly called the Euler function, denoted by Φ(t). P ∞ (t) = ∞ n=0 p(n)t n\\n\\nacts as a generating function for the number of unordered partitions of n, p(n). Note that p N (n) = p(n) for n ≤ N , i.e. the number of partitions of n into no more than N parts is the same as the total number of partitions of n so long as n ≤ N .\\n\\nThis makes explicit the connection between Z U(N) (t) and the partition numbers p N (n).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The SU (2) Gauge Group Case\",\n \"text\": \"Here we first consider f (t) = k j=1 t j in (2.1) so that the variables 0 ≤ t i < 1 represent k bosons in the single particle partition function. For such fields transforming in the adjoint representation of SU (2) then (2.2) simplifies significantly. For any U ∈ SU (2) we may write U = V ΘV † , where V is unitary and Θ = diag.(e iθ , e -iθ ), for 0 ≤ θ < 2π, so that for F (U ) = F (θ), then in usual Weyl parametrisation,\\n\\nSU(2) dµ SU(2) (U )F (U ) = 1 π 2π 0 dθ sin 2 θ F (θ) = 1 4π 4π 0 dθ 1 -cos θ F ( θ 2 ) = 1 2π 2π 0 dθ 1 -cos θ F ( θ 2 ) , (2.29)\\n\\nwhere F (θ) = F (θ + π) is assumed in writing the last line. In the present case,\\n\\nF (U ) = F (θ) = n≥1 f (t n )χ Adj. (U n )/n, where χ Adj. (U ) = tr(U )tr(U † ) -1 = e 2iθ + e -2iθ + 1 = 2 cos 2θ + 1, so that Z SU(2) (t) = 1 2π 2π 0 dθ (1 -cos θ) k j=1 1 (1 -t j )(1 -t j e iθ )(1 -t j e -iθ )\\n\\n.\\n\\n(2.30)\\n\\nMaking the variable change z = e iθ , and using that F (θ) = F (-θ) is even, then\\n\\nZ SU(2) (t) = 1 2πi dz z (1 -z) k j=1 1 (1 -t j )(1 -t j z)(1 -t j z -1 ) , (2.31)\\n\\nwhere the integral is around the unit circle |z| = 1. The residues in (2.31) may be easily computed since all the relevant (simple) poles in the disc |z| < 1 occur at the points z = t j . Thus\\n\\nZ SU(2) (t) = k i=1 t i k-1 1 -t i 2 k j=1 j =i 1 (t i -t j )(1 -t i t j )(1 -t j ) . (2.32)\\n\\nThis partition function has an interesting interpretation from a group theory perspective. We may write,\\n\\n1 + t (1 -tz)(1 -tz -1 ) = ∞ n=0 χ n (z)t n , (2.33)\\n\\nwhere\\n\\nχ j (z) = z j+ 1 2 -z -j-1 2 z 1 2 -z -1 2 , j ∈ 1 2 Z , (2.34)\\n\\nis an SU (2) character, corresponding to the spin j irreducible representation, R j . Now the integral in (2.31) acts as an SU (2) inner product,\\n\\nχ j , χ k = 1 2πi dz z (1 -z)χ j (z)χ k (z -1 ) = δ jk , (2.35)\\n\\nfor j, k ∈ N. Thus, from (2.31) with (2.33) and (2.35),\\n\\nk j=1 (1 -t j 2 )Z SU(2) (t) = ∞ n 1 ,...,n k =0 χ n 1 • • • χ n k , 1 t 1 n 1 • • • t k n k , (2.36)\\n\\nacts as a generating function for the number of singlets in the decomposition of the 5 By using that χ n (z) = n j=-n z j we may use the Cauchy residue theorem to compute explicitly that\\n\\nSU (2) representation R n 1 ⊗ • • • ⊗ R n k .\\n\\nχ n 1 • • • χ n k , 1 = 2n 1 j 1 =0 • • • 2n k j k =0 (δ j 1 +•••+j k ,n 1 +•••+n k -δ j 1 +•••+j k ,n 1 +•••+n k +1 ) .\\n\\n(2.37)\\n\\nIf we modify the one particle partition function to include k bosons and k fermions and hence consider (2.1) in the form f (t, t) = k j=1 t j -k =1 t then we may similarly as above evaluate\\n\\nZ SU(2) (t, t) = 1 2πi dz z (1 -z) 1≤j≤k 1≤≤ k (1 -t )(1 -t z)(1 -t z -1 ) (1 -t j )(1 -t j z)(1 -t j z -1 ) , (2.38)\\n\\nwhere the contour is around the unit disc |z| = 1. So long as k > k then (2.38) receives contributions from only those simple poles at z = t j so that for this case we obtain\\n\\nZ SU(2) (t, t) k> k = k i=1 t i k-k-1 1 -t i 2 1≤≤ k 1≤j≤k,j =i (t i -t )(1 -t i t )(1 -t ) (t i -t j )(1 -t i t j )(1 -t j ) . (2.39)\\n\\nFor k ≤ k then (2.38) also receives contributions from poles at z = 0. For instance\\n\\nZ SU(2) (t, t) k= k = k i=1 1 t i (1 -t i 2 ) 1≤j,≤k j =i (t i -t )(1 -t i t )(1 -t ) (t i -t j )(1 -t i t j )(1 -t j ) + 1≤j,≤k t (1 -t ) t j (1 -t j ) , (2.40)\\n\\n5 Generating functions for products of Lie algebra representations have been considered elsewhere, in [39] for instance. A generating function for the number of singlets in n products of the fundamental times n products of the anti-fundamental representations for SU (N ) was found by Gessel [40] in terms of Toeplitz determinants involving Bessel functions. See also [41] for a nice physics oriented discussion of similar issues. The special case of R 1 2\\n\\n⊗ • • • ⊗ R 1 2\\n\\n(2n products of the fundamental) for SU (2), contains a Catalan number, 1 n+1 2n n , of singlet representations.\\n\\nwhere the last term on the right-hand side of (2.40) comes from the simple pole at z = 0. These formulae should be useful for computing the multi-trace partition functions, for fundamental fields transforming in an SU (2) gauge group, in other sectors of N = 4 super Yang Mills. For instance, after a suitable identification of the variables t j , t with variables in single particle partition functions for semi-short sectors of N = 4 super Yang Mills, described in detail in [33] , then (2.40) should allow for an explicit expression for corresponding multi-trace partition functions. They may also be useful for computing the N = 4 superconformal index of [1] for SU (2) gauge group, or at least for restrictions of it such as described in [33] or [42] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Counting Operators in Free N = 4 Super Yang Mills\",\n \"text\": \"In this section, the counting of half and quarter BPS operators for free N = 4 super Yang Mills, when the fundamental fields transform in the adjoint representation of U (N ), is discussed in some detail.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Counting Operators Directly\",\n \"text\": \"We may, of course, proceed to count multi-trace half and quarter BPS primary operators directly, in terms of the fundamental fields, Z, for half BPS operators and Z, Y , for quarter BPS operators.\\n\\n(Z, Y ) forms a U (2) doublet, where U (2) has generators given by a subset of the SU (4) R generators, H i , E i± , 1 ≤ i ≤ 3, where H i are the Cartan sub-algebra generators and E i± are ladder operators satisfying (in the Chevalley-Serre basis)\\n\\n[H i , E j± ] = ±K ij E j± , with [K ij ] being the usual SU (4) Cartan matrix. The U (2) generators consist of the SU (2) generators H 2 , E 2± , where explicitly [(H 1 , H 2 , H 3 ), E 2± ] = ∓(1, -2, 1)E 2± , along with the generator H 1 +H 2 +H 3 , whose eigenvalues give the conformal dimensions in this case, [H 1 +H 2 +H 3 , (Z, Y )] = (Z, Y ) [33]. Explicitly, we have that [E 2+ , Z] = 0, [E 2-, Z] = Y , [(H 1 , H 2 , H 3 ), Z] = (0, 1, 0)Z, [(H 1 , H 2 , H 3 ), Y ] = (1, -1, 1)Y so that an operator involving n Z's and m Y 's transforms in the [m, n-m, m] SU (4) R R-symmetry representation.\\n\\nFor k-trace half BPS primary operators transforming in the [0, n, 0] SU (4) R Rsymmetry representation, with conformal dimension n, then in terms of the fundamental field Z a basis is provided by,\\n\\ntr(Z n 1 ) • • • tr(Z n k ) , k i=1 n i = n . (3.1)\\n\\nWe have that, due to trace identities for finite N , tr(Z n ) for n > N is expressible in terms of a sum over multi-trace operators of the form (3.1), for k > 1, and thus, a minimal basis for multi-trace half BPS primary operators consists of (3.1) for all 1 ≤ k ≤ n and with every n i ≤ N , ordered so that n 1 ≥ n 2 ≥ . . . ≥ n k ≥ 0, i.e. so that (n 1 , . . . , n k ) is a partition of n where each part n i ≤ N . With this restriction, the number of multi-trace half BPS primary operators for a given n is\\n\\nN (n) = p N (n) , (3.2)\\n\\nsince the number p N (n), in (2.27), of partitions of n into ≤ N parts is the same as the number of partitions of n in which each part is ≤ N -see [43] for a simple proof employing generating functions.\\n\\nFor quarter BPS operators belonging to the [m, n-m, m] SU (4) R R-symmetry representation, a basis for k-trace operators is\\n\\ntr j Z n 1j Y m 1j • • • tr j Z n kj Y m kj , i,j n ij = n , i,j m ij = m , (3.3)\\n\\nwhere there is a choice of ordering in each trace. (Note that the m = 0 case corresponds to the half BPS case already considered.) Using the basis provided by (3.3) for all allowable k, then to avoid over-counting of multi-trace quarter BPS operators, the cyclicity of each trace and also trace identities for finite N must be accounted for. Assuming that this is done, let M (n,m) denote the number of elements in this minimal basis for multi-trace operators of the form (3. The number of such U (2) descendants coincides with N (n+m-j,j) , the number of corresponding primary operators. In this way, we obtain that\\n\\nM (n,m) = N (n,m) + N (n+1,m-1) + . . . + N (n+m-1,1) + N (n+m) , (3.4)\\n\\nso that N (n,m) = M (n,m) -M (n+1,m-1) may be obtained recursively for each m.\\n\\nWe may illustrate by counting all multi-trace quarter BPS primary operators in the [1, n-1, 1] R-symmetry representation. In this case a basis for k+1-trace operators is provided by\\n\\ntr(Z n 1 ) • • • tr(Z n k )tr(Z j Y ) , k i=1 n i = n -j . (3.5)\\n\\nCyclicity of traces implies that we may arrange Y as shown, to avoid over-counting. U (N ) trace identities imply, similarly as for the half BPS case, that a minimal basis for multitrace operators requires j < N and each n i ≤ N in (3.5) for every 1 ≤ k ≤ n-j, so that (n 1 , . . . , n k ) forms a partition of n-j, with every part ≤ N . Thus, by a similar argument as for the half BPS case, M (n,1) = N-1 j=0 p N (nj). Finally, to ensure that only primary operators are counted then we must subtract off contributions from descendants of half BPS primary operators in the [0, n+1, 0] SU (4) R representation, of which there are p N (n+1). Using (3.4) with (3.2) we then conclude that\\n\\nN (n,1) = N-1 j=0 p N (n -j) -p N (n+1) , (3.6)\\n\\ngives the number of multi-trace quarter BPS primary operators in the [1, n-1, 1] Rsymmetry representation.\\n\\nCounting in this fashion becomes more difficult for greater m and now a procedure is described employing symmetric polynomials to find a generating function for the numbers of multi-trace quarter BPS primary operators in the [m, n-m, m] SU (4) R representation, for m = 0, 1, 2 at finite N and for any n, m at large N . This generating function is subsequently used to provide asymptotic counting for fixed m, large n in the large N limit.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Counting Operators via Expansion of Partition Functions in Schur Polynomials\",\n \"text\": \"For k bosonic fundamental fields, we may take f (t) = k j=1 t j in (2.1) so that (2.5) may be written as\\n\\nZ U(N) (t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) k j=1 N r,s=1 1 1 -t j z r z s -1 . (3.7)\\n\\nOften it is the case that such partition functions should be expanded in terms of s λ (t), the k variable Schur polynomial labelled by partitions λ. An example is provided by (1.1) for counting multi-trace quarter BPS operators. We may use the Cauchy-Littlewood formula (2.23) to expand in this way, to obtain\\n\\nZ U(N) (t) = λ ℓ(λ)≤min.{k,N 2 } N λ s λ (t) , (3.8)\\n\\nwhere\\n\\nN λ = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s λ (zz -1 ) , (3.9)\\n\\nwhere zz -1 has components z i z j -1 , 1 ≤ i, j ≤ N .\\n\\nFrom Macdonald [24] we have that\\n\\ns λ (xy) = µ,ν⊢|λ| γ λ µν s µ (x)s ν (y) , (3.10)\\n\\nin terms of Kronecker coefficients,\\n\\nγ λ µν = 1 |λ|! σ∈S |λ| χ λ (σ)χ µ (σ)χ ν (σ) = ρ⊢|λ| 1 z ρ χ λ ρ χ µ ρ χ ν ρ , (3.11)\\n\\nbeing a sum over irreducible S |λ| characters evaluated at σ ∈ S |λ| , related to a sum over irreducible S |λ| characters evaluated on the conjugacy classes labelled by the partitions ρ in the second line. Using (2.6) along with the orthonormality property of Schur polynomials we find that\\n\\nN λ = µ⊢|λ| ℓ(µ)≤N γ λ µµ .\\n\\n(3.12)\\n\\nThe situation becomes much more involved if we include also k fermionic fields, so that (2.1) may be written in the form f (t, t) = k j=1 t j -k =1 t , and attempt to expand Z U(N) (t, t) in terms of products of Schur polynomials s λ (t)s µ ( t). Such expansion is required for counting, for instance, for the free field partition function in the eighth BPS sector of N = 4 super Yang Mills. In this case the partition function is expanded, analogous to (1.1), in terms of SU (2|3) characters, which may be expressed in terms of a linear combination of products of two-variable and three-variable Schur polynomials. (See [33] for a discussion of counting for the eighth BPS sector along these lines.) Including fermions, (3.7) becomes modified by\\n\\nZ U(N) (t, t) = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) 1≤j≤k 1≤≤ k N r,s=1 1 -t z r z s -1 1 -t j z r z s -1 . (3.13)\\n\\nTo achieve the expansion, we may use the Cauchy-Littlewood formula (2.23) along with another formula of Littlewood,\\n\\nL i=1 M j=1 (1 + x i y j ) = λ ℓ(λ)≤L,ℓ( λ)≤M s λ (x 1 , . . . , x L ) s λ(y 1 , . . . , y M ) , (3.14)\\n\\nwhere λ is the partition conjugate to λ (where the rows and columns of the Young diagram corresponding to λ are interchanged) and where the sum is restricted to those λ whereby the corresponding Young diagrams have at most L rows, ℓ(λ) ≤ L, and M columns, ℓ( λ) ≤ M . We may thus write\\n\\nZ U(N) (t, t) = λ ℓ(λ)≤min.{k,N 2 } µ ℓ(µ)≤ k,ℓ( μ)≤N 2 N λ,µ s λ (t) s µ ( t) , (3.15)\\n\\nwhere\\n\\nN λ,µ = (-1) |µ| (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s λ (zz -1 )s μ(zz -1 ) . (3.16)\\n\\nObviously these numbers are considerably more involved than those in (3.9). We may of course use (3.10) again to interpret (3.16) in terms of Kronecker coefficients.\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"Counting Quarter BPS Operators by Symmetric Polynomial Methods\",\n \"text\": \"The two bosonic fundamental field case is now focused upon. 6 In particular, the numbers N (n,m) in (1.1) are evaluated using results of the last sub-section.\\n\\nWe may proceed to evaluate N λ recursively. The simplest case is for N λ = N (n) , whereby introducing a formal variable t then it is clear, by (2.23) with (2.22), (2.25) and (2.26), that ∞ n=0\\n\\nN (n) t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) N j,k=1 1 1 -tz j z k -1 = P N (t) , (3.17)\\n\\nso that, by (2.28), N (n) is given by (3.2).\\n\\nMore generally to evaluate N (n,m) from (3.9) we may use, for y = zz -1 ,\\n\\ns (m) (y)s (n) (y) = s (n,m) (y) + s (n+1,m-1) (y) + . . . + s (n+m-1,1) (y) + s (n+m) (y) , s (m) (zz -1 ) = µ⊢m ℓ(µ)≤N s µ (z)s µ (z -1 ) , (3.18)\\n\\n6 The two boson case leads to an interesting generalisation of an identity in [44] involving Littlewood-Richardson coefficients c ν λµ , the coefficients that appear in the decomposition s λ (x)s µ (x) = ν c ν λµ s ν (x). With f (t) in (2.1) given by f (t) = t 1 + t 2 , and expanding appropriately the corresponding integrand in (3.7) using (2.23); then using (2.6), the orthonormality of Schur polynomials and the result (2.14), we obtain (note that\\n\\nc ν λµ = 0 if |ν| = |λ| + |µ|) Z U(∞) (t 1 , t 2 ) = λ,µ t 1 |λ| t 2 |µ| s λ s µ , s λ s µ ∞ = λ,µ,ν ν⊢|λ|+|µ| t 1 |λ| t 2 |µ| (c ν λµ ) 2 = n≥1 1 1 -t 1 n -t 2 n\\n\\nwhich reduces to Theorem 4.1 of [44] if we take t 1 = t 2 = t.\\n\\nwhere the expression in the first line of (3.18) may be easily seen using Young tableaux multiplication rules while (2.23) determines the expression in the second line. From (3.9) with (3.18), we may find a useful generating function, in terms of a formal variable t, for the numbers in (3.4) as follows,\\n\\nF (m) N (t) = ∞ n=0 M (n,m) t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) s (m) (zz -1 ) ∞ n=0 s (n) (zz -1 )t n = 1 (2πi) N N ! N i=1 dz i z i ∆(z)∆(z -1 ) µ⊢m ℓ(µ)≤N s µ (z)s µ (z -1 ) N j,k=1 1 1 -tz j z k -1 = µ⊢m ℓ(µ)≤N λ t |λ| s λ s µ , s λ s µ N , (3.19)\\n\\nso that we may write\\n\\nN (n,m) = 1 2πi dt t 1 t n F (m) N (t) - 1 t n+1 F (m-1) N (t) , (3.20)\\n\\nwhich allows for recursive determination of N (n,m) .\\n\\nApplying this to the case of N λ = N (n,1) we have, from (3.18)\\n\\ns (1) (zz -1 ) = s (1) (z)s (1) (z -1 ) , (3.21) so that, from (3.19), F (1)\\n\\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = λ ℓ(λ)≤N t |λ| s (1) s λ , s (1) s λ N . (3.22)\\n\\nUsing (again, this may be easily seen from Young tableaux multiplication rules)\\n\\ns (1) (z)s λ (z) = N r=1 s λ+e r (z) , (3.23)\\n\\nfor {e r ; 1 ≤ r ≤ N, e r • e s = δ rs } being usual orthonormal vectors, we find that\\n\\nF (1)\\n\\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = λ ℓ(λ)≤N t |λ| N r,s=1\\n\\ns λ+e r , s λ+e s N .\\n\\n(3.24)\\n\\nNow for any partition λ, s λ+e r , s λ+e s N vanishes unless e r = e s for any r, s and λ r-1 -λ r > 0 for r = 2, . . . , N , due to\\n\\ns (λ 1 ,...,λ r-1 ,λ r +1,...,λ N ) (z) = 0 for λ r-1 = λ r , r > 1 . (3.25)\\n\\nChanging summation variables to those in (2.25) then we have, with the definition (2.26), F\\n\\nN (t) = ∞ n=0 N (n,1) + N (n+1) t n = a 1 ,...,a N ≥0 t a 1 +...+Na N + N-1 r=1 a 1 ,...,a N ≥0 a r ≥1 t a 1 +...+Na N = N-1 i=0 t i a 1 ,...,a N ≥0 t a 1 +...+Na N = 1 1 -t P N-1 (t) . (1)\\n\\n(3.26) Thus, using (2.28), (3.2) with (3.26), 7\\n\\nN (n,1) = n j=0 p N-1 (j) -N (n+1) = n j=0 p N-1 (j) -p N (n + 1) . (3.27)\\n\\nFor the case of N λ = N (n,2) we have that, from (3.18),\\n\\ns (2) (zz -1 ) = s (2) (z)s (2) (z -1 ) + s (1,1) (z)s (1,1) (z -1 ) , (3.28)\\n\\nso that, from (3.19), we have\\n\\nF (2) N (t) = λ ℓ(λ)≤N t |λ| s (2) s λ , s (2) s λ N + s (1,1) s λ , s (1,1) s λ N . (3.29) Using s (2) (z)s λ (z) = N r=1 s λ+2e r (z) + 1≤r<s≤N s λ+e r +e s (z) , s (1,1) (z)s λ (z) = 1≤r<s≤N s λ+e r +e s (z) , (3.30)\\n\\n7 This formula agrees with (3.6) due to\\n\\nN -1 j=0 p N (n -j) = n j=0 p N -1 (\\n\\nj) which follows because the corresponding generating functions match,\\n\\n∞ n=0 N -1 j=0 p N (n -j)t n = (1 + t + . . . + t N -1 )P N (t) = 1 1 -t P N -1 (t) = ∞ n=0 n j=0 p N -1 (j)t n .\\n\\nalong with (3.25) and\\n\\ns (λ 1 ,...,λ r-1 ,λ r +2,...,λ N ) (z) = -s (λ 1 ,...,λ r +1,λ r-1 +1,...,λ N ) (z) , (3.31)\\n\\nfor the cases where λ r-1 = λ r , we may obtain, with the definition (2.26), F\\n\\nN (t) = 1-t N+1 (1-t)(1-t 2 ) P N-1 (t) + 1 (1-t)(1-t 2 ) P N-2 (t) , (2)\\n\\nwhere the first contribution comes from λ t |λ| s (2) s λ , s (2) s λ N while the second comes from λ t |λ| s (1,1) s λ , s (1,1) s λ N . Since the partition number p k (-n) = 0 for n = 1, 2, . . . we may write, using (2.28), 8\\n\\n1 (1 -t)(1 -t 2 ) P k (t) = ∞ n,i,j=0 p k (n -i -2j)t n . (3.33)\\n\\nThus, from (3.32) with (3.27),\\n\\nN (n,2) = - n+1 j=0 p N-1 (j) + ∞ i,j=0 (p N-2 (n-i-2j) + p N-1 (n-i-2j)) if n ≤ N , ∞ i,j=0 (p N-2 (n-i-2j) + p N-1 (n-i-2j) -p N-1 (N +1-n-i-2j)) if n ≥ N +1. (3.34)\\n\\nTables of the numbers (3.2), (3.27) and (3.34) are given in appendix B for some few cases of n, N . Notice from these tables that the numbers N (n,m) below the diagonal line N ≥ n + m for a given n are the same for all N . This is a general feature that derives from values of N (n,m) for N ≥ n + m, which numbers may be obtained from a corresponding generating function that is now constructed.\\n\\nUsing these techniques, we may provide a consistency check of (3.17), (3.26), (3.32) along with a general result for N (n,m) for high enough values of N , N ≥ m + n. This employs the orthogonality property of power symmetric polynomials p λ (z) (in the large N 8 This is a special case of the following: for any f (n), n ∈ Z, that satisfies f (-n) = 0, n = 1, 2, . . ., then we may (at least formally) write\\n\\nP k (t) ∞ n=0 f (n)t n = ∞ n,i 1 ,...,i k =0 f (n -i 1 -2i 2 -. . . -ki k )t n . limit) along with s (n) (z) = λ⊢n 1 z λ p λ (z) = ∞ i 1 ,...,i n =0 1 i 1 !i 2 ! • • • i n ! δ i 1 +2i 2 +...+ni n ,n p 1 (z) i 1 1 2 p 2 (z) i 2 • • • 1 n p n (z) i n . (3.35)\\n\\nUsing the trivial identity p λ (xy) = p λ (x)p λ (y) then from (2.6), (3.19) with (3.35) we have\\n\\nF (m) ∞ (t) = ∞ n=0 λ⊢n µ⊢m 1 z λ z µ p λ p µ , p λ p µ ∞ t n = ∞ n=0 λ⊢n µ⊢m 1 z λ z µ p ν , p ν ∞ t n = ∞ n=0 λ⊢n µ⊢m z ν z λ z µ t n , (3.36)\\n\\nwhere for (1 a 1 , 2 a 2 , . . .) being the frequency representation of λ and (1 b 1 , 2 b 2 , . . .) being that of µ then ν has frequency representation (1 a 1 +a 2 , 2 a 2 +b 2 , . . .) so that |ν| = n + m. This agrees with F\\n\\nN (t) in a series expansion up to O(t N-m ) (since the last equation in (3.36) is also valid for finite N so long as |ν| = n + m ≤ N , by a result of appendix A). Now, since\\n\\nz ν z λ z µ = ∞ j=1 (a j +b j )! a j !b j ! , (3.37)\\n\\nwe obtain from (3.36) that,\\n\\nF (m) ∞ (t) = ∞ n=0 ∞ a 1 ,...,a n =0 ∞ b 1 ,...,b m =0 δ a 1 +•••+na n ,n δ b 1 +•••+mb m ,m m j=1 (a j + b j )! a j !b j ! t n = ∞ b 1 ,...,b m =0 δ b 1 +•••+mb m ,m m j=1 ∞ a j =0 (a j + b j )! a j !b j ! t ja j j>m 1 1 -t j = ∞ b 1 ,...,b m =0 δ b 1 +•••+mb m ,m m j=1 1 (1 -t j ) b j +1 j>m 1 1 -t j . (3.38)\\n\\nFor the first few cases we have that, with P N (t) as defined in (2.26),\\n\\nF (m) ∞ (t) = P ∞ (t) for m = 0 1 1-t P ∞ (t) for m = 1 2 (1-t)(1-t 2 ) P ∞ (t) for m = 2 , (3.39)\\n\\nwhose series expansion agrees with (3.17), (3.26), (3.32) up to O(t N-m ) for, respectively, m = 0, 1, 2. We may use (3.20) with (3.36) to determine N (n,m) exactly for N ≥ n + m.\\n\\nAsymptotic Counting of Quarter BPS Operators at Large N Asymptotic counting for the one boson case in the large N limit, for which, with P N (t) as defined in (2.26), with p(n) being the total number of (unordered) partitions of n,\\n\\nZ U(∞) (t) = P ∞ (t) = ∞ n=1 p(n)t n , (3.40)\\n\\nis the multi-trace partition function, entails finding an asymptotic value for the partition number p(n) for 'large' n. This may be achieved by performing a saddle point approxima-\\n\\ntion of p(n) = 1 2πi dtP ∞ (t)t -n-1 .\\n\\nThe function P ∞ (t) has a 'large' singularity at t = 1, but in addition has singularities at all other roots of unity -see [45] on the validity of ignoring these contributions asymptotically. This method was used by Hardy and Ramanujan to find their celebrated formula, here given in a less detailed form as,\\n\\np(n) ∼ 1 4n √ 3 exp π 2 3 n , (3.41)\\n\\nwhich was improved by Rademacher to give p(n) exactly. Their method relied crucially on the modular properties of P ∞ (t).\\n\\nFocusing now on the two bosonic fundamental field case for which, in the large N limit, (3.20) with (3.38) gives exact counting, at issue is first finding asymptotic values for the numbers\\n\\nQ(n, m, b) = Q(n, m, b 1 , . . . , b m ), with constraint equation m j=1 jb j = m, defined by m j=1 1 (1 -t j ) b j +1 j>m 1 1 -t j = 1 + ∞ n=1 Q(n, m, b)t n . (3.42)\\n\\nHaving found these we may then attempt to find the dominant contribution to (3.20) with (3.38) for large N . In order to give asymptotic values for Q(n, m, b) we may follow [46] and apply a formula due to Meinardus which gives a general result for the generating function\\n\\n∞ n=1 (1 -t n ) -a n = 1 + ∞ n=1 r(n)t n . (3.43)\\n\\nA detailed version of Meinardus' theorem may be found in [46] but for purposes of brevity we may note that it implies that, as n → ∞,\\n\\nr(n) ∼ C n κ exp AΓ(α + 1)ζ(α + 1)n α 1/(α+1) (α + 1)/α , (3.44)\\n\\nwhere ζ(s) = ∞ j=1 j -s is the Riemann zeta function and the constants C, κ, α, A are determined by the auxiliary Dirichlet series,\\n\\nD(s) = ∞ j=1 a j j s , (3.45)\\n\\nwhich must converge for Re(s) > α, a positive real number, and possess an analytic continuation in the region Re(s) ≥ c, -1 < c < 0, such that, in this region, D(s) is analytic except at a simple pole at s = α where it has residue A. In terms of α, A then\\n\\nC = 1 2π(1+α) AΓ(α + 1)ζ(α + 1) (1-2D(0))/2(α+1) exp D ′ (0) , κ = (D(0) -1 -1 2 α)/(α + 1) . (3.46)\\n\\nApplying Meinardus' theorem to the case of (3.42), clearly we have\\n\\nD(s) = m j=1 b j j s + ζ(s) , (3.47)\\n\\nso that, assuming m = m j=1 jb j is fixed, D(s) has a simple pole at s = α = 1 where it has residue A = 1. Using\\n\\nD(0) = -1 2 + m j=1 b j , exp D ′ (0) = 1 √ 2π m j=1 1 j b j , (3.48)\\n\\nthen, from (3.44) with (3.46), we may easily determine that, as n → ∞,\\n\\nQ(n, m, b) ∼ 1 4n √ 3 √ 6n π m j=1 b j m j=1 1 j b j exp π 2 3 n . (3.49)\\n\\nThis reduces to (3.41) when b j = 0, 1 ≤ j ≤ m, whereby Q(n, 0, . . . , 0) = p(n). Using (3.20) for (3.38) with (3.42) and (3.49) then, as n → ∞,\\n\\nN (n,m) = b 1 ,...,b m ≥0 m j=1 jb j =m Q(n, m, b) - b 1 ,...,b m-1 ≥0 m-1 j=1 jb j =m-1 Q(n+1, m-1, b) ∼ 1 4n √ 3 √ 6n π m exp π 2 3 n , (3.50)\\n\\nsince Q(n, m, m, 0, . . . , 0), for b 1 = m, b j = 0, j > 1, dominates over all other terms in (3.50) . This gives asymptotic values for the numbers in (1.1), for counting quarter BPS operators, transforming in [m, n-m, m] SU (4) R representations, in the large N limit of free N = 4 super Yang Mills, as previously described.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Counting Operators in the Chiral Ring of N = 4 Super Yang Mills\",\n \"text\": \"For the purposes of counting operators in the chiral ring of N = 4 super Yang Mills, we denote corresponding multi-trace partition functions by C U(N) (t).\\n\\nThe generating function for C U(N) (t) for the case of one bosonic fundamental field has been written in the form [1, 3, 9 ]\\n\\nC(ν, t) = ∞ n=0 1 1 -νt n = ∞ N=0 ν N C U(N) (t) , (4.1)\\n\\nso that ν acts as a chemical potential for the rank of the gauge group U (N ). The equivalence C U(N) (t) = Z U(N) (t) = P N (t), with Z U(N) (t) as in (2.25) , is actually a special case of the q-Binomial theorem. Writing -see [43] for notation -\\n\\n(a; q) k = (1 -a)(1 -aq) • • • (1 -aq k-1 ) , (4.2)\\n\\nthen the q-Binomial theorem is, for\\n\\n|x|, |q| < 1, ∞ k=0 (a; q) k (q; q) k x k = (ax; q) ∞ (x; q) ∞ . (4.3)\\n\\n(Identifying ν = x and q = t and setting a = 0 in (4.3), so that 1/(ν; t) ∞ = C(ν, t) above and 1/(t; t) N = P N (t) in (2.26), then C U(N) (t) = Z U(N) (t) = P N (t) straightforwardly. This special case of the q-Binomial theorem is due to Euler.)\\n\\nFor the two boson case, so that the single particle partition function is given by f (t, u) = t + u for some t, u, then the generating function for the finite N chiral ring partition function C U(N) (t, u) is given by [1, 3, 9]\\n\\nC(ν, t, u) = ∞ n,m=0 1 1 -νt n u m = ∞ N=0 ν N C U(N) (t, u) . (4.4)\\n\\nThis function is more difficult to analyse in terms of counting though has been investigated by Stanley [47] in relation to partitions -there it has been dubbed the 'double Eulerian' generating function. Through use of the Cauchy-Littlewood formula, then we may expand C U(N) (t, u) in terms of partitions of N as,\\n\\nC U(N) (t, u) = λ⊢N h λ (t)h λ (u) , (4.5)\\n\\nwhere h λ (t) = s λ (1, t, t 2 , . . .) , (4.6) so that using an identity for Schur polynomials to be found in [24, 47] then\\n\\nC U(N) (t, u) = λ⊢N 1≤i<j≤N (1 -t λ i -λ j +j-i )(1 -u λ i -λ j +j-i ) N i=1 (t; t) λ i +N -i (u; u) λ i +N -i (tu) N i=1 (i-\\n\\n1)λ i . (4.7) (4.5) with (4.6) has a natural interpretation in terms of plane partitions in that, for π being all column-strict plane partitions of shape λ, |π| = i,j π ij , 9 h λ (t) = s λ (1, t, t 2 , . . .) = π t |π| . (4.8) Obviously, (4.5) with (4.8) generalise for other chiral ring sectors. (For a different connection between the 'double Eulerian' generating function and major indices of permutations see [47], p. 385.) As an illustration of (4.5) with (4.8), we may consider the case N = 2 whereby λ = (2, 0), (1, 1) gives the two possible partitions of 2. For λ = (2, 0) (corresponding to a Young diagram with a single row of two boxes) π 11 ≥ π 12 ≥ 0 gives all column-strict plane partitions of shape (2, 0), while for λ = (1, 1) (corresponding to a Young diagram with a single column of two boxes) then π 11 > π 21 ≥ 0 gives all columnstrict plane partitions of shape (1, 1). Thus, h (2,0) (t) = π 11 ,π 12 ≥0 π 11 ≥π 12 t π 11 +π 12 = 1 (1t)(1t 2 ) , h (1,1) (t) = π 11 ,π 21 ≥0 π 11 >π 21 t π 11 +π 21 = t (1t)(1t 2 ) ,\\n\\nso that, from (4.5) for N = 2,\\n\\nC U(2) (t, u) = h (2,0) (t)h (2,0) (u) +h (1,1) (t)h (1,1) (u) = 1 + ut (1 -t)(1 -t 2 )(1 -u)(1 -u 2 ) , (4.10)\\n\\nwhich is the correct result as may be verified by extracting the ν 2 coefficient in an expansion of (4.4) up to O(ν 2 ).\\n\\n9 See [47] for a detailed description of plane partitions. Briefly, a column-strict plane partition of shape λ is an array π = (π ij ) of non-negative integers with finitely many non-zero entries, that is arranged in a Young tableaux with shape λ -see appendix A -such that the numbers π ij are weakly decreasing along each row, π ij ≥ π i j+1 ≥ 0, and strictly decreasing down each column, π ij > π i+1 j ≥ 0. The sum of the parts of π is given by |π| = i,j π ij . (Note that in contrast to the definition in [47] , here we are allowing π ij = 0, for some i, j, to be a part of the plane partition π with shape λ.)\\n\\nIn the large N limit,\\n\\nC U(∞) (t, u) = n 1 ,n 2 ≥0 n 1 +n 2 >0 1 1 -t n 1 u n 2 , (4.11)\\n\\nupon which attention is shortly focused.\\n\\nFor the numbers N (n,m) → N (n,m) counting quarter BPS primary operators for the chiral ring of N = 4 super Yang Mills, belonging to [m, nm, m] SU (4) R R-symmetry representations, as in (1.1), we have foot_1\\n\\nN (n,m) = 1 8π 2 dt du C U(N) (t, u) s (n,m) (t -1 , u -1 ) (t -1 -u -1 ) 2 . (4.12)\\n\\nThese may be more conveniently evaluated in terms of the numbers in (3.4) M (n,m) → M (n,m) , counting all chiral ring quarter BPS operators in the [m, n-m, m] SU (4) R representation, given by\\n\\nM (n,m) = 1 (2πi) 2 dt du C U(N) (t, u) t -n-1 u -m-1 , (4.13)\\n\\nso that N (n,m) = M (n,m) -M (n+1,m-1) . Defining P λ (n) to be the number of column-strict plane partitions π of shape λ so that |π| = i,j π ij = n, then, from (4.5) with (4.8) and (4.13), M (n,m) = λ⊢N P λ (n)P λ (m). Thus,\\n\\nN (n,m) = λ⊢N (P λ (n)P λ (m) -P λ (n+1)P λ (m-1)) , (4.14)\\n\\ncounts chiral ring quarter BPS primary operators in SU (4) R representations [m, nm, m] for any n, m at finite N .\\n\\nAsymptotic Counting for Chiral Ring BPS Operators at Large N For asymptotic counting of operators in the chiral ring of N = 4 super Yang Mills at large N , a relatively crude method is employed here which nevertheless captures the exponential behaviour of counting numbers of interest. This method is based on saddle point approximations of functions near a dominant singularity -see [45] for a useful summary. (Often for physical applications in thermodynamics, e.g. for entropy formulae, we are interested only in the exponential behaviour of such numbers anyhow.)\\n\\nTo illustrate, we consider the one boson case in the large N limit again. We first find a convenient 'approximating function' as follows,\\n\\nP ∞ (t) = ∞ n=1 1 1 -t n = exp - ∞ n=1 ln(1 -t n ) ∼ exp - ∞ 0 ds ln(1 -t s ) = exp - π 2 6 ln t , (4.15)\\n\\nwhich has an 'easier' singularity structure. (The approximation in the second step may be justified by the Euler-Maclaurin formula for approximating sums by integrals.) Using (4.15) then for large enough n,\\n\\np(n) = 1 2πi dt P ∞ (t)t -n-1 ∼ 1 2πi dt e g(t) , g(t) = - π 2 6 ln t -n ln t . (4.16)\\n\\nWe may approximate the latter integral for large n by noting that the dominant contribution is at the saddle point t ′ = e -π/ √ 6n ∼ 1 for which (4.18) Thus, ln p(n) ∼ π 2 3 n ,\\n\\ng(t ′ ) = π 2 3 n , g ′ (t ′ ) = 0 , g ′′ (t ′ ) = 2 π √ 6n 3 e π √ 2/3n = α , (4.17\\n\\nwhich captures the correct behaviour of ln p(n) for large n, according to (3.41).\\n\\nWe may proceed analogously for the quarter BPS chiral ring multi-trace partition function at large N , (4.11), which we approximate by\\n\\nC U(∞) (t, u) = n 1 ,n 2 ≥0 n 1 +n 2 >0 1 1 -t n 1 u n 2 = exp - n 1 ,n 2 ≥0 n 1 +n 2 >0 ln(1 -t n 1 u n 2 ) ∼ exp - ∞ 0 ∞ 0 dv dw ln(1 -t v u w ) = exp ζ(3) ln t ln u . (4.20)\\n\\nIn this case we have, from (4.13),\\n\\nM (n,m) ∼ 1 (2πi) 2\\n\\ndt du e g(t,u) , (\\n\\nwhere\\n\\ng(t, u) = ζ(3) ln t ln u -n ln t -m ln u , (4.22)\\n\\nfor n, m large. The dominant contribution to the integral, for n, m both large and of the same order, occurs about the point (t ′ , u ′ ) ∼ (1, 1) where\\n\\nt ′ = e -(ζ(3)mn -2 ) 1/3 , u ′ = e -(ζ(3)nm -2 ) 1/3 , (4.23)\\n\\nfor which\\n\\ng(t ′ , u ′ ) = 3 3 ζ(3)nm , ∂ ∂t g(t, u) (t ′ ,u ′ ) = ∂ ∂u g(t, u) (t ′ ,u ′ ) = 0 , ∂ 2 ∂t 2 g(t, u) (t ′ ,u ′ ) = 2(ζ(3) -1 n 5 m -1 ) 1 3 e 2(ζ(3)mn -2 ) 1/3 = α , ∂ 2 ∂u 2 g(t, u) (t ′ ,u ′ ) = 2(ζ(3) -1 m 5 n -1 ) 1 3 e 2(ζ(3)nm -2 ) 1/3 = β , ∂ 2 ∂t∂u g(t, u) (t ′ ,u ′ ) = (ζ(3) -1 n 2 m 2 ) 1 3 e (ζ(3)mn -2 ) 1/3 +(ζ(3)nm -2 ) 1/3 = γ . (4.24)\\n\\nSo long as m, n are both large and of the same order, the saddle point approximation is justified and we obtain\\n\\nM (n,m) ∼ e 3 3 √ ζ(3)nm 1 4π 2 ∞ -∞ ∞ -∞ dv dw e -1 2 (αv 2 +βw 2 +2γvw) = h(α, β, γ)e 3 3 √ ζ(3)nm , (4.25) where h(α, β, γ) = 1 2π αβ -γ 2 = 1 2π √ 3 (ζ(3)m -2 n -2 ) 1 3 e -(ζ(3)mn -2 ) 1/3 -(ζ(3)nm -2 ) 1/3 . (4.26)\\n\\nWe thus have that for n, m both comparably large,\\n\\nln M (n,m) ∼ 3 3 ζ(3)nm , (4.27) so that ln N (n,m) = ln M (n,m) -M (n+1,m-1) ∼ ln M (n,m) ∼ 3 3 ζ(3)nm . (4.28)\\n\\nIt is difficult to check the consistency of this result given the dearth of literature on these types of multi-variable generating functions and their asymptotic behaviour, however, we may consider the simpler function, also considered in [9] ,\\n\\nC U(∞) (t, t) = ∞ n=1 1 (1 -t n ) n+1 = ∞ r=0 E(r) t r , (4.29)\\n\\nwhere, in terms of the counting numbers N (n,m) , from (1.1), By considering (4.29) directly, we may employ Meinardus' theorem (described in the third section) to find the behaviour of ln E(r) as r → ∞. Note, however that Meinardus' theorem may not be applied directly to (4.29) since the corresponding auxiliary Dirichlet series (3.45), with a j = j+1, has two simple poles. To overcome this difficulty we split (4.29) into a product of two functions, both separately amenable to application of Meinardus' theorem. One is the reciprocal of the Euler function, P ∞ (t) in (4.15) . The other, the MacMahon function, is given by\\n\\nC U(∞) (t, t) = ∞ n=0 n m=0 (n-m+1) N (n,m) t n+m ⇒ E(r) = [ 1 2 r] m=0 (r-2m+1) N (r-m,m) . ( 4\\n\\nM (t) = ∞ n=1 1 (1 -t n ) n = ∞ r=0 q(r) t r , (4.32)\\n\\nand has been considered in a similar context as here in [3] . 11 Writing\\n\\nC U(∞) (t, t) = P ∞ (t)M (t) , (4.33)\\n\\nwith P ∞ (t) as in (4.15), so that, using (4.29),\\n\\nE(r) = r 1 ,r 2 ≥0 r 1 +r 2 =r p(r 1 )q(r 2 ) , (4.34)\\n\\nwe may find the asymptotic behaviour of ln E(r), as r → ∞, by extracting the dominant contribution from (4.34) using the asymptotic behaviour of p(r), q(r). The auxiliary Dirichlet series for M (t) in (4.32) is, from (3.45) with a j = j,\\n\\nD(s) = ζ(s -1) , (4.35)\\n\\n11\\n\\nThe relation of M (t) in (4.32) to plane partitions is given a description in [46] . Briefly, q(r) gives the number of ordinary plane partitions π, so that π ij ≥ π i+1 j > 0, π ij ≥ π i j+1 > 0, with |π| = i,j π ij = r. p(r) < q(r) as ordinary partitions λ are a special case of plane partitions.\\n\\nIn fact, the formula for ln q(r) found here is a special case of a more exact asymptotic formula first found by Wright [48] for the number of plane partitions q(r) of the number r.\\n\\nat the saddle point (t ′ 1 , . . . , t ′ k-1 ) ∼ (1, . . . , 1), where (ln t ′ 1 , . . . , ln t ′ k-1 ) = -k ζ(k) m 1 • • • m k-1 (1/m 1 , . . . , 1/m k-1 ) , (4.41) so that ∂ ∂t j g(t 1 , . . . , t k-1 ) (t ′ 1 ,...,t ′ k-1 ) = 0 , j = 1, . . . , k-1 . (4.42) (4.39) is consistent with a result implied by Meinardus' theorem. The function, 13 C U(∞) (t, . . . , t) ∼ j=1 m j = r, for which the solution is m j = m M = r/(k -1) (relaxing the constraint that m j be non-negative integers, which is irrelevant asymptotically), so that ln M (m M ,...,m M ) ∼ ln c(k, r). This is applicable to counting multi-trace operators in the eighth BPS chiral ring sector for N = 4 super Yang Mills with fundamental fields Z, Y, X involving m 1 Z's, m 2 Y 's, m 3 X's. Expanding the corresponding partition function (4.37), with k = 4, in terms of Schur polynomials s (m 1 ,m 2 ,m 3 ) (t), m 1 ≥ m 2 ≥ m 3 ≥ 0, similar to (1.1), the expansion coefficients N (m 1 ,m 2 ,m 3 ) count spinless multi-trace primary operators transforming in the [m 2 + m 3 , m 1m 2 , m 2m 3 ] SU (4) R R-symmetry representation, with conformal dimensions m 1 +m 2 +m 3 [33] . Just as in (4.28), asymptotically ln N (m 1 ,m 2 ,m 3 ) ∼ ln M (m 1 ,m 2 ,m 3 ) . This counting, however, ignores contributions of the fermionic fields λ, λ, which it may be important to include in order to give correct counting of eighth BPS chiral ring operators. 13 This may be easily seen from (4.37), as the number of solutions to k-1 j=1 m j = n, where m j are non-negative integers, is the binomial number n+k-2 k-2 which, to leading order in large n, behaves like n k-2 /(k-2)!. More properly, we should split the product C U(∞) (t, . . . , t) into pieces separately amenable to Meinardus' theorem, as for the prior case for k = 3, however, just as for that case, the numbers c(k, r) dominate, and so other contributions are ignored here.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions\",\n \"text\": \"There are some obvious questions not answered by this work. The first is whether or not the approach in the second section using symmetric polynomials can give insight into thermodynamics at finite N , such as for the Hagedorn transition, for example. While it gives the large N expression (2.14) in an elementary way, its wider applicability or usefulness to such questions is unclear. The approach is undoubtedly useful for finding exact expressions for counting numbers (as in (3.2), (3.27) and (3.34) for quarter BPS operators) and (3.9), (3.16 ) may be useful for analysing counting for more complicated sectors of N = 4 super Yang Mills, with gauge group U (N ).\\n\\nThe second question is how the arguments employing symmetric polynomial techniques here may be extended to other gauge groups, the most pertinent being perhaps SU (N ). Arguments here employing (2.6) and the orthonormality property of Schur polynomials should remain largely unaffected for SU (N ). Exact values for counting numbers obtained here should require some modification for SU (N ), though asymptotic values may be unchanged.\\n\\nThe third question concerns asymptotic values for counting numbers and how these may be improved. The asymptotic counting formulae given in such papers as [3, 9] for chiral ring sectors are special cases of formulae such as those of Hardy and Ramanujan, Meinardus, etc., all of which derive from single variable generating functions. It is hoped that the expressions (3.50), (4.28), (4.39), given here for asymptotic counting of BPS operators, that distinguishes between differing R-symmetry charges, represents a serious attempt at going beyond consideration of single variable generating functions. 14 Improving upon these formulae will require more sophisticated techniques, perhaps along the lines used to find those of Hardy and Ramanujan or Meinardus and employing any modular properties of the multi-variable functions involved. This issue may also be important for microscopic counting for Black Holes, as the BPS solutions found thus far, for N = 4 superconformal symmetry, depend on special values of R-symmetry charges [17, 18, 19, 20] -see [50] for a related detailed discussion.\\n\\nThus far, the elegant results for finite N partition functions for chiral ring sectors have been interpreted from a largely geometric perspective -it may be interesting to investigate more how such results are related to the theory of random matrices and/or symmetric polynomials.\"\n }\n]","isConversational":false}}},{"rowIdx":183,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1040"},"arxiv_id":{"kind":"string","value":"0704.1040"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"d006cfb78bb11b0580d2546a119242f89886581647bc1485d955750afa5f30f4"},"title":{"kind":"string","value":"Recent Results on Thermal Casimir Force between Dielectrics and Related\n Problems"},"abstract":{"kind":"string","value":"We review recent results obtained in the physics of the thermal Casimir force\nacting between two dielectrics, dielectric and metal, and between metal and\nsemiconductor. The detailed derivation for the low-temperature behavior of the\nCasimir free energy, pressure and entropy in the configuration of two real\ndielectric plates is presented. For dielectrics with finite static dielectric\npermittivity it is shown that the Nernst heat theorem is satisfied. Hence, the\nLifshitz theory of the van der Waals and Casimir forces is demonstrated to be\nconsistent with thermodynamics. The nonzero dc conductivity of dielectric\nplates is proved to lead to a violation of the Nernst heat theorem and, thus,\nis not related to the phenomenon of dispersion forces. The low-temperature\nasymptotics of the Casimir free energy, pressure and entropy are derived also\nin the configuration of one metal and one dielectric plate. The results are\nshown to be consistent with thermodynamics if the dielectric plate possesses a\nfinite static dielectric permittivity. If the dc conductivity of a dielectric\nplate is taken into account this results in the violation of the Nernst heat\ntheorem. We discuss both the experimental and theoretical results related to\nthe Casimir interaction between metal and semiconductor with different charge\ncarrier density. Discussions in the literature on the possible influence of\nspatial dispersion on the thermal Casimir force are analyzed. In conclusion,\nthe conventional Lifshitz theory taking into account only the frequency\ndispersion remains the reliable foundation for the interpretation of all\npresent experiments."},"authors":{"kind":"list like","value":["B. Geyer","G.L. Klimchitskaya","V.M. Mostepanenko"],"string":"[\n \"B. Geyer\",\n \"G.L. Klimchitskaya\",\n \"V.M. Mostepanenko\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph","cond-mat.stat-mech","hep-th"],"string":"[\n \"quant-ph\",\n \"cond-mat.stat-mech\",\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-08"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The Casimir effect 1 is the force and also the specific polarization of the vacuum arising in restricted quantization volumes and originating from the zero-point oscillations of quantized fields. This force acts between two closely spaced macrobodies, between an atom or a molecule and macrobody or between two atoms or molecules. During more than fifty years, passed after the discovery of the Casimir effect, it has attracted much theoretical attention because of numerous applications in quantum field theory, atomic physics, condensed matter physics, gravitation and cosmology, 1 October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 4 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThese findings on the application of the Lifshitz theory to real metals have inspired a renewed interest in the Casimir force between dielectrics. As was mentioned above, at nonzero temperature dielectrics possess an although small but not equal to zero dc conductivity. In Ref. 57, 58 the van der Waals force arising from the dc conductivity of a dielectric plate was shown to lead to large effect in noncontact atomic friction, 59 a phenomenon having so far no satisfactory theoretical explanation. This brings up the question: Is it necessary or possible to take into account the dc conductivity of dielectrics in the Lifshitz theory? Recall that in the case of a positive answer the static dielectric permittivity of a dielectric material would be infinitely large. It is amply clear that the resolution of the above issue should be in accordance with the fundamentals of thermodynamics. For this reason, it is desirable to investigate the low-temperature behavior of the Casimir free energy and entropy for two dielectric plates both with neglected and included effects of the dc conductivity.\n\nA major breakthrough in the investigation of this problem was achieved in the year 2005. In Ref. 60 a new variant of perturbation theory was developed in a small parameter proportional to the product of the separation distance between the plates and the temperature. As a result, the behavior of the Casimir free energy, entropy and pressure at low temperatures was found analytically. If the static dielectric permittivity is finite, the thermal correction was demonstrated to be in accordance with thermodynamics. This solves positively the fundamental problem about the agreement between the Lifshitz theory and thermodynamics for the case of two dielectric plates. In Ref. 60 it was shown that, on the contrary, the formal inclusion of a small conductivity of dielectric plates at low frequencies into the model of their dielectric response leads to a violation of the Nernst heat theorem. This result gives an important guidance on how to extrapolate the tabulated optical data for the complex refractive index to low frequencies in numerous applications of the van der Waals and Casimir forces. All these problems and related ones arising for semiconductor materials are discussed in this review.\n\nIn Sec. 2 we derive the analytical behavior of the Casimir free energy, entropy and pressure in the configuration of two parallel dielectric plates at both low and high temperature. It is demonstrated that if the static dielectric permittivity is finite the Lifshitz theory is in agreement with thermodynamics. Sec. 3 contains the derivation of the low-temperature behavior for the Casimir free energy and entropy between two dielectric plates with included dc conductivity. In this case the Lifshitz theory is found to be in contradiction with the Nernst heat theorem. The conclusion is made that the conductivity properties of a dielectric material at a constant current are unrelated to the van der Waals and Casimir forces and must not be included into the model of dielectric response. In Sec. 4 we consider the thermal Casimir force between metal and dielectric. This problem was first investigated in Ref. 61. It was found that for dielectric plate with finite static dielectric permittivity the Nernst heat theorem is satisfied but the Casimir entropy may take negative values. Here we not only reproduce an analytical proof of the Nernst heat theorem but also October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 5 find the next perturbation orders in the expansion of the Casimir free energy and entropy in powers of a small parameter. The results obtained in high-temperature limit are also provided. Sec. 5 is devoted to the Casimir interaction between metal and dielectric plates with included dc conductivity of the dielectric material. We demonstrate that in this case the Nernst heat theorem is violated. Sec. 6 contains the discussion of semiconductors which present a wide variety of electric properties varying from metallic to dielectric. We consider the Casimir interaction between metal and semiconductor test bodies and formulate the criterion when it is appropriate to include the dc conductivity of a semiconductor into the model of dielectric response. In doing so, the results of recent experiments 62,63 on the measurement of the Casimir force between metal and semiconductor test bodies are taken into account. To this point the assumption has been made that metal, dielectric and semiconductor materials of the Casimir plates possess only temporal dispersion, i.e., can be described by the dielectric permittivity depending only on frequency. In Sec. 7 we discuss recent controversial results by different authors (see, e.g., Refs. 64 -68) attempting to take into account also spatial dispersion. As is shown in this section, the way of inclusion of spatial dispersion into the Lifshitz theory, used in Refs. 64 -68, is unjustified. We argue that the account of spatial dispersion cannot influence theoretical results obtained with the help of usual, spatially local, Lifshitz theory within presently used ranges of experimental separations. Sec. 8 contains our conclusions and discussion.\n\nWe consider two thick dielectric plates (semispaces) described by the frequencydependent dielectric permittivity ε(ω) and restricted by the parallel planes z = ±a/2 with a separation a between them, in thermal equilibrium at temperature T . The Lifshitz formula for the free energy of the van der Waals and Casimir interaction between the plates is given by 2,3,4,5,6,36,37,\n\n38 F (a, T ) = k B T 2π ∞ l=0 1 - δ l0 2 ∞ 0 k ⊥ dk ⊥ ( 1\n\n) × ln 1 -r 2 (ξ l , k ⊥ )e -2aq l + ln 1 -r 2 ⊥ (ξ l , k ⊥ )e -2aq l ,\n\nwhere the reflection coefficients for two independent polarizations of electromagnetic field are defined as\n\nr (ξ l , k ⊥ ) = ε l q l -k l ε l q l + k l , r ⊥ (ξ l , k ⊥ ) = k l -q l k l + q l . ( 2\n\n)\n\nHere k ⊥ is the magnitude of the wave vector in the plane of plates, ξ l = 2πk B T l/ are the Matsubara frequencies, k B is the Boltzmann constant, ε l = ε(iξ l ), and q l = ξ 2 textDr 6 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThe problems in the application of the Lifshitz theory to real materials discussed in the Introduction are closely connected with the values of the reflection coefficients at zero Matsubara frequency. For later use we discuss it for the various cases.\n\n• For ideal metals 40 it holds r (0, k ⊥ ) = r ⊥ (0, k ⊥ ) = 1.\n\n(\n\n) 4\n\n• For real metals described by the dielectric function of the Drude model,\n\nε(iξ l ) = 1 + ω 2 p ξ l [ξ l + ν(T )] , ( 5\n\n)\n\nwhere ω p is the plasma frequency and ν(T ) is the relaxation parameter, it holds 41 r (0, k ⊥ ) = 1, r ⊥ (0, k ⊥ ) = 0. (6) Eq. ( 6 ) results in the discontinuity between the cases of ideal and real metals and leads to the violation of the Nernst heat theorem for metallic plates having perfect crystal lattices. 52,53,54,56 • For real metals described by the dielectric function of the plasma model, ε(iξ l ) = 1 + ω 2\n\np ξ 2 l , ( 7\n\n)\n\nfrom Eq. (2) it follows: 44,45 r (0, k ⊥ ) = 1, r ⊥ (0, k ⊥ ) =\n\nc 2 k 2 ⊥ + ω 2 p -ck ⊥ c 2 k 2 ⊥ + ω 2 p + ck ⊥ . ( 8\n\n)\n\nHere, in the limit of ideal metals (ω p → ∞) the continuity is preserved because r ⊥ (0, k ⊥ ) in Eq. ( 8 ) goes to unity. The free energy (1) calculated with the permittivity (8) is also consistent with thermodynamics. • For dielectrics and semiconductors the dielectric permittivities at the imaginary Matsubara frequencies are given by the Ninham-Parsegian representation, 69,70 ε(iξ l ) = 1 + j\n\nC j 1 + ξ 2 l /ω 2 j , ( 9\n\n)\n\nwhere the parameters C j are the absorption strengths satisfying the condition j\n\nC j = ε 0 -1 ( 10\n\n)\n\nand ω j are the characteristic absorption frequencies. Here, the static dielectric permittivity ε 0 ≡ ε(0) is supposed to be finite. Although Eq. (10) is an October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 7 approximate one, it gives a very accurate description for many materials. 71 By the substitution of Eq. (9) in Eq. ( 2 ) one arrives at r (0, k ⊥ ) ≡ r 0 = ε 0 -1 ε 0 + 1 , r ⊥ (0, k ⊥ ) = 0. (11)\n\nNote that the vanishing of the transverse reflection coefficient for dielectrics at zero frequency in Eq. (11) has another meaning than for the Drude metals in Eq. ( 6 ). For Drude metal the parallel reflection coefficient is equal to the physical value for real photons at normal incidence, i.e., to unity, and the transverse one vanishes instead of taking unity, its physical value. This results in the violation of the Nernst heat theorem for perfect crystal lattices. In the case of dielectrics both reflection coefficients at zero frequency in Eq. (11) depart from the physical value for real photons which is equal to ( √ ε 0 -1)/( √ ε 0 + 1). In this case, however, one of them is larger and the other one is smaller than the physical value. As we will see below, this leads to the preservation of Nernst's heat theorem confirming that Eq. (9), despite being approximate, describes the material properties of dielectric and semiconductor plates in a thermodynamic consistent way.\n\nNow we derive the analytic representation for the Casimir free energy in Eq. (1) at low temperatures. For convenience in calculations, we introduce the dimensionless variables\n\nζ l = ξ l ξ c = 2aξ l c = τ l, y = 2aq l , ( 12\n\n)\n\nwhere ξ c = c/(2a) is the characteristic frequency, τ = 4πk B aT /( c), and q l was defined in Eq. ( 3 ). Then the Lifshitz formula (1) takes the form\n\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l dy f (ζ l , y) , ( 13\n\n)\n\nwhere f (ζ, y) = f (ζ, y) + f ⊥ (ζ, y), (14) f ,⊥ (ζ, y) = y ln 1 -r 2 ,⊥ (ζ, y)e -y , (15) and reflection coefficients (2), in terms of variables (12) , being given by\n\nr (ζ l , y) = ε l y -y 2 + ζ 2 l (ε l -1) ε l y + y 2 + ζ 2 l (ε l -1) , r ⊥ (ζ l , y) = y 2 + ζ 2 l (ε l -1) -y y 2 + ζ 2 l (ε l -1) + y . ( 16\n\n)\n\nTo separate the temperature independent contribution and thermal correction in Eq. ( 13 ) we apply the Abel-Plana formula, 3,6\n\n∞ l=0 1 - δ l0 2 F (l) = ∞ 0 F (t) dt + i ∞ 0 dt F (it) -F (-it) e 2πt -1 , ( 17\n\n)\n\nwhere F (z) is an analytic function in the right half-plane. Here, taking it as\n\nF (x) = ∞ x dy f (x, y) ( 18\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 8 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nand using Eq. (17), we can identically rearrange Eq. (13) to the form F (a, T ) = E(a) + ∆F (a, T ) , (19) where E(a) is the energy of the van der Waals or Casimir interaction at zero temperature,\n\nE(a) = c 32π 2 a 3 ∞ 0 dζ ∞ ζ dy f (ζ, y) , ( 20\n\n)\n\nand ∆F (a, T ) is the thermal correction to this energy, ∆F (a, T ) = i cτ 32π 2 a 3 ∞ 0 dt F (iτ t) -F (-iτ t) e 2πt -1 . (21) Note that, in fact, Eq. ( 21 ) describes the dependence of the free energy on the temperature arising from the dependence on temperature of the Matsubara frequencies. Thus, ∆F (a, T ) in (21) coincides with the thermal correction to the energy, defined as F (a, T ) -F (a, 0), only for plate materials with temperature independent properties. The asymptotic expressions for the energy E(a) at both short and large separations are well known. 6,37,38 Below we find the asymptotic expressions for the thermal correction (21) under the conditions τ ≪ 1 and τ ≫ 1. Taking into account the definition of τ in Eq. ( 12 ), the asymptotic expressions at τ ≪ 1 are applicable both at small and large separations if the temperature is sufficiently low.\n\nWe begin with condition τ ≪ 1. Let us substitute Eq. (9) in Eqs. (14) -(16), expand the function f (x, y) in powers of x = τ t, and than integrate the obtained expansion with respect to y from x to infinity in order to find F (x) in Eq. (18) and F (ix) -F (-ix) in Eq. (21) .\n\nIt is easy to check that f ⊥ (ζ, y) does not contribute to the leading, second, order in the expansion of F (ix) -F (-ix) in powers of x. Thus, we can restrict ourselves by the consideration of the expansion f (x, y) = y ln 1 -r 2 0 e -y + 2ε 0 r 2 0 ε 0 + 1\n\nx 2 y (e y -r 2 0 ) + 4ξ 2 c r 2 0 (ε 0 + 1)ω 2 1 yx 2 e y -r 2 0 + O(x 3 ), ( 22\n\n)\n\nwhere r 0 was defined in Eq. (11) . Note that for simplicity we consider here only one oscillator in Eq. ( 9 ) and put ω j = ω 1 . The case of several oscillator modes can be considered in an analogous way.\n\nAs a next step, we integrate Eq. ( 22 ) term by term according to Eq. ( 18 ), expand the partial results in powers of x and sum up the obtained series. Thereby we obtain the following expressions:\n\nZ 1 (x) ≡ ∞ x y dy ln 1 -r 2 0 e -y = - ∞ n=1 (1 + nx)e -nx n 3 r 2n 0 = -Li 3 (r 2 0 ) - x 2 2 ln(1 -r 2 0 ) + O(x 3 ) , ( 23\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 9\n\nZ 2 (x) ≡ 2ε 0 r 2 0 x 2 ε 0 + 1 ∞ x dy y (e y -r 2 0 ) = - 2ε 0 x 2 ε 0 + 1 ∞ n=1 r 2n 0 Ei(-nx) , ( 24\n\n) Z 3 (x) ≡ 4r 2 0 ξ 2 c x 2 (ε 0 + 1)ω 2 1 ∞ x ydy e y -r 2 0 = 4ξ 2 c x 2 (ε 0 + 1)ω 2 1 ∞ n=1 r 2n 0 (1 + nx)e -nx n 2 = 4ξ 2 c (ε 0 + 1)ω 2 1 x 2 Li 2 (r 2 0 ) - x 4 2 r 2 0 1 -r 2 0 + O(x 5 ) , ( 25\n\n)\n\nwhere Li n (z) is the polylogarithm function and Ei(z) is the exponential integral function. From these equations it follows\n\nZ 1 (ix) -Z 1 (-ix) = O(x 3 ) , Z 3 (ix) -Z 3 (-ix) = O(x 5 ) , ( 26\n\n) Z 2 (ix) -Z 2 (-ix) = 2iπ ε 0 ε 0 + 1 r 2 0 1 -r 2 0 x 2 + O(x 3 ) , ( 27\n\n)\n\nand, thus, Z 1 and Z 3 do not contribute to the leading order in the expansion of F (ix) -F (-ix). The latter is determined by Z 2 only. As a result, we arrive at\n\nF (ix) -F (-ix) = iπ (ε 0 -1) 2 2(ε 0 + 1) x 2 -iαx 3 + O(x 4 ), ( 28\n\n)\n\nwhere r 0 was substituted from Eq. ( 11 ) and α was introduced for the still unknown real coefficient of the next to leading order resulting from Z 1 and Z 2 as well as, possibly, from f ⊥ (ζ, y). At this stage it is difficult to determine the value of this coefficient because all powers in the expansion of f (x, y) contribute to it. Remarkably, the two leading orders depend only on the static dielectric permittivity ε 0 and are not influenced by the dependence of the dielectric permittivity on the frequency contained in Z 3 . Substituting Eq. (28) in Eq. (21) and using Eq. ( 19 ), we obtain\n\nF (a, T ) = E(a) - c 32π 2 a 3 ζ(3) 8π 2 (ε 0 -1) 2 ε 0 + 1 τ 3 -C 4 τ 4 + O(τ 5 ) , ( 29\n\n)\n\nwhere C 4 ≡ α/240 and ζ(z) is the Riemann zeta function. So far we have considered the free energy. The thermal pressure is obtained as\n\nP (a, T ) = - ∂F (a, T ) ∂a = P 0 (a) - c 32π 2 a 4 C 4 τ 4 + O(τ 5 ) , ( 30\n\n)\n\nwhere P 0 (a) = -∂E(a)/∂a is the Casimir pressure at zero temperature. In order to determine the value of the coefficient C 4 of the leading term, we express the pressure directly through the Lifshitz formula\n\nP (a, T ) = - cτ 32π 2 a 4 ∞ l=0 1 - δ l0 2 ∞ ζ l y 2 dy r 2 (ζ l , y) e y -r 2 (ζ l , y) + r 2 ⊥ (ζ l , y) e y -r 2 ⊥ (ζ l , y) . ( 31\n\n)\n\nAgain, applying the Abel-Plana formula (17), we represent the pressure as follows,\n\nP (a, T ) = P 0 (a) + ∆P (a, T ), ( 32\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 10 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nwhere the thermal correction to P 0 (a), the pressure at zero temperature, is\n\n∆P (a, T ) = - i cτ 32π 2 a 4 ∞ 0 dt Φ(iτ t) -Φ(-iτ t) e 2πt -1 ( 33\n\n)\n\nand the function Φ(x) is given by\n\nΦ(x) ≡ Φ (x) + Φ ⊥ (x) , Φ ,⊥ (x) = ∞ x dy y 2 r 2 ,⊥ (x, y) e y -r 2 ,⊥ (x, y) . ( 34\n\n)\n\nFirst, we determine the leading term of the expansion of Φ ⊥ (x) in powers of x. For this purpose, let us introduce the new variable v = y/x and note that the reflection coefficient r ⊥ (x, v) depends on x only through the frequency dependence of ε given by Eq. ( 9 ). Thus, we can rewrite and expand Eq. (34) as follows:\n\nΦ ⊥ (x) = x 3 ∞ 1 dv v 2 r 2 ⊥ (x, v) e vx -r 2 ⊥ (x, v) = x 3 ∞ 1 dv v 2 r 2 ⊥ (v) 1 -r 2 ⊥ (v) + O(x 4 ), ( 35\n\n)\n\nwhere, according to Eq. ( 16 ),\n\nr ⊥ (v) ≡ r ⊥ (0, v) = √ v 2 + ε 0 -1 -v √ v 2 + ε 0 -1 + v . ( 36\n\n)\n\nIntegration in Eq. ( 35 ) with account of Eq. (36) results in\n\nΦ ⊥ (x) = 1 - ε 0 (3 -ε 0 ) 2 √ ε 0 x 3 6 + O(x 4 ) , ( 37\n\n)\n\nfrom which it follows:\n\nΦ ⊥ (ix) -Φ ⊥ (-ix) = -i 1 - 1 2 √ ε 0 (3 -ε 0 ) x 3 3 + O(x 5 ). ( 38\n\n)\n\nThe expansion of Φ (x) from Eq. (34) in powers of x is somewhat more cumbersome. It can be performed in the following way. As is seen from the second equality in Eq. ( 26 ), the dependence of the dielectric permittivity on frequency contributes to F (ix) -F (-ix) starting from only the 5th power in x. Bearing in mind the connection between free energy and pressure, we can conclude that the dependence on the frequency contributes to Φ (ix) -Φ (-ix) starting from the 4th order. We are looking for the lowest (third) order expansion term of Φ (ix) -Φ (-ix). Because of this, it is permissible to disregard the frequency dependence of ε and describe the dielectric by its static dielectric permittivity.\n\nTo begin with, we identically rearrange Φ (x) in Eq. ( 34 ) by subtracting and adding the two first expansion terms of the function under the integral in powers of x,\n\nΦ (x) = ∞ x dy y 2 r 2 (x, y) e y -r 2 (x, y) -y 2 r 2 0 e y -r 2 0 + x 2 2ε 0 ε 0 + 1 r 2 0 e -y (1 -r 2 0 e -y ) 2 + ∞ x y 2 dy r 2 0 e y -r 2 0 -x 2 2ε 0 ε 0 + 1 ∞ x dy r 2 0 e -y (1 -r 2 0 e -y ) 2 , ( 39\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 11 and consider these three integrals separately. The first integral in terms of the new variable v = y/x reads\n\nQ 1 (x) ≡ x 3 ∞ 1 dv v 2 r 2 (v) e vx -r 2 (v) -v 2 r 2 0 e vx -r 2 0 + 2ε 0 ε 0 + 1 r 2 0 e -vx (1 -r 2 0 e -vx ) 2 , ( 40\n\n)\n\nwhere, in accordance with Eq. ( 16 ),\n\nr (v) ≡ r (0, v) = ε 0 v - √ v 2 + ε 0 -1 ε 0 v + √ v 2 + ε 0 -1 . ( 41\n\n)\n\nExpanding Q 1 (x) in powers of x and explicitly calculating the remaining integrals for the lowest, third, power of x results in\n\nQ 1 (x) = x 3 ∞ 1 dv v 2 r 2 (v) 1 -r 2 (v) -v 2 r 2 0 1 -r 2 0 + 2ε 0 ε 0 + 1 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ) ( 42\n\n) = x 3 6 1 - √ ε 0 2 1 + 3ε 0 -2 ε 2 0 + 2 r 2 0 1 -r 2 0 - 12 ε 0 ε 0 + 1 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ).\n\nThe second and third integrals on the right-hand side of Eq. ( 39 ) are simply determined with the following result:\n\nQ 2 (x) ≡ ∞ x y 2 dy r 2 0 e y -r 2 0 = ∞ n=1 r 2n 0 (2 + 2nx + n 2 x 2 )e -nx n 3 = 3Li 3 (r 2 0 ) - x 3 3 r 2 0 1 -r 2 0 + O(x 4 ) , ( 43\n\n)\n\nQ 3 (x) ≡ - 2ε 0 x 2 ε 0 + 1 ∞ x dy r 2 0 e -y (1 -r 2 0 e -y ) 2 = - 2ε 0 x 2 ε 0 + 1 r 2 0 e -x 1 -r 2 0 e -x = - 2ε 0 ε 0 + 1 x 2 r 2 0 1 -r 2 0 -x 3 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ) . ( 44\n\n) Substituting Eqs. (42), (43) and (44) into Φ (x) = Q 1 (x) + Q 2 (x) + Q 3 (x), we arrive at Φ (ix) -Φ (-ix) = -i 1 + √ ε 0 2 2ε 2 0 -3ε 0 -1 x 3 3 + O(x 4 ). ( 45\n\n)\n\nThen, by summing Eqs. (38) and (45), the result is obtained\n\nΦ(ix) -Φ(-ix) = -i 2 + ε 5/2 0 -ε 3/2 0 -2 √ ε 0 x 3 3 + O(x 4 ). ( 46\n\n)\n\nNow we substitute Eq. (46) in Eq. ( 33 ) and perform integration. Finally, from Eq. (32) the desired expression for the Casimir pressure is derived\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 √ ε 0 -1 ε 2 0 + ε 3/2 0 -2 720 τ 4 + O(τ 5 ) . ( 47\n\n)\n\nBy comparison with Eq. ( 30 ) the explicit form of the coefficient C 4 is found as\n\nC 4 = √ ε 0 -1 ε 2 0 + ε 3/2 0 -2 /720 ( 48\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 12 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and, thus, both two first perturbation orders in the expansion for the free energy (29) are determined. Equations (29), (47) and (48) solve the fundamental problem of the thermodynamic consistency of the Lifshitz theory in the case of two dielectric plates. From Eqs. (29) and (48) the entropy of the van der Waals and Casimir interaction between plates takes the form\n\nS(a, T ) = - ∂F (a, T ) ∂T = 3k B ζ(3)(ε 0 -1) 2 64π 3 a 2 (ε 0 + 1) τ 2 ( 49\n\n) × 1 - 2π 2 (ε 0 + 1) ε 3/2 0 + 2ε 0 + 2 √ ε 0 + 2 135ζ(3)( √ ε 0 + 1) 2 τ + O(τ 2 ) .\n\nAs is seen from Eq. (49), in the limit τ → 0 (T → 0) the lower order contributions to the entropy are of the second and the third powers in the small parameter τ . Thus, the entropy vanishes when the temperature goes to zero as it must be in accordance with the third law of thermodynamics (the Nernst heat theorem).\n\nA similar behavior was obtained for ideal metals 40,72,73 and for real metals described by the plasma model. 45,52 For example, in the case of plates made of ideal metal the entropy at low temperatures is given by 40,72,73\n\nS(a, T ) = 3k B ζ(3) 32π 3 a 2 τ 2 1 - 2π 2 135ζ(3) τ + O(τ 2 ) . ( 50\n\n)\n\nNote, however, that the expansion coefficients in Eq. ( 50 ) cannot be obtained as a straightforward limit |ε 0 | → ∞ in Eq. ( 49 ) and the above equations for the free energy and pressure. The mathematical reason is that it is impermissible to interchange the limiting transitions τ → 0 and |ε 0 | → ∞ in the power expansions of functions depending on ε 0 as a parameter. Remarkably, the low-temperature behavior of the free energy, pressure and entropy of nonpolar dielectrics in Eqs. (29), (47) and (49) is universal, i.e., is determined only by the static dielectric permittivity. The absorption bands included in Eq. ( 9 ) do not influence the low-temperature behavior. A more simple derivation of the results (29), (47)-(49) for dielectrics with constant ε is contained in Ref. 74. As was demonstrated above, all these results remain unchanged if the dependence of dielectric permittivity on frequency is taken into account.\n\nIn Ref. 60 more general results were obtained related to two dissimilar dielectric plates with dielectric permittivities ε (1) (ω) and ε (2) (ω). For brevity here we present only the final expressions for the low-temperature behavior of the Casimir free energy, pressure and entropy between dissimilar plates. They are as follows: 60\n\nF (a, T ) = E(a) - c 32π 2 a 3 τ 3 × ζ(3) 8π 2 ε ( 1\n\n) 0 + ε ( 2\n\n) 0 + 2ε (1) 0 ε (2) 0 (ε ( 1\n\n) 0 + 1)(ε ( 2\n\n) 0 + 1) (ε ( 1\n\n) 0 -1)(ε (2) 0 -1) (ε (1) 0 + ε (2) 0 ) -C 4 τ + O(τ 2 ) ,\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 13 1,2) (0) and the coefficient C 4 is given by 60\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 C 4 τ 4 + O(τ 5 ) , ( 51\n\n) S(a, T ) = k B 2πa 2 τ 2 × 3ζ(3) 32π 2 ε ( 1\n\n) 0 + ε ( 2\n\n) 0 + 2ε (1) 0 ε (2) 0 (ε ( 1\n\n) 0 + 1)(ε ( 2\n\n) 0 + 1) (ε ( 1\n\n) 0 -1)(ε ( 2\n\n) 0 -1) (ε (1) 0 + ε (2) 0 ) -C 4 τ + O(τ 2 ) . Here ε (1,2) 0 ≡ ε (\n\nC 4 = 1 720 2 + 1 ε ( 1\n\n) 0 + ε (2) 0 ε (1) 0 + ε (2) 0 2 ( 52\n\n) × -ε ( 1\n\n) 0 + ε (2) 0 2 2ε ( 1\n\n) 0 + 2ε ( 2\n\n) 0 + ε (1) 0 ε (2) 0 -ε (1) 0 ε (2) 0 + ε (1) 0 ε (2) 0 ε (1) 0 ε (2) 0 5ε (1) 0 ε (2) 0 -3ε (1) 0 -3ε ( 2\n\n) 0 + 1 + ε (1) 0 ε (2) 0 ε (1) 0 -ε (2) 0 2 ε ( 1\n\n) 0 ε (2) 0 ε (1) 0 ε ( 2\n\n) 0 -ε ( 1\n\n) 0 ε ( 2\n\n) 0 -ε ( 1\n\n) 0 -ε (2) 0 - 3 ε ( 1\n\n) 0 ε (2) 0 2 ε ( 1\n\n) 0 -1 ε ( 2\n\n) 0 -1 ε ( 1\n\n) 0 -ε (2) 0 ε ( 1\n\n) 0 + ε (2) 0 Artanh ε ( 1\n\n) 0 + ε (2) 0 ε (1) 0 -ε (2) 0 ε (1) 0 ε ( 2\n\n) 0 -ε ( 1\n\n) 0 -ε (2) 0 .\n\nIt is easily seen that in the limit ε 52) coincide with equations (29), (47)-(49) having obtained above. Note that in the application region of low-temperature asymptotic expressions the entropy of the Casimir interaction between dielectric plates is nonnegative.\n\n(1) 0 = ε ( 2\n\n) 0 = ε 0 equations (51), (\n\nThe obtained analytic behavior of the free energy, pressure and entropy at low temperatures can be compared with the results of numerical computations using the Lifshitz formula. Dielectric properties of the plates can be described by the static dielectric permittivity or more precisely using the optical tabulated data for the complex index of refraction. As an example, in Fig. 1 we present the thermal corrections to the Casimir energy (a) and pressure (b) at a separation a = 400 nm as functions of temperature in the configuration of two dissimilar plates made of high-resistivity Si and SiO 2 . The dielectric permittivities of both materials along the imaginary frequency axis were computed in Ref. 75 using the optical data of Ref. 76. The precise thermal corrections computed by taking into account these permittivities are shown by the solid lines and corrections computed by our analytical asymptotic expressions are shown by the long-dashed lines. Short-dashed lines indicate the results computed by the Lifshitz formula with constant dielectric permittivities of Si and SiO 2 equal to ε (1) 0 = 11.67 and ε (2) 0 = 3.84, respectively. As is seen in Fig. 1a ,b, at T < 60 K the results obtained using the analytical asymptotic October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 14 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 25 50 75 100 125 150 175 200 1 2 3 4 T (K) -∆F (pJ) 25 50 75 100 125 150 175 200 1 2 3 4 5 6 7 T (K) -∆P (µPa) (b) (a)\n\nFig. 1. Magnitudes of the thermal corrections to the energy (a) and pressure (b) in configuration of two plates, one made of Si and another one of SiO 2 , at a separation a = 400 nm as a function of temperature calculated by the use of different approaches: by the Lifshitz formula and tabulated optical data (solid lines), by the Lifshitz formula and static dielectric permittivities (short-dashed lines), by the asymptotic expressions in Eqs. (51) and (52) (long-dashed lines).\n\nexpressions practically coincide with the solid lines computed using the tabulated optical data for the materials of the plates.\n\nNow we return to the case of two similar dielectric plates and consider the asymptotic expressions under the condition τ ≫ 1, i.e., at high temperatures (large separations). It is well known 37,38,77 that in this case the approximation of static October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 15 dielectric permittivity works good and the main contribution is given by the zerofrequency term of the Lifshitz formula (13)\n\nF (a, T ) = cτ 64π 2 a 3 ∞ 0 ydy ln 1 -r 2 0 e -y ( 53\n\n)\n\n(the other terms being exponentially small). Performing the integration in Eq. (53) we obtain\n\nF (a, T ) = - k B T 16πa 2 Li 3 r 2 0 . ( 54\n\n)\n\nIn a similar manner for the Casimir pressure and entropy at τ ≫ 1 it follows\n\nP (a, T ) = - k B T 8πa 3 Li 3 r 2 0 , S(a, T ) = k B 16πa 2 Li 3 r 2 0 . ( 55\n\n)\n\nEquations (54) and (55) are simply generalized 60 for the case of two dissimilar dielectric plates by performing the replacement r 2 0 → r (1) 0 r (2) 0 , where r (1,2) 0 are defined by Eq. (11) with the static dielectric permittivities of dissimilar plates ε (1,2) 0 .\n\nAs was discussed in the previous section, the zero-frequency term in formula (13), i.e., the contribution with l = 0, is of prime importance and determines many of the basic properties of the Casimir interaction. In the above consideration we have described dielectric materials by Eq. ( 9 ) with finite static dielectric permittivity ε 0 . This resulted in Eq. (11) where one reflection coefficient at zero frequency is larger and the other one is smaller than the physical value for real photons at normal incidence. However, in the Lifshitz theory, the departure of both coefficients from their physical values is coordinated in such a way that the Nernst heat theorem remains valid. It is common knowledge that at nonzero temperatures dielectric materials possess a negligibly small but not equal to zero dc conductivity. From physical intuition it is reasonable to expect that the influence of this conductivity on the van der Waals and Casimir forces should be also negligible. In Ref. 57 it was shown that, on the contrary, the inclusion of small dielectric dc conductivity in the model of dielectric response leads to a large effect in dispersion forces. This raises the question if dc conductivity is related to dispersion forces or if, on the contrary, the zero-frequency contribution should be understood not literally but as an analytic continuation from the region of high frequencies determining the physical phenomenon of dispersion forces.\n\nTo illustrate this problem in more details, we consider the asymptotic behavior of the free energy and entropy at low temperature with included dc conductivity. What this means is that, instead of the dielectric permittivity ε(iξ l ) given in Eq. ( 9 ), one uses 57,58\n\nε(iξ l ) = ε(iξ l ) + 4πσ 0 ξ l = ε l (iξ) + β(T ) l , ( 56\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 16 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nwhere σ 0 is the dc conductivity of the plate material and β(T ) = 2 σ 0 /(k B T ). The conductivity of dielectrics depends on temperature as σ 0 ∼ exp(-b/T ) where b is determined by the energy gap ∆ which differs for different materials. 78 The smallness of the dc conductivity of dielectrics can be illustrated 79 by the example of SiO 2 where at T = 300 K it holds β ∼ 10 -12 . Thus, the role of the dc conductivity is really negligible for all l ≥ 1. In addition, β(T ) quickly decreases with decrease of T and, as a consequence, remains negligible at any T . In spite of this, the substitution of Eq. (56) into the reflection coefficients (16) leads to different result than in Eq. (11): r (0, y) = 1, r (0, y) = 0. (57) Equation (57) is in some analogy to Eq. ( 6 ), obtained for metals described by the Drude model, which leads to the violation of the Nernst heat theorem in the case of perfect crystal lattices. Now we substitute the dielectric permittivity εl ≡ ε(ξ l ) in Eqs. (13) -(16) instead of ε l and find the Casimir free energy F (a, T ) with included dc conductivity. For convenience, we separate the zero-frequency term, subtract and add the usual zero-frequency contribution for dielectric without dc conductivity. The result is\n\nF (a, T ) = k B T 16πa 2 ∞ 0 ydy ln 1 -e -y -ln 1 -r 2 0 e -y + k B T 16πa 2 ∞ 0 ydy ln 1 -r 2 0 e -y ( 58\n\n) + k B T 8πa 2 ∞ l=1 ∞ ζ l ydy ln 1 -r2 (ζ l , y)e -y + ln 1 -r2 ⊥ (ζ l , y)e -y ,\n\nwhere r 0 was defined in Eq. ( 11 ). Let us expand the last, third, integral on the righthand side of Eq. (58) in powers of the small parameters β/l. Then, we combine the zero-order contribution in this expansion with the second integral on the righthand side of Eq. ( 58 ) and obtain the Casimir free energy F (a, T ) calculated with the dielectric permittivities ε l . The first integral on the right-hand side of Eq. (58) is calculated explicitly. Then, Eq. ( 58 ) can be rewritten as\n\nF (a, T ) = F (a, T ) - k B T 16πa 2 ζ(3) -Li 3 r 2 0 + R(a, T ). ( 59\n\n)\n\nHere, R(a, T ) is of order O(β/l). It represents the first and higher-order contributions in the expansion of the third integral on the right-hand side of Eq. (58) in powers of β/l. Restricting its explicit form to the first order contribution we get\n\nR(a, T ) = R 1 (a, T ) + O (β/l) 2 , ( 60\n\n) R 1 (a, T ) = k B T 4πa 2 ∞ l=1 β l ∞ ζ l dy y 2 e -y y 2 + ζ 2 l (ε l -1) (2 -ε l ) ζ 2 l -2y 2 y 2 + ζ 2 l (ε l -1) + ε l y 2 × r (ζ l , y) 1 -r 2 (ζ l , y)e -y - ζ 2 l y 2 + ζ 2 l (ε l -1) + y 2 r ⊥ (ζ l , y) 1 -r 2 ⊥ (ζ l , y)e -y . ( 61\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 17\n\nCalculating the entropy by the first equality in Eq. ( 49 ), we arrive at\n\nS(a, T ) = S(a, T ) + k B 16πa 2 ζ(3) -Li 3 r 2 0 - ∂R(a, T ) ∂T , ( 62\n\n)\n\nwhere S(a, T ) is the entropy for the plates with the dielectric permittivity ε l given by Eq. ( 49 ).\n\nLet us now show that the quantity R(a, T ) exponentially goes to zero with the decrease of T . First we consider only the integral in Eq. ( 61 ), expand the integrated function in powers of τ (we recall that ζ l = τ l), restrict ourselves to the main contribution, resulting for τ = 0, and rearrange it appropriately:\n\n-2 ∞ ζ l dy ye -y (ε 0 + 1) 2 r 0 1 -r 2 0 e -y = - 2 (ε 2 0 -1) ∞ ζ l dy y r 2 0 e -y 1 -r 2 0 e -y = - 2 (ε 2 0 -1) ∞ n=1 r 2n 0 ∞ ζ l dy ye -ny = - 2 (ε 2 0 -1) ∞ n=1 r 2n 0 1 + nζ l n 2 e -nζ l . ( 63\n\n)\n\nSubstituting this in Eq. ( 61 ), we find\n\nR 1 (a, T ) = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r 2n 0 n 2 ∞ l=1 e -nτ l l + nτ ∞ l=1 e -nτ l = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r 2n 0 n 2 -ln 1 -e -nτ + nτ e nτ -1 = k B Li 2 r 2 0 2πa 2 (ε 2 0 -1) T β ln τ + T β O(τ 0 ) . ( 64\n\n)\n\nHere, the last line is obtained by using the equality\n\n-ln 1 -e -nτ + nτ e nτ -1 = -ln τ + 1 -ln n + O(τ 2 ) , ( 65\n\n)\n\nsubstituting only its leading term and observing the definition of the integral log-\n\narithm, Li 2 (z) = (1/2) ∞ n=1 z n /n 2 .\n\nTaking into account that β ∼ (1/T ) exp(-b/T ), we get the conclusion that the temperature dependence of R 1 (a, T ) is given by\n\nR 1 (a, T ) ∼ e -b/T ln T. ( 66\n\n)\n\nThus, both R 1 (a, T ) and its derivative with respect to T in Eqs. (59) and (62) go to zero. The terms of the second and higher powers in β in Eq. (60) go to zero even faster than R 1 when T → 0. Finally, in the limit T → 0 from Eq. (62) it follows\n\nS(a, 0) = k B 16πa 2 ζ(3) -Li 3 r 2 0 > 0. ( 67\n\n)\n\nThe right-hand side of this equation depends on the parameter of the system under consideration (the separation distance a) and implies a violation of the Nernst heat theorem. An analogous result was obtained 60 in the case of two dissimilar dielectrics.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 18 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThe violation of the Nernst heat theorem in the Casimir interaction for dielectrics originates from the inclusion of the dc conductivity in the model of dielectric response. This violation is, however, of different nature than the one discussed above in the case of Drude metals. In the case of dielectrics the entropy at zero temperature is positive but in the case of Drude metals it is negative. In the case of metals the violation is caused by the vanishing contribution from the transverse electric mode at zero frequency whereas the other reflection coefficient takes the physical value 1 [see Eq. ( 6 )]. For dielectrics the situation is quite opposite. In this case the transverse reflection coefficient at zero frequency is always equal to zero [compare Eqs. (11) and (57) in the absence and in the presence of the contribution from dc conductivity]. Here the violation occurs due to the unity value of the parallel reflection coefficient at zero frequency in Eqs. (57) which departs from the value r 0 = (ε 0 -1)/(ε 0 + 1) coordinated with the zero value of the transverse coefficient in Eq. (11) .\n\nOne can conclude that the dc conductivity of a dielectric is not related to the nature of the van der Waals and Casimir forces and must not be included in the model of dielectric response. Ignoring this rule results in a violation of thermodynamics. Physically it is amply clear that there is no fluctuating field of zero frequency and that for such high-frequency phenomena as the van der Waals and Casimir forces the low-frequency behavior should be obtained by analytic continuation from the region of high frequencies. This permits to conclude that the correct procedure consists in the substitution of the finite static dielectric permittivities into the zero-frequency term of the Lifshitz formula, as Lifshitz and his collaborators really did. 36,37,38\n\nThe Casimir interaction between metal and dielectric plates suggests the interesting opportunity to verify the thermodynamic consistency of the Lifshitz theory with different models of the dielectric response. This configuration was first investigated in Ref. 61 where it was proved that the Casimir entropy is in accordance with the demands of the Nernst heat theorem if the static permittivity of the dielectric plate is finite. In Ref. 61, however, only the first leading terms in the lowtemperature asymptotic expressions for the free energy and entropy were obtained and the Casimir pressure was derived only in the dilute approximation. Here we derive the more precise low-temperature behavior for the Casimir free energy, pressure and entropy in the configuration of one plate made of ideal metal and another plate made of dielectric with any finite static dielectric permittivity. For the configuration of metal and dielectric plates the Lifshitz formula takes the form 80 analogical to Eq. (13)\n\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l ydy ( 68\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 19\n\n× ln 1 -r M (ζ l , y) r D (ζ l , y) e -y + ln 1 -r M ⊥ (ζ l , y) r D ⊥ (ζ l , y) e -y .\n\nHere the reflection coefficients r M,D ,⊥ for metal and dielectric, respectively, are given by Eqs. (16) where ε l should be changed for ε M,D l\n\n= ε M,D (iξ l ).\n\nFor an ideal metal r M ,⊥ (ζ l , y) = 1 and Eq. ( 68 ) takes the more simple form\n\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l ydy ( 69\n\n) × ln 1 -r (ζ l , y)e -y + ln 1 -r ⊥ (ζ l , y)e -y\n\n(here and below we omit the index D near the reflection coefficient and permittivity of a dielectric plate). We admit that the dielectric permittivity calculated at Matsubara frequencies ε l ≡ ε 0 , i.e., is equal to its static value and find the asymptotic behavior of Eq. (69) at small τ . [In analogy with Sec. 2 it is possible to prove that the deviations of ε(iξ l ) from ε 0 at high frequencies do not influence the lowtemperature behavior of the Casimir free energy, pressure and entropy. It can be shown also that the results of this section are valid not only for ideal metal plate but for plate made of real metal as well.] The free energy (69) can be represented by Eqs. (18) -(21), with the function f (ζ, y) replaced by\n\nf (ζ, y) = y ln 1 -r (ζ, y)e -y + y ln 1 -r ⊥ (ζ, y)e -y ≡ f (ζ, y) + f⊥ (ζ, y). ( 70\n\n)\n\nIn the case of one dielectric and one metal plate both f and f⊥ contribute to F (ix) -F (-ix). The expansion of f (x, y) in powers of x takes the form\n\nf (x, y) = y ln(1 -r 0 e -y ) - ε 0 -1 4y e -y - ε 0 ε 0 + 1 ∞ n=1 r n 0 e -ny y x 2 + O(x 3 ). ( 71\n\n)\n\nNow we integrate Eq. (71) in accordance with Eq. (18) to find the function F (x). The integral of the first term on the right-hand side of Eq. (71) is evaluated using the new variable v = y -x:\n\n∞ x ydy ln(1 -r 0 e -y ) = ∞ 0 vdv ln(1 -r 0 e -v ) + O(x 2 ), ( 72\n\n)\n\nwhere the coefficient near the first-order contribution in x vanishes. As a result, this term could contribute to F (ix) -F (-ix) only starting from the third expansion order. The integrals of the second-order terms on the right-hand side of Eq. (71) are simply calculated using the formulas\n\n∞ x dy e -y y = -Ei(-x), ∞ x dy e -ny y = -Ei(-nx). ( 73\n\n)\n\nFinally, we obtain\n\nF (ix) -F (-ix) = iπ (ε 0 -1) 2 4 (ε 0 + 1) x 2 -iγ x 3 + O(x 4 ), ( 74\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 20 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nwhere the unknown third order expansion coefficient is designated as γ. Substituting Eq. (74) in Eq. (21) and using Eq. ( 19 ), we find the free energy in the system metal-dielectric in the form\n\nF (a, T ) = E(a) - c 32π 2 a 3 ζ(3) 16π 2 (ε 0 -1) 2 ε 0 + 1 τ 3 -K 4 τ 4 + O(τ 5 ) , ( 75\n\n)\n\nwhere K 4 ≡ γ/240. The Casimir pressure in the configuration of metal and dielectric plates obtained from Eq. (75) is equal to\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 K 4 τ 4 + O(τ 5 ) . ( 76\n\n)\n\nThe direct application of the Lifshitz formula gives the expression for the pressure analogical to Eq. (31),\n\nP (a, T ) = - cτ 32π 2 a 4 ∞ l=0 1 - δ l0 2 ∞ ζ l y 2 dy r (ζ l , y) e y -r (ζ l , y) + r ⊥ (ζ l , y) e y -r ⊥ (ζ l , y) . ( 77\n\n)\n\nUsing the Abel-Plana formula (17), Eq. ( 77 ) can be represented in the form of Eqs. (32), (33) where\n\nΦ ,⊥ (x) = ∞ x dy y 2 r ,⊥ (x, y) e y -r ,⊥ (x, y) . ( 78\n\n)\n\nAgain, we deal first with Φ ⊥ (x). By adding and subtracting the asymptotic behavior of the integrated function at small x,\n\ny 2 r ⊥ (x, y) e y -r ⊥ (x, y) = 1 4 (ε 0 -1)x 2 e -y + O(x 3 ) , ( 79\n\n)\n\nand introducing the new variable v = y/x, the function Φ ⊥ (x) can be identically rearranged and expanded in powers of x as follows:\n\nΦ ⊥ (x) = 1 4 (ε 0 -1)x 2 e -x + x 3 ∞ 1 dv v 2 ∞ n=1 r n ⊥ (v)e -nvx - 1 4 (ε 0 -1)e -vx = 1 4 (ε 0 -1)x 2 (1 -x) + x 3 ∞ 1 dv v 2 r ⊥ (v) 1 -r ⊥ (v) - ε 0 -1 4 + O(x 4 ). ( 80\n\n)\n\nThe integral on the right-hand side of Eq. (80) is converging and can be simply calculated with the result\n\nΦ ⊥ (x) = ε 0 -1 4 x 2 - 1 6 (ε 0 √ ε 0 -1) x 3 + O(x 4 ). ( 81\n\n)\n\nTo deal with Φ (x) we add and subtract in Eq. ( 78 ) the two first expansion terms of the integrated function in powers of x,\n\nΦ (x) = ∞ x y 2 dy r 0 e y -r 0 - ε 0 r 0 e -y x 2 y 2 (ε 0 + 1) (1 -r 0 e -y ) 2 ( 82\n\n) + ∞ x y 2 dy r (x, y) e y -r (x, y) - r 0 e y -r 0 + ε 0 r 0 e -y x 2 y 2 (ε 0 + 1) (1 -r 0 e -y ) 2 .\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 21\n\nThe asymptotic expansion of the first integral on the right-hand side of Eq. (82) is given by\n\n2Li 3 (r 0 ) - ε 0 (ε 0 -1) 2(ε 0 + 1) x 2 + 1 12 (ε 0 -1)(3ε 0 -2)x 3 + O(x 4 ), ( 83\n\n)\n\nand of the second one by\n\n- 1 4 ε 0 (ε 0 -1) - 1 6 ε 0 (ε 0 √ ε 0 -1) + 1 2 ε 0 (ε 0 -1) √ ε 0 x 3 + O(x 4 ). ( 84\n\n)\n\nBy summing Eqs. (83) and (84) we find\n\nΦ (x) = 2Li 3 (r 0 ) - ε 0 (ε 0 -1) 2(ε 0 + 1) x 2 ( 85\n\n) - 1 6 (ε 0 -1) + (ε 0 √ ε 0 -1) -3ε 0 (ε 0 -1) √ ε 0 x 3 + O(x 4 ).\n\nFinally we add Eq. (85) to Eq. ( 81 ) and arrive at Φ(ix\n\n) -Φ(-ix) = -i 2 3 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 x 3 + O(x 4 ). ( 86\n\n)\n\nSubstituting this in Eq. (33) and using Eq. (32), the asymptotic expression for the Casimir pressure is obtained\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 1 360 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 τ 4 + O(τ 5 ) . ( 87\n\n)\n\nComparing Eqs. (87) and (76), the explicit expression for the coefficient K 4 reads\n\nK 4 = 1 360 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 . ( 88\n\n)\n\nNow we are in a position to find the asymptotic behavior of the entropy in the limit of low temperatures in the configuration of two parallel plates one of which is metallic and the other one dielectric. By calculating the negative derivative of Eq. (75) with respect to temperature, one arrives at\n\nS(a, T ) = 3k B ζ(3)(ε 0 -1) 2 128π 3 a 2 (ε 0 + 1) τ 2 ( 89\n\n) × 1 - 8π 2 (ε 0 + 1) 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 135ζ(3)(ε 0 -1) 2 τ + O(τ 2 ) .\n\nThis equation is in analogy to Eq. (49) obtained for the case of two dielectrics. As is seen from Eq. ( 89 ), the entropy of the Casimir and van der Waals interactions between metal and dielectric plates vanishes when the temperature goes to zero, i.e., the Nernst heat theorem is satisfied. Note that the first term of order τ 2 on the right-hand side of Eq. (89) was already obtained in Ref. 61. It is notable also that at low temperatures the entropy goes to zero remaining positive. At the same time, as was shown in Ref. 61, at larger temperatures entropy is nonmonotonous and may take negative values. This interesting property distinguishes the configuration of metal and dielectric plates from two dielectric plates. In the latter configuration the October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 22 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko negative entropy appears only for nonphysical dielectrics with anomalously large and frequency independent dielectric permittivities. 53 Now let us consider the case τ ≫ 1, i.e., high temperatures (large separations). In the same way as for two dielectric plates, here the main contribution to the free energy is given by the term with l = 0 in Eq. (69),\n\nF (a, T ) = cτ 64π 2 a 3 ∞ 0 ydy ln 1 -r 0 e -y . ( 90\n\n)\n\nPerforming the integration in Eq. ( 90 ), we obtain\n\nF (a, T ) = - k B T 16πa 2 Li 3 (r 0 ) . ( 91\n\n)\n\nFrom this equation, for the Casimir pressure and entropy at τ ≫ 1 it follows\n\nP (a, T ) = - k B T 8πa 3 Li 3 (r 0 ) , S(a, T ) = k B 16πa 2 Li 3 (r 0 ) . ( 92\n\n)\n\nThe results (91) and (92) are analogous to (54) and (55) for two dielectric plates.\n\nIn the previous section it was supposed that the static dielectric permittivity ε 0 of the dielectric plate is finite. Now we will deal with the configuration of metal and dielectric plates with included dc conductivity of the dielectric material. In doing so the permittivity of the dielectric plate is given by\n\nε(iξ l ) = ε 0 + β(T ) l , ( 93\n\n)\n\nwhere all notations were introduced in and below Eq. (56). Thus, the reflection coefficients at zero frequency satisfy Eq. ( 4 ) for a plate made of ideal metal and Eq. (57) for a plate made of dielectric with included dc conductivity. Let us find the low-temperature behavior of the Casimir entropy and verify the consistency of the Lifshitz theory with thermodynamics in this nonstandard situation. For this purpose we substitute the dielectric permittivity (93) in Eq. (69) instead of ε 0 and find the Casimir energy F (a, T ) with included dc conductivity of a dielectric plate. In the same way as in Sec. 3, it is convenient to separate the zero-frequency term of F (a, T ) and subtract and add the usual zero-frequency contribution for metal-dielectric plates computed with the dielectric permittivity ε 0 ,\n\nF (a, T ) = k B T 16πa 2 ∞ 0 ydy ln 1 -e -y -ln 1 -r 0 e -y + k B T 16πa 2 ∞ 0 ydy ln 1 -r 0 e -y ( 94\n\n) + k B T 8πa 2 ∞ l=1 ∞ ζ l ydy ln 1 -r (ζ l , y)e -y + ln 1 -r⊥ (ζ l , y)e -y .\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 23\n\nHere the reflection coefficients r ,⊥ are calculated with the permittivity (93). We expand the third integral on the right-hand side of Eq. (94) in powers of the small parameter β/l. The zero-order contribution in this expansion together with the second integral of Eq. ( 94 ) form the Casimir free energy F (a, T ) calculated with dielectric permittivity ε 0 . The first integral on the right-hand side of Eq. (94) is calculated explicitly. As a result, Eq. ( 94 ) is rearranged to\n\nF (a, T ) = F (a, T ) - k B T 16πa 2 [ζ(3) -Li 3 (r 0 )] + Q(a, T ), ( 95\n\n)\n\nwhere Q(a, T ) contains the first and higher-order contributions in the expansion of the third integral on the right-hand side of Eq. (94) in powers of β/l. The explicit form of the main first-order term in Q(a, T ) is the following:\n\nQ 1 (a, T ) = k B T 8πa 2 ∞ l=1 β l ∞ ζ l dy y 2 e -y y 2 + ζ 2 l (ε 0 -1) (2 -ε 0 )ζ 2 l -2y 2 y 2 + ζ 2 l (ε 0 -1) + ε 0 y 2 × 1 1 -r (ζ l , y)e -y - ζ 2 l y 2 + ζ 2 l (ε 0 -1) + y 2 1 1 -r ⊥ (ζ l , y)e -y . ( 96\n\n)\n\nIn the same way as in Sec. 3, we expand the integrated function in Eq. (96) in powers of τ (bearing in mind that ζ l = τ l) and preserve only the main contribution at τ = 0:\n\nQ 1 (a, T ) = - k B T 4πa 2 ∞ l=1 β l ∞ ζ l dy yr 0 e -y (ε 2 0 -1) (1 -r 0 e -y ) ( 97\n\n) = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r n 0 n 2 ∞ l=1 e -nτ l l + nτ ∞ l=1 e -nτ l .\n\nDealing with this expression in the same way as with Eq. ( 64 ), we arrive at\n\nQ 1 (a, T ) ∼ e -b/T ln T. ( 98\n\n)\n\nThe Casimir entropy in the configuration of metal and dielectric plates with included dc conductivity of the dielectric plate is obtained as minus derivative of Eq. (95) with respect to temperature,\n\nS(a, T ) = S(a, T ) + k B 16πa 2 [ζ(3) -Li 3 (r 0 )] - ∂Q(a, T ) ∂T . ( 99\n\n)\n\nUsing Eq. ( 98 ), the calculation of the limiting value at T → 0 is straightforward:\n\nS(a, 0) = k B 16πa 2 [ζ(3) -Li 3 (r 0 )] > 0. ( 100\n\n)\n\nFrom this equation it follows that the inclusion of the dc conductivity of dielectric plate in the configuration metal-dielectric results in a violation of the Nernst heat theorem. In the above this result was obtained for a metallic plate made of ideal metal. It can be shown that it remains valid for a metallic plate made of real metal with finite conductivity.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 24 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThus, both configurations (two dielectric plates or one metal plate and one dielectric) lead to the same conclusion that when the dc conductivity is included in the model of dielectric response of the dielectric plate, the Lifshitz theory loses its consistency with thermodynamics. This confirms the conclusion made in Sec. 3 that the actual properties of dielectric materials at very low frequencies are in fact not related to the van der Waals and Casimir forces.\n\nAs was mentioned in the Introduction, semiconductors possess a wide variety of electric and optical properties ranging from metallic to dielectric. This opens the possibility to modulate the van der Waals and Casimir forces by changing the charge carrier density. Bearing in mind the discussed above problems on the consistency of the Lifshitz theory with thermodynamics, semiconductors can provide us with a test for the validity of different approaches. Thus, if for good dielectric the dc conductivity does not play any real role in the van der Waals and Casimir forces, the question arises on how much it should be increased in order to become a relevant factor in the description of dispersion forces. In Ref. 81 the Casimir force acting between two Si plates was calculated using the simple analytic expression for Si dielectric permittivity as a function of frequency. The complete tabulated optical data of Si were used in Ref. 75 to calculate the van der Waals interaction of different atoms with a Si wall. The first attempt to measure the van der Waals force between a glass lens and a Si plate and to modify it by light due to the change of carrier density was undertaken in Ref. 82. However, glass is a dielectric and therefore the electric forces due to localized point charges could not be controlled. This might explain that no force change occured on illumination at small separations where the effect should be most pronounced.\n\nThe first measurements of the Casimir force between a gold coated sphere and a single crystal Si plate were performed in Refs. 62, 63 by means of the atomic force microscope. The experiments used a p-type B doped Si plate of resistivity ρ = 0.0035 Ω cm. The chosen resistivity of the plate is in some sense intermediate between the resistivity of metals (which is usually two or three orders of magnitude lower) and the resistivity of dielectrics (it can be by about a factor of 10 5 larger; for instance, high-resistivity \"dielectric\" Si has the resistivity ρ 0 = 1000 Ω cm). Thus, the used Si plate had a relatively large absorption typical for semiconductors but it was also enough conductive to avoid the accumulation of charges. 63 In Fig. 2 , taken from Ref. 63, the differences of the theoretical and mean experimental Casimir forces acting between Au sphere and Si plate are presented as functions of separation. In Fig. 2a the theoretical force F theor is computed using the Lifshitz formula and the dielectric permittivity of a Si plate with the relatively low resistivity ρ used in experiment. This dielectric permittivity goes to infinity as ξ -1 with decreasing frequency (see the solid line in Fig. 3 ). In Fig. 2b the theo-October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 25 80 100 120 140 -10 -5 0 5 10 15 (nm) F theor -F expt (pN) 80 100 120 140 -10 -5 0 5 10 15\n\n(nm) F theor -F expt (pN) (b)\n\n(a) a a Fig. 2. Differences of the theoretical and mean experimental Casimir forces versus separation. Theoretical forces are computed (a) for the Si plate used in experiment and (b) for dielectric Si. Solid and dashed lines indicate 95 and 70% confidence intervals, respectively.\n\nretical Casimir force F theor is computed using the dielectric permittivity of the Si plate made of \"dielectric\" Si with high resistivity ρ 0 . The dielectric permittivity of high-resistivity Si is shown by the dashed line in Fig. 3 . It is characterized by the finite static value ε Si (0) = 11.67. The solid and dashed lines in Fig. 2a ,b indicate the 95% and 70% confidence intervals, respectively. As is seen from Fig. 2 , the theoretical approach using the dielectric permittivity of high-resistivity \"dielectric\" Si is excluded by experiment within the separation range from 60 to 110 nm at 70% confidence. At the same time, the theory using the dielectric permittivity of Si with a low resistivity ρ is consistent with experiment.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 26 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\n13.5 14 14.5 15 15.5 16 16.5 17 5 10 15 20 25 Log 10 [ξ (rad/s)] ε (iξ) D Fig. 3. Dielectric permittivity of Si plate used in experiment along the imaginary frequency axis (solid line). Dashed line shows the dielectric permittivity of dielectric Si.\n\nThe above results suggest an approach on how to correctly determine the possible role of the low-frequency conductivity properties in dispersion forces. As is seen from Fig. 3 , the dielectric permittivity of low-resistivity Si (solid line) significantly departs from the dielectric permittivity of \"dielectric\" Si in the region around the important dimensional parameter of the problem, the characteristic frequency c/(2a) ∼ 10 14 -10 15 rad/s. That is why the low-resistivity sample cannot be described at low frequencies by the static dielectric permittivity of Si equal to ε Si (0) = 11.67. To describe it, the term β(T )/l, like in Eq. ( 93 ), should be added to ε Si (0). Note that in this case β(T )/l > 1 at the first Matsubara frequencies with l = 1, 2, 3, . . . and, thus, this quantity cannot be considered as a small parameter. At the same time, for a high-resistivity sample the inclusion of the dc conductivity would lead to deviations from the dashed line in Fig. 3 only at frequencies ξ < 10 8 rad/s, which are much less than the characteristic frequency. This comparison permits to make a conclusion in what experimental situations the conductivity properties of semiconductors at low frequencies should be taken into account and when they should be omitted as being not related to dispersion forces.\n\nThe future experiments on the modification of semiconductor charge carrier density by laser light 83 will bring a more clear understanding of this problem on the connection between the low-frequency material properties and dispersion forces.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 27\n\nThe presented above new analytic results on the low-temperature behavior of the Casimir free energy, pressure and entropy between dielectrics or between dielectric and metal are based on the conventional Lifshitz theory which describes dielectric materials by means of the frequency dependent dielectric permittivity. In fact, the assumption that the material of the plates possesses only the frequency dispersion means that the components of electric displacement are connected with the components of electric field by the relation\n\nD k (r, ω) = ε kl (r, ω)E l (r, ω). ( 101\n\n)\n\nThis equation is central in all different derivations of the Lifshitz formula (see, e.g., Refs. 2, 6, 36-38, 69, 70, 84). The effects of spatial nonlocality (spatial dispersion) are in fact essential only at shortest separations between the plates comparable with atomic dimensions and also for metals at sufficiently large separations (typically of about 2-3 µm) in the frequency region of the anomalous skin effect. The Casimir force in the latter region was described by the Lifshitz theory reformulated in terms of the Leontovich impedance. 85 Recently the spatial dispersion came to the attention in connection with the thermal Casimir force. 64, 65, 66, 67, 68 In particular, it was claimed 64, 65 that for real metals at any separation the account of spatial dispersion leads to practically the same result (6) for the reflection coefficients at zero frequency as was obtained earlier using the Drude model dielectric function (5). This conclusion, if it is correct, not only returns us to the contradiction with experiment 30,32,33 but also casts doubts on all results obtained by means of the conventional Lifshitz theory accounting for only the frequency dispersion. It is natural when the spatial dispersion contributes a small fraction of a percent as it is generally believed in numerous applications of the Lifshitz theory. It is, however, quite another matter when the account of the spatial dispersion results in some \"dramatic effects\", 64 i.e., in several hundred times larger thermal correction than is obtained in the local case. Below we demonstrate that the conclusions of Refs. 64 -68 are in fact not reliable because they use the Lifshitz theory of dispersion forces outside of its application range. 86 To find the electromagnetic modes associated with an empty gap between the plates, Refs. 64 -68 use the standard continuity boundary conditions,\n\nE 1t = E 2t , B 1n = B 2n , D 1n = D 2n , B 1t = B 2t , ( 102\n\n)\n\nwhich are commonly applied in the derivation of the Lifshitz formula for spatially local nonmagnetic materials. Here B is the magnetic induction, n is the normal to the boundary directed inside the medium, the subscripts n, t refer to the normal and tangential components, respectively, the subscript 1 refers to the vacuum and subscript 2 to the plate material. In Refs. 64 -68 the spatial dispersion is described by the longitudinal and transverse dielectric permittivities depending on the wave vector and frequency: ε kl = ε kl (q, ω). However, as is shown below, in the theory October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 28 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko of the Casimir effect both the boundary conditions (102) and permittivities ε kl = ε kl (q, ω) are inapplicable. We start from the boundary conditions and recall the set of Maxwell equations in a metal describing the Casimir effect, rotE + 1 c ∂B ∂t = 0, divD = 0, rotB -1 c ∂D ∂t = 0 divB = 0. (103) Equations (103) do not contain any external, i.e., independent on E, D and B, current or charge densities. The definition of the electric displacement is\n\n∂D ∂t = ∂E ∂t + 4πi, ( 104\n\n)\n\nwhere the volume current i is induced by E and B and takes into account the conduction electrons.\n\nIn electrodynamics with spatial dispersion the electric field and magnetic induction are finite at the boundary surfaces, whereas the electric displacement can tend to infinity. 87 Then, integrating Eqs. (103) over the thickness of the boundary layer as is done in Ref. 88, we reproduce the first two conditions in Eq. ( 102 ) and arrive at the modified third and fourth conditions, 87,89\n\nE 1t = E 2t , B 1n = B 2n , D 2n -D 1n = 4πσ, [n × (B 2 -B 1 )] = 4π c j, ( 105\n\n)\n\nwhere the induced surface charge and current densities are given by\n\nσ = 1 4π 2 1 div[n × [D × n]]dl, j = 1 4π 2 1 ∂D ∂t dl. ( 106\n\n)\n\nNote that the boundary conditions (105), (106) are obtained from the macroscopic Maxwell equations for physical fields. They should not be mixed with the boundary conditions arising in perturbative theories and for the fictitious fields (see below). In linear electrodynamics for a medium with time-independent properties without spatial dispersion the material equation connecting the electric displacement and electric field takes the form\n\nD k (r, t) = t -∞ εkl (r, t -t ′ )E l (r, t ′ )dt ′ . ( 107\n\n)\n\nAccording to this equation, the electric displacement at a point r and moment t is determined by the electric field at the same point r at different moments t ′ ≤ t (the spatial dispersion is absent but the temporal may be present). It is easily seen that the substitution of Eq. (107) in Eq. (106) leads to σ = 0, j= 0 and, as a result, the boundary conditions (105) coincide with the standard continuity conditions (102). It is unjustified, however, to use conditions (102) in the presence of spatial dispersion. In Refs. 87, 90 a few examples are presented illustrating that in this case neither σ nor j is equal to zero. We now turn to a discussion of the use of dielectric permittivity ε kl (q, ω) in the theory of the Casimir effect with account of spatial dispersion. In the presence of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 29 only frequency dispersion, it is possible to perform the Fourier transformation of the fields E(r, t) = ∞ -∞ E(r, ω)e -iωt dω, D(r, t) = ∞ -∞ D(r, ω)e -iωt dω, (108) in Eq. ( 107 ) and arrive at Eq. (101) where\n\nε kl (r, ω) = ∞ 0 εkl (r, τ )e iωτ dτ ( 109\n\n)\n\nis the frequency-dependent dielectric permittivity and τ ≡ t -t ′ . In fact Eqs. (101), (108) and (109) are used in parallel with the boundary conditions (102) in all derivations of the Lifshitz formula. 2,6,36,37,38,69,70, 84 If the material of the plates is characterized not only by temporal but also spatial dispersion, Eq. (107) is generalized to\n\nD k (r, t) = t -∞ dt ′ dr ′ εkl (r, r ′ , t -t ′ )E l (r ′ , t ′ ). ( 110\n\n)\n\nIf the material medium were uniform in space (i.e., all points were equivalent), the kernel ε would not depend on r and r ′ separately, as in Eq. ( 110 ), but on the difference R ≡ rr ′ . In this case, by performing the Fourier transformation, E(r, t) = ∞ -∞ dω dqE(q, ω)e i (qr-ωt) , D(r, t) = ∞ -∞ dω dqD(q, ω)e i (qr-ωt) , (111) and substituting it in Eq. ( 110 ), one could introduce the dielectric permittivities\n\nε kl (q, ω) = ∞ 0 dτ dR εkl (R, τ )e -i(qR-ωτ ) , ( 112\n\n)\n\nas Refs. 64-68 do, and rearrange Eq. (110) to the form\n\nD k (q, ω) = ε kl (q, ω)E l (q, ω). ( 113\n\n)\n\nIn the Casimir effect, however, the material medium is not uniform due to the presence of a macroscopic gap between the two plates (half spaces). Because of this, the assumption that the kernel ε depends only on R and τ is wrong. As a result, it is not possible to introduce the dielectric permittivity ε kl (q, ω) depending on the wave vector and frequency. In fact, for systems with spatial dispersion in the presence of boundaries the kernel ε depends not only on R and τ but also on the distance from the boundary. 87 In this complicated situation the following approximate phenomenological approach is sometimes applicable. 87 For electromagnetic waves with a wavelength λ the kernel ε(r, r ′ , τ ) in Eq. (110) differs essentially from zero only in a certain vicinity of the point r with characteristic dimensions l ≪ λ (for nonmetallic condensed media l is of the order of the lattice constant). Then it is reasonable to assume that ε is a function of R=r-r ′ , except for a layer of thickness l adjacent to the boundary surface. If one is mostly interested in bulk phenomena October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 30 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and neglects the role and influence of a subsurface layer, the quantity ε kl (q, ω) may be employed as a reasonable approximation. This approximate phenomenological approach is widely applied in the theoretical investigation of the anomalous skin effect (see, e.g., Ref. 91). Note that in Ref. 91 some kind of fictitious infinite system was introduced and electromagnetic fields in this system are discontinuous on the boundary surface. This discontinuity should not be confused with the discontinuity of physical fields of a real system in the presence of spatial dispersion described by Eqs. (105) and (106) . (There is also another approach to the description of the anomalous skin effect in polycrystals using the generalizations of the local Leontovich impedance which takes into account the shape of Fermi surface 92 ). The frequency and wave vector dependent dielectric permittivity in the presence of boundaries is also approximately applied in the theory of radiative heat transfer 93 or in the study of electromagnetic interaction of molecules with metal surfaces. 94 In all these applications the boundary effects are usually taken into account by the boundary conditions (105) supplemented by so called \"additional boundary conditions\".\n\nIt is unlikely, however, that the approximate phenomenological approach using such quantity as ε kl (q, ω) in the presence of boundaries would be applicable in the theory of the Casimir force where the boundary effects on the zero-point electromagnetic oscillations are of prime importance. It is notable also that this approach faces serious theoretical difficulties including the violation of the law of conservation of energy. 95 It is then not surprising that the application of this approach in Refs. 64, 65 results in the contribution to the Casimir free energy from the transverse electric mode which is in contradiction with experiment. 30,32,33 One more shortcoming of Refs. 64-68 is that they substitute the dielectric permittivity ε kl (q, ω), depending on both wave vector and frequency, into the conventional Lifshitz formula derived in the presence of only temporal dispersion. In the famous review paper 96 it has been noticed, however, that with the inclusion of spatial dispersion the free energy of a fluctuating field takes the form F = F L + ∆F , where F L is given by the conventional Lifshitz expression derived in a spatially local case and written in terms of the Fresnel reflection coefficients, and ∆F is an additional term which can be expressed in terms of the thermal Green's function of the electromagnetic field and polarization operator. Review 96 calls as not reliable the results of, e.g., Ref. 97 obtained by the substitution of dielectric permittivity ε kl (q, ω), taking account of spatial dispersion, into the conventional Lifshitz formula. It can be true that the Lifshitz formula written in terms of general reflection coefficients is applicable in both spatially local and nonlocal situations. However, as far as the exact reflection coefficients in a spatially nonlocal case are unknown, the use of some approximate phenomenological models, elaborated in literature for applications different than the Casimir effect, may lead to incorrect results for ∆F and create inconsistencies with experiment.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 31\n\nTo conclude this section, the results of Refs. 64-68 on the influence of spatial nonlocality on the Casimir interaction are shown to be not reliable. Although at present there is no fundamental theory of the thermal Casimir force incorporating spatial dispersion, there is no reason to expect that it can play any significant role in the frequency region of infrared optics (experimental separations) or normal skin effect (i.e., at separations between plates greater than 4-5 µm).\n\nIn the foregoing we have presented the derivation of analytic asymptotic expressions for the free energy, pressure and entropy of the Casimir interaction between two dielectric plates and between metal and dielectric plates at low and high temperatures. It was shown that the low-temperature behavior of the Casimir interaction between dielectrics and between dielectric and metal is determined by the static dielectric permittivities of nonpolar dielectrics. The obtained results were shown to be in agreement with thermodynamics when the static dielectric permittivities of dielectrics are finite. In particular, the entropy of the Casimir interaction goes to zero when the temperature vanishes, i.e., the Nernst heat theorem is satisfied. This demonstrates the consistency of the original Lifshitz's approach to the van der Waals forces between dielectrics which disregards the small conductivity of dielectrics at constant current.\n\nThe second important result shown above is that the inclusion of the dc conductivity of dielectrics into the model of dielectric response leads not to some small corrections to the characteristics of the Casimir interaction, as one could expect, but makes the Lifshitz theory inconsistent with thermodynamics leading to the violation of the Nernst heat theorem. This reveals that real material properties at very low, quasistatic frequencies are in fact not related to the phenomenon of the van der Waals and Casimir forces which is actually determined by sufficiently high frequencies. In this case the zero-frequency contribution to the Casimir force should be understood not literally but as analytic continuation to zero of the material physical behavior in the region around the characteristic frequency.\n\nThe presented results provide a basis for the calculation of the van der Waals and Casimir forces between real materials. Such calculations are much needed for numerous applications of the Casimir force discussed in the Introduction, in particular for the applications in nanotechnology and for constraining predictions of fundamental physical theories beyond the Standard Model. Bearing in mind that semiconductors are the main constituent materials in nanotechnological devices, it is a subject of high priority to understand the Casimir effect with semiconductor boundaries. In this connection we have discussed new experimental results and theoretical ideas on the Casimir interaction between a metal sphere and semiconductor plate. It was stressed that by changing the charge carrier density in the semiconductor it is possible to bring it in different intermediate states between metallic and dielectric. In this case the problem arises when the conductivity properties of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 32 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko semiconductor are not related to dispersion forces and when they are becoming relevant. A criterion for the resolution of this problem was formulated based on the relation between the typical frequency at which the dc conductivity properties come into play and the characteristic frequency of the Casimir effect. In fact the thermal Casimir interaction between semiconductors remains an open question and much work should be done both in experiment and theory to gain a better insight into this subject.\n\nThe last major problem discussed in this review is whether or not the spatial dispersion influences essentially the thermal Casimir force between real materials. In the present literature there is no agreement on this subject. We adduced arguments in favor of the statement that in the region of experimental separations the influence of the spatial dispersion on the Casimir force is negligible small. The statements on the opposite, contained in the literature, were shown to be not reliable because they are obtained by the application of the Lifshitz theory outside of its application range. At the same time it was ascertained that at the moment there is no consistent fundamental theory of the van der Waals and Casimir forces taking the spatial dispersion into account. This is the problem to solve in the foreseeable future. ACKNOWLEDGMENTS G.L.K. and V.M.M. are grateful to the Center of Theoretical Studies and the Institute for Theoretical Physics, Leipzig University for their kind hospitality. This work was supported by Deutsche Forschungsgemeinschaft grant 436 RUS 113/789/0-2. References 1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948). 2. P. W. Milonni, The Quantum Vacuum (Academic Press, San Diego, 1994). 3. V. M. Mostepanenko and N. N. Trunov, The Casimir Effect and its Applications (Clarendon Press, Oxford, 1997). 4. K. A. Milton, The Casimir Effect (World Scientific, Singapore, 2001). 5. M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999). 6. M. Bordag, U. Mohideen and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001). 7. S. K. Lamoreaux, Rep. Progr. Phys. 68, 201 (2005). 8. P. Candelas and S. Weinberg, Nucl. Phys. B237, 397 (1984). 9. M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D58, 075003 (1998); D60, 055004 (1999); D62, 011701(R) (2000). 10. J. C. Long, H. W. Chan and J. C. Price, Nucl. Phys. B539, 23 (1999). 11. E. Fischbach, D. E. Krause, V. M. Mostepanenko and M. Novello, Phys. Rev. D64, 075010 (2001). 12. V. M. Mostepanenko, Int. J. Mod. Phys. A17, 722 (2002); A17, 4143 (2002). 13. G. L. Klimchitskaya and U. Mohideen, Int. J. Mod. Phys. A17, 4143 (2002). 14. J. F. Babb, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A70, 042901 (2004). 15. M. Antezza, L. P. Pitaevskii and S. Stringari, Phys. Rev. A70, 053619 (2004). 16. M. Krech, The Casimir Effect in Critical Systems (World Scientific, Singapore, 1994).\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 33 17. H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D.J. Bishop and F. Capasso, Science 291, 1941 (2001); Phys. Rev. Lett. 87, 211801 (2001). 18. E. V. Blagov, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B71, 235401 (2005). 19. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer, Berlin, 1995). 20. S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997). 21. U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998). 22. G. L. Klimchitskaya, A. Roy, U. Mohideen and V. M. Mostepanenko, Phys. Rev. A60, 3487 (1999). 23. A. Roy and U. Mohideen, Phys. Rev. Lett. 82, 4380 (1999). 24. A. Roy, C.-Y. Lin and U. Mohideen, Phys. Rev. D60, 111101(R) (1999). 25. B. W. Harris, F. Chen and U. Mohideen, Phys. Rev. A62, 052109 (2000). 26. T. Ederth, Phys. Rev. A62, 062104 (2000). 27. G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, Phys. Rev. Lett. 88, 041804 (2002). 28. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. Lett. 88, 101801 (2002). 29. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A66, 032113 (2002). 30. R. S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. López and V. M. Mostepanenko, Phys. Rev. D68, 116003 (2003). 31. F. Chen, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Phys. Rev.\n\nA69, 022117 (2004). 32. R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya, D. E. Krause and V. M. Mostepanenko, Ann. Phys. (N.Y.) 318, 37 (2005). 33. G. L. Klimchitskaya, R. S. Decca, D. López, E. Fischbach, D. E. Krause and V. M. Mostepanenko, Int. J. Mod. Phys. A20, 2205 (2005). 34. Y. Srivastava, A. Widom and M. H. Friedman, Phys. Rev. Lett. 55, 2246 (1985). 35. Y. Srivastava and A. Widom, Phys. Rep. 148, 1 (1987). 36. E. M. Lifshitz, Sov. Phys. JETP 2, 73 (1956). 37. I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, Sov. Phys. Usp. 4, 153 (1961). 38. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part II (Pergamon, Oxford, 1980). 39. J. Schwinger, L. L. DeRaad and K. A. Milton, Ann. Phys. (N.Y.) 115, 1 (1978). 40. L. S. Brown and G. J. Maclay, Phys. Rev. 184, 1272 (1969). 41. M. Boström and B. E. Sernelius, Phys. Rev. Lett. 84, 4757 (2000). 42. J. Feinberg, A. Mann and M. Revzen, Ann. Phys. (N.Y.) 288, 103 (2001). 43. A. Scardiccio and R. L. Jaffe, Nucl. Phys. B743, 249 (2006). 44. C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A62, 012110 (2000). 45. M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. Lett.\n\n85, 503 (2000). 46. J. S. Høye, I. Brevik, J. B. Aarseth and K. A. Milton, Phys. Rev. E67, 056116 (2003). 47. I. Brevik, J. B. Aarseth, J. S. Høye and K. A. Milton, Phys. Rev. E71, 056101 (2005). 48. V. B. Bezerra, G. L. Klimchitskaya and C. Romero, Phys. Rev. A65, 012111 (2002). 49. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A67, 062102 (2003). 50. J. R. Torgerson and S. K. Lamoreaux, Phys. Rev. E70, 047102 (2004). 51. S. K. Lamoreaux and W. T. Buttler, Phys. Rev. E71, 036109 (2005). 52. V. B. Bezerra, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A66, 022112 (2002).\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 34 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 53. V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko and C. Romero, Phys. Rev. A69, 022119 (2004). 54. V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. López, V. M. Mostepanenko and C. Romero, Phys. Rev. E73, 028101 (2006). 55. I. Brevik, S. A. Ellingsen and K. A. Milton, New J. Phys. 8, 236 (2006). 56. V. M. Mostepanenko, V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. López and C. Romero, J. Phys. A39, 6589 (2006). 57. J. R. Zurita-Sánchez, J.-J. Greffet and L. Novotny, Phys. Rev. A69, 022902 (2004). 58. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati and J.-J. Greffet, Surf. Sci. Rep.\n\n57, 59 (2005). 59. B. S. Stipe, H. J. Mamin, T. D. Stowe, T. W. Kenny and D. Rugar, Phys. Rev. Lett.\n\n87, 096801 (2001). 60. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D72, 085009 (2005). 61. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A72, 022111 (2005). 62. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A72, 020101(R) (2005); A73, 019905(E) (2006). 63. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev.\n\nA74, 022103 (2006). 64. B. E. Sernelius, Phys. Rev. B71, 235114 (2005). 65. V. B. Svetovoy and R. Esquivel, Phys. Rev. E72, 036113 (2005). 66. R. Esquivel and V. B. Svetovoy, Phys. Rev. A69, 062102 (2004). 67. R. Esquivel, C. Villarreal and W. L. Mochán, Phys. Rev. A68, 052103 (2003). 68. A. M. Contreras-Reyes and W. L. Mochán, Phys. Rev. A72, 034102 (2005). 69. J. Mahanty and B. W. Ninham, Dispersion Forces (Academic, New York, 1976). 70. V. A. Parsegian and B. W. Ninham, Nature 224, 1197 (1969). 71. L. Bergström, Adv. Coll. Interface Sci. 70, 125 (1997). 72. H. Mitter and D. Robaschik, Eur. Phys. J. B13, 335 (2000). 73. B. W. Ninham and J. Daicic, Phys. Rev. A57, 1870 (1998). 74. G. L. Klimchitskaya, B. Geyer and V. M. Mostepanenko, J. Phys. A39, 6495 (2006). 75. A. O. Caride, G. L. Klimchitskaya, V. M. Mostepanenko and S. I. Zanette, Phys. Rev. A71, 042901 (2005). 76. Handbook of Optical Constants of Solids, ed. E. D. Palik (Academic, New York, 1985). 77. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Int. J. Mod. Phys. A16, 3291 (2001). 78. J. C. Slater, Insulators, Semiconductors and Metals. Quantum Theory of Molecules and Solids, Vol. 3, (McGraw-Hill, New York, 1967). 79. Materials Science and Engineering Handbook, eds. J. F. Shackelford and W. Alexander (CRC, Boca Raton, 2001). 80. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A65, 062109 (2002). 81. N. Inui, J. Phys. Soc. Jpn. 72, 2198 Jpn. 72, (2003);); 73, 2198 (2004); 75, 024004 (2006) . 82. W. Arnold, S. Hunklinger and K. Dransfeld, Phys. Rev. B19, 6049 (1979). 83. F. Chen and U. Mohideen, J. Phys. A39, 6233 (2006). 84. C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A67, 043811 (2003). 85. E. I. Kats, Sov. Phys. JETP 46, 109 (1977). 86. G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B75, 036101 (2007). 87. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer, Berlin, 1984).\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 35 88. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). 89. V. L. Ginzburg, Physics and Astrophysics (Pergamon Press, Oxford, 1985). 90. V. M. Agranovich and V. L. Ginzburg, Sov. Phys. JETP 36, 440 (1973). 91. K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968). 92. I. M. Kaganova and M. I. Kaganov, Phys. Rev. B63, 054202 (2001). 93. S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 3 (Springer, Berlin, 1989). 94. G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984). 95. J. T. Foley and A. J. Devaney, Phys. Rev. B12, 3104 (1975). 96. Yu. S. Barash and V. L. Ginzburg, Sov. Phys. Usp. 18, 305 (1975). 97. G. G. Kleinman and U. Landman, Phys. Rev. Lett. 33, 524 (1974)."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"INTRODUCTION","text":"The Casimir effect 1 is the force and also the specific polarization of the vacuum arising in restricted quantization volumes and originating from the zero-point oscillations of quantized fields. This force acts between two closely spaced macrobodies, between an atom or a molecule and macrobody or between two atoms or molecules. During more than fifty years, passed after the discovery of the Casimir effect, it has attracted much theoretical attention because of numerous applications in quantum field theory, atomic physics, condensed matter physics, gravitation and cosmology, 1 October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 4 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThese findings on the application of the Lifshitz theory to real metals have inspired a renewed interest in the Casimir force between dielectrics. As was mentioned above, at nonzero temperature dielectrics possess an although small but not equal to zero dc conductivity. In Ref. 57, 58 the van der Waals force arising from the dc conductivity of a dielectric plate was shown to lead to large effect in noncontact atomic friction, 59 a phenomenon having so far no satisfactory theoretical explanation. This brings up the question: Is it necessary or possible to take into account the dc conductivity of dielectrics in the Lifshitz theory? Recall that in the case of a positive answer the static dielectric permittivity of a dielectric material would be infinitely large. It is amply clear that the resolution of the above issue should be in accordance with the fundamentals of thermodynamics. For this reason, it is desirable to investigate the low-temperature behavior of the Casimir free energy and entropy for two dielectric plates both with neglected and included effects of the dc conductivity.\n\nA major breakthrough in the investigation of this problem was achieved in the year 2005. In Ref. 60 a new variant of perturbation theory was developed in a small parameter proportional to the product of the separation distance between the plates and the temperature. As a result, the behavior of the Casimir free energy, entropy and pressure at low temperatures was found analytically. If the static dielectric permittivity is finite, the thermal correction was demonstrated to be in accordance with thermodynamics. This solves positively the fundamental problem about the agreement between the Lifshitz theory and thermodynamics for the case of two dielectric plates. In Ref. 60 it was shown that, on the contrary, the formal inclusion of a small conductivity of dielectric plates at low frequencies into the model of their dielectric response leads to a violation of the Nernst heat theorem. This result gives an important guidance on how to extrapolate the tabulated optical data for the complex refractive index to low frequencies in numerous applications of the van der Waals and Casimir forces. All these problems and related ones arising for semiconductor materials are discussed in this review.\n\nIn Sec. 2 we derive the analytical behavior of the Casimir free energy, entropy and pressure in the configuration of two parallel dielectric plates at both low and high temperature. It is demonstrated that if the static dielectric permittivity is finite the Lifshitz theory is in agreement with thermodynamics. Sec. 3 contains the derivation of the low-temperature behavior for the Casimir free energy and entropy between two dielectric plates with included dc conductivity. In this case the Lifshitz theory is found to be in contradiction with the Nernst heat theorem. The conclusion is made that the conductivity properties of a dielectric material at a constant current are unrelated to the van der Waals and Casimir forces and must not be included into the model of dielectric response. In Sec. 4 we consider the thermal Casimir force between metal and dielectric. This problem was first investigated in Ref. 61. It was found that for dielectric plate with finite static dielectric permittivity the Nernst heat theorem is satisfied but the Casimir entropy may take negative values. Here we not only reproduce an analytical proof of the Nernst heat theorem but also October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 5 find the next perturbation orders in the expansion of the Casimir free energy and entropy in powers of a small parameter. The results obtained in high-temperature limit are also provided. Sec. 5 is devoted to the Casimir interaction between metal and dielectric plates with included dc conductivity of the dielectric material. We demonstrate that in this case the Nernst heat theorem is violated. Sec. 6 contains the discussion of semiconductors which present a wide variety of electric properties varying from metallic to dielectric. We consider the Casimir interaction between metal and semiconductor test bodies and formulate the criterion when it is appropriate to include the dc conductivity of a semiconductor into the model of dielectric response. In doing so, the results of recent experiments 62,63 on the measurement of the Casimir force between metal and semiconductor test bodies are taken into account. To this point the assumption has been made that metal, dielectric and semiconductor materials of the Casimir plates possess only temporal dispersion, i.e., can be described by the dielectric permittivity depending only on frequency. In Sec. 7 we discuss recent controversial results by different authors (see, e.g., Refs. 64 -68) attempting to take into account also spatial dispersion. As is shown in this section, the way of inclusion of spatial dispersion into the Lifshitz theory, used in Refs. 64 -68, is unjustified. We argue that the account of spatial dispersion cannot influence theoretical results obtained with the help of usual, spatially local, Lifshitz theory within presently used ranges of experimental separations. Sec. 8 contains our conclusions and discussion."},{"section_type":"RESULTS","section_title":"NEW ANALYTICAL RESULTS FOR THE THERMAL CASIMIR FORCE BETWEEN DIELECTRICS","text":"We consider two thick dielectric plates (semispaces) described by the frequencydependent dielectric permittivity ε(ω) and restricted by the parallel planes z = ±a/2 with a separation a between them, in thermal equilibrium at temperature T . The Lifshitz formula for the free energy of the van der Waals and Casimir interaction between the plates is given by 2,3,4,5,6,36,37,\n\n38 F (a, T ) = k B T 2π ∞ l=0 1 - δ l0 2 ∞ 0 k ⊥ dk ⊥ ( 1\n\n) × ln 1 -r 2 (ξ l , k ⊥ )e -2aq l + ln 1 -r 2 ⊥ (ξ l , k ⊥ )e -2aq l ,\n\nwhere the reflection coefficients for two independent polarizations of electromagnetic field are defined as\n\nr (ξ l , k ⊥ ) = ε l q l -k l ε l q l + k l , r ⊥ (ξ l , k ⊥ ) = k l -q l k l + q l . ( 2\n\n)\n\nHere k ⊥ is the magnitude of the wave vector in the plane of plates, ξ l = 2πk B T l/ are the Matsubara frequencies, k B is the Boltzmann constant, ε l = ε(iξ l ), and q l = ξ 2 textDr 6 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThe problems in the application of the Lifshitz theory to real materials discussed in the Introduction are closely connected with the values of the reflection coefficients at zero Matsubara frequency. For later use we discuss it for the various cases.\n\n• For ideal metals 40 it holds r (0, k ⊥ ) = r ⊥ (0, k ⊥ ) = 1.\n\n(\n\n) 4\n\n• For real metals described by the dielectric function of the Drude model,\n\nε(iξ l ) = 1 + ω 2 p ξ l [ξ l + ν(T )] , ( 5\n\n)\n\nwhere ω p is the plasma frequency and ν(T ) is the relaxation parameter, it holds 41 r (0, k ⊥ ) = 1, r ⊥ (0, k ⊥ ) = 0. (6) Eq. ( 6 ) results in the discontinuity between the cases of ideal and real metals and leads to the violation of the Nernst heat theorem for metallic plates having perfect crystal lattices. 52,53,54,56 • For real metals described by the dielectric function of the plasma model, ε(iξ l ) = 1 + ω 2\n\np ξ 2 l , ( 7\n\n)\n\nfrom Eq. (2) it follows: 44,45 r (0, k ⊥ ) = 1, r ⊥ (0, k ⊥ ) =\n\nc 2 k 2 ⊥ + ω 2 p -ck ⊥ c 2 k 2 ⊥ + ω 2 p + ck ⊥ . ( 8\n\n)\n\nHere, in the limit of ideal metals (ω p → ∞) the continuity is preserved because r ⊥ (0, k ⊥ ) in Eq. ( 8 ) goes to unity. The free energy (1) calculated with the permittivity (8) is also consistent with thermodynamics. • For dielectrics and semiconductors the dielectric permittivities at the imaginary Matsubara frequencies are given by the Ninham-Parsegian representation, 69,70 ε(iξ l ) = 1 + j\n\nC j 1 + ξ 2 l /ω 2 j , ( 9\n\n)\n\nwhere the parameters C j are the absorption strengths satisfying the condition j\n\nC j = ε 0 -1 ( 10\n\n)\n\nand ω j are the characteristic absorption frequencies. Here, the static dielectric permittivity ε 0 ≡ ε(0) is supposed to be finite. Although Eq. (10) is an October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 7 approximate one, it gives a very accurate description for many materials. 71 By the substitution of Eq. (9) in Eq. ( 2 ) one arrives at r (0, k ⊥ ) ≡ r 0 = ε 0 -1 ε 0 + 1 , r ⊥ (0, k ⊥ ) = 0. (11)\n\nNote that the vanishing of the transverse reflection coefficient for dielectrics at zero frequency in Eq. (11) has another meaning than for the Drude metals in Eq. ( 6 ). For Drude metal the parallel reflection coefficient is equal to the physical value for real photons at normal incidence, i.e., to unity, and the transverse one vanishes instead of taking unity, its physical value. This results in the violation of the Nernst heat theorem for perfect crystal lattices. In the case of dielectrics both reflection coefficients at zero frequency in Eq. (11) depart from the physical value for real photons which is equal to ( √ ε 0 -1)/( √ ε 0 + 1). In this case, however, one of them is larger and the other one is smaller than the physical value. As we will see below, this leads to the preservation of Nernst's heat theorem confirming that Eq. (9), despite being approximate, describes the material properties of dielectric and semiconductor plates in a thermodynamic consistent way.\n\nNow we derive the analytic representation for the Casimir free energy in Eq. (1) at low temperatures. For convenience in calculations, we introduce the dimensionless variables\n\nζ l = ξ l ξ c = 2aξ l c = τ l, y = 2aq l , ( 12\n\n)\n\nwhere ξ c = c/(2a) is the characteristic frequency, τ = 4πk B aT /( c), and q l was defined in Eq. ( 3 ). Then the Lifshitz formula (1) takes the form\n\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l dy f (ζ l , y) , ( 13\n\n)\n\nwhere f (ζ, y) = f (ζ, y) + f ⊥ (ζ, y), (14) f ,⊥ (ζ, y) = y ln 1 -r 2 ,⊥ (ζ, y)e -y , (15) and reflection coefficients (2), in terms of variables (12) , being given by\n\nr (ζ l , y) = ε l y -y 2 + ζ 2 l (ε l -1) ε l y + y 2 + ζ 2 l (ε l -1) , r ⊥ (ζ l , y) = y 2 + ζ 2 l (ε l -1) -y y 2 + ζ 2 l (ε l -1) + y . ( 16\n\n)\n\nTo separate the temperature independent contribution and thermal correction in Eq. ( 13 ) we apply the Abel-Plana formula, 3,6\n\n∞ l=0 1 - δ l0 2 F (l) = ∞ 0 F (t) dt + i ∞ 0 dt F (it) -F (-it) e 2πt -1 , ( 17\n\n)\n\nwhere F (z) is an analytic function in the right half-plane. Here, taking it as\n\nF (x) = ∞ x dy f (x, y) ( 18\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 8 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nand using Eq. (17), we can identically rearrange Eq. (13) to the form F (a, T ) = E(a) + ∆F (a, T ) , (19) where E(a) is the energy of the van der Waals or Casimir interaction at zero temperature,\n\nE(a) = c 32π 2 a 3 ∞ 0 dζ ∞ ζ dy f (ζ, y) , ( 20\n\n)\n\nand ∆F (a, T ) is the thermal correction to this energy, ∆F (a, T ) = i cτ 32π 2 a 3 ∞ 0 dt F (iτ t) -F (-iτ t) e 2πt -1 . (21) Note that, in fact, Eq. ( 21 ) describes the dependence of the free energy on the temperature arising from the dependence on temperature of the Matsubara frequencies. Thus, ∆F (a, T ) in (21) coincides with the thermal correction to the energy, defined as F (a, T ) -F (a, 0), only for plate materials with temperature independent properties. The asymptotic expressions for the energy E(a) at both short and large separations are well known. 6,37,38 Below we find the asymptotic expressions for the thermal correction (21) under the conditions τ ≪ 1 and τ ≫ 1. Taking into account the definition of τ in Eq. ( 12 ), the asymptotic expressions at τ ≪ 1 are applicable both at small and large separations if the temperature is sufficiently low.\n\nWe begin with condition τ ≪ 1. Let us substitute Eq. (9) in Eqs. (14) -(16), expand the function f (x, y) in powers of x = τ t, and than integrate the obtained expansion with respect to y from x to infinity in order to find F (x) in Eq. (18) and F (ix) -F (-ix) in Eq. (21) .\n\nIt is easy to check that f ⊥ (ζ, y) does not contribute to the leading, second, order in the expansion of F (ix) -F (-ix) in powers of x. Thus, we can restrict ourselves by the consideration of the expansion f (x, y) = y ln 1 -r 2 0 e -y + 2ε 0 r 2 0 ε 0 + 1\n\nx 2 y (e y -r 2 0 ) + 4ξ 2 c r 2 0 (ε 0 + 1)ω 2 1 yx 2 e y -r 2 0 + O(x 3 ), ( 22\n\n)\n\nwhere r 0 was defined in Eq. (11) . Note that for simplicity we consider here only one oscillator in Eq. ( 9 ) and put ω j = ω 1 . The case of several oscillator modes can be considered in an analogous way.\n\nAs a next step, we integrate Eq. ( 22 ) term by term according to Eq. ( 18 ), expand the partial results in powers of x and sum up the obtained series. Thereby we obtain the following expressions:\n\nZ 1 (x) ≡ ∞ x y dy ln 1 -r 2 0 e -y = - ∞ n=1 (1 + nx)e -nx n 3 r 2n 0 = -Li 3 (r 2 0 ) - x 2 2 ln(1 -r 2 0 ) + O(x 3 ) , ( 23\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 9\n\nZ 2 (x) ≡ 2ε 0 r 2 0 x 2 ε 0 + 1 ∞ x dy y (e y -r 2 0 ) = - 2ε 0 x 2 ε 0 + 1 ∞ n=1 r 2n 0 Ei(-nx) , ( 24\n\n) Z 3 (x) ≡ 4r 2 0 ξ 2 c x 2 (ε 0 + 1)ω 2 1 ∞ x ydy e y -r 2 0 = 4ξ 2 c x 2 (ε 0 + 1)ω 2 1 ∞ n=1 r 2n 0 (1 + nx)e -nx n 2 = 4ξ 2 c (ε 0 + 1)ω 2 1 x 2 Li 2 (r 2 0 ) - x 4 2 r 2 0 1 -r 2 0 + O(x 5 ) , ( 25\n\n)\n\nwhere Li n (z) is the polylogarithm function and Ei(z) is the exponential integral function. From these equations it follows\n\nZ 1 (ix) -Z 1 (-ix) = O(x 3 ) , Z 3 (ix) -Z 3 (-ix) = O(x 5 ) , ( 26\n\n) Z 2 (ix) -Z 2 (-ix) = 2iπ ε 0 ε 0 + 1 r 2 0 1 -r 2 0 x 2 + O(x 3 ) , ( 27\n\n)\n\nand, thus, Z 1 and Z 3 do not contribute to the leading order in the expansion of F (ix) -F (-ix). The latter is determined by Z 2 only. As a result, we arrive at\n\nF (ix) -F (-ix) = iπ (ε 0 -1) 2 2(ε 0 + 1) x 2 -iαx 3 + O(x 4 ), ( 28\n\n)\n\nwhere r 0 was substituted from Eq. ( 11 ) and α was introduced for the still unknown real coefficient of the next to leading order resulting from Z 1 and Z 2 as well as, possibly, from f ⊥ (ζ, y). At this stage it is difficult to determine the value of this coefficient because all powers in the expansion of f (x, y) contribute to it. Remarkably, the two leading orders depend only on the static dielectric permittivity ε 0 and are not influenced by the dependence of the dielectric permittivity on the frequency contained in Z 3 . Substituting Eq. (28) in Eq. (21) and using Eq. ( 19 ), we obtain\n\nF (a, T ) = E(a) - c 32π 2 a 3 ζ(3) 8π 2 (ε 0 -1) 2 ε 0 + 1 τ 3 -C 4 τ 4 + O(τ 5 ) , ( 29\n\n)\n\nwhere C 4 ≡ α/240 and ζ(z) is the Riemann zeta function. So far we have considered the free energy. The thermal pressure is obtained as\n\nP (a, T ) = - ∂F (a, T ) ∂a = P 0 (a) - c 32π 2 a 4 C 4 τ 4 + O(τ 5 ) , ( 30\n\n)\n\nwhere P 0 (a) = -∂E(a)/∂a is the Casimir pressure at zero temperature. In order to determine the value of the coefficient C 4 of the leading term, we express the pressure directly through the Lifshitz formula\n\nP (a, T ) = - cτ 32π 2 a 4 ∞ l=0 1 - δ l0 2 ∞ ζ l y 2 dy r 2 (ζ l , y) e y -r 2 (ζ l , y) + r 2 ⊥ (ζ l , y) e y -r 2 ⊥ (ζ l , y) . ( 31\n\n)\n\nAgain, applying the Abel-Plana formula (17), we represent the pressure as follows,\n\nP (a, T ) = P 0 (a) + ∆P (a, T ), ( 32\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 10 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nwhere the thermal correction to P 0 (a), the pressure at zero temperature, is\n\n∆P (a, T ) = - i cτ 32π 2 a 4 ∞ 0 dt Φ(iτ t) -Φ(-iτ t) e 2πt -1 ( 33\n\n)\n\nand the function Φ(x) is given by\n\nΦ(x) ≡ Φ (x) + Φ ⊥ (x) , Φ ,⊥ (x) = ∞ x dy y 2 r 2 ,⊥ (x, y) e y -r 2 ,⊥ (x, y) . ( 34\n\n)\n\nFirst, we determine the leading term of the expansion of Φ ⊥ (x) in powers of x. For this purpose, let us introduce the new variable v = y/x and note that the reflection coefficient r ⊥ (x, v) depends on x only through the frequency dependence of ε given by Eq. ( 9 ). Thus, we can rewrite and expand Eq. (34) as follows:\n\nΦ ⊥ (x) = x 3 ∞ 1 dv v 2 r 2 ⊥ (x, v) e vx -r 2 ⊥ (x, v) = x 3 ∞ 1 dv v 2 r 2 ⊥ (v) 1 -r 2 ⊥ (v) + O(x 4 ), ( 35\n\n)\n\nwhere, according to Eq. ( 16 ),\n\nr ⊥ (v) ≡ r ⊥ (0, v) = √ v 2 + ε 0 -1 -v √ v 2 + ε 0 -1 + v . ( 36\n\n)\n\nIntegration in Eq. ( 35 ) with account of Eq. (36) results in\n\nΦ ⊥ (x) = 1 - ε 0 (3 -ε 0 ) 2 √ ε 0 x 3 6 + O(x 4 ) , ( 37\n\n)\n\nfrom which it follows:\n\nΦ ⊥ (ix) -Φ ⊥ (-ix) = -i 1 - 1 2 √ ε 0 (3 -ε 0 ) x 3 3 + O(x 5 ). ( 38\n\n)\n\nThe expansion of Φ (x) from Eq. (34) in powers of x is somewhat more cumbersome. It can be performed in the following way. As is seen from the second equality in Eq. ( 26 ), the dependence of the dielectric permittivity on frequency contributes to F (ix) -F (-ix) starting from only the 5th power in x. Bearing in mind the connection between free energy and pressure, we can conclude that the dependence on the frequency contributes to Φ (ix) -Φ (-ix) starting from the 4th order. We are looking for the lowest (third) order expansion term of Φ (ix) -Φ (-ix). Because of this, it is permissible to disregard the frequency dependence of ε and describe the dielectric by its static dielectric permittivity.\n\nTo begin with, we identically rearrange Φ (x) in Eq. ( 34 ) by subtracting and adding the two first expansion terms of the function under the integral in powers of x,\n\nΦ (x) = ∞ x dy y 2 r 2 (x, y) e y -r 2 (x, y) -y 2 r 2 0 e y -r 2 0 + x 2 2ε 0 ε 0 + 1 r 2 0 e -y (1 -r 2 0 e -y ) 2 + ∞ x y 2 dy r 2 0 e y -r 2 0 -x 2 2ε 0 ε 0 + 1 ∞ x dy r 2 0 e -y (1 -r 2 0 e -y ) 2 , ( 39\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 11 and consider these three integrals separately. The first integral in terms of the new variable v = y/x reads\n\nQ 1 (x) ≡ x 3 ∞ 1 dv v 2 r 2 (v) e vx -r 2 (v) -v 2 r 2 0 e vx -r 2 0 + 2ε 0 ε 0 + 1 r 2 0 e -vx (1 -r 2 0 e -vx ) 2 , ( 40\n\n)\n\nwhere, in accordance with Eq. ( 16 ),\n\nr (v) ≡ r (0, v) = ε 0 v - √ v 2 + ε 0 -1 ε 0 v + √ v 2 + ε 0 -1 . ( 41\n\n)\n\nExpanding Q 1 (x) in powers of x and explicitly calculating the remaining integrals for the lowest, third, power of x results in\n\nQ 1 (x) = x 3 ∞ 1 dv v 2 r 2 (v) 1 -r 2 (v) -v 2 r 2 0 1 -r 2 0 + 2ε 0 ε 0 + 1 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ) ( 42\n\n) = x 3 6 1 - √ ε 0 2 1 + 3ε 0 -2 ε 2 0 + 2 r 2 0 1 -r 2 0 - 12 ε 0 ε 0 + 1 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ).\n\nThe second and third integrals on the right-hand side of Eq. ( 39 ) are simply determined with the following result:\n\nQ 2 (x) ≡ ∞ x y 2 dy r 2 0 e y -r 2 0 = ∞ n=1 r 2n 0 (2 + 2nx + n 2 x 2 )e -nx n 3 = 3Li 3 (r 2 0 ) - x 3 3 r 2 0 1 -r 2 0 + O(x 4 ) , ( 43\n\n)\n\nQ 3 (x) ≡ - 2ε 0 x 2 ε 0 + 1 ∞ x dy r 2 0 e -y (1 -r 2 0 e -y ) 2 = - 2ε 0 x 2 ε 0 + 1 r 2 0 e -x 1 -r 2 0 e -x = - 2ε 0 ε 0 + 1 x 2 r 2 0 1 -r 2 0 -x 3 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ) . ( 44\n\n) Substituting Eqs. (42), (43) and (44) into Φ (x) = Q 1 (x) + Q 2 (x) + Q 3 (x), we arrive at Φ (ix) -Φ (-ix) = -i 1 + √ ε 0 2 2ε 2 0 -3ε 0 -1 x 3 3 + O(x 4 ). ( 45\n\n)\n\nThen, by summing Eqs. (38) and (45), the result is obtained\n\nΦ(ix) -Φ(-ix) = -i 2 + ε 5/2 0 -ε 3/2 0 -2 √ ε 0 x 3 3 + O(x 4 ). ( 46\n\n)\n\nNow we substitute Eq. (46) in Eq. ( 33 ) and perform integration. Finally, from Eq. (32) the desired expression for the Casimir pressure is derived\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 √ ε 0 -1 ε 2 0 + ε 3/2 0 -2 720 τ 4 + O(τ 5 ) . ( 47\n\n)\n\nBy comparison with Eq. ( 30 ) the explicit form of the coefficient C 4 is found as\n\nC 4 = √ ε 0 -1 ε 2 0 + ε 3/2 0 -2 /720 ( 48\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 12 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and, thus, both two first perturbation orders in the expansion for the free energy (29) are determined. Equations (29), (47) and (48) solve the fundamental problem of the thermodynamic consistency of the Lifshitz theory in the case of two dielectric plates. From Eqs. (29) and (48) the entropy of the van der Waals and Casimir interaction between plates takes the form\n\nS(a, T ) = - ∂F (a, T ) ∂T = 3k B ζ(3)(ε 0 -1) 2 64π 3 a 2 (ε 0 + 1) τ 2 ( 49\n\n) × 1 - 2π 2 (ε 0 + 1) ε 3/2 0 + 2ε 0 + 2 √ ε 0 + 2 135ζ(3)( √ ε 0 + 1) 2 τ + O(τ 2 ) .\n\nAs is seen from Eq. (49), in the limit τ → 0 (T → 0) the lower order contributions to the entropy are of the second and the third powers in the small parameter τ . Thus, the entropy vanishes when the temperature goes to zero as it must be in accordance with the third law of thermodynamics (the Nernst heat theorem).\n\nA similar behavior was obtained for ideal metals 40,72,73 and for real metals described by the plasma model. 45,52 For example, in the case of plates made of ideal metal the entropy at low temperatures is given by 40,72,73\n\nS(a, T ) = 3k B ζ(3) 32π 3 a 2 τ 2 1 - 2π 2 135ζ(3) τ + O(τ 2 ) . ( 50\n\n)\n\nNote, however, that the expansion coefficients in Eq. ( 50 ) cannot be obtained as a straightforward limit |ε 0 | → ∞ in Eq. ( 49 ) and the above equations for the free energy and pressure. The mathematical reason is that it is impermissible to interchange the limiting transitions τ → 0 and |ε 0 | → ∞ in the power expansions of functions depending on ε 0 as a parameter. Remarkably, the low-temperature behavior of the free energy, pressure and entropy of nonpolar dielectrics in Eqs. (29), (47) and (49) is universal, i.e., is determined only by the static dielectric permittivity. The absorption bands included in Eq. ( 9 ) do not influence the low-temperature behavior. A more simple derivation of the results (29), (47)-(49) for dielectrics with constant ε is contained in Ref. 74. As was demonstrated above, all these results remain unchanged if the dependence of dielectric permittivity on frequency is taken into account.\n\nIn Ref. 60 more general results were obtained related to two dissimilar dielectric plates with dielectric permittivities ε (1) (ω) and ε (2) (ω). For brevity here we present only the final expressions for the low-temperature behavior of the Casimir free energy, pressure and entropy between dissimilar plates. They are as follows: 60\n\nF (a, T ) = E(a) - c 32π 2 a 3 τ 3 × ζ(3) 8π 2 ε ( 1\n\n) 0 + ε ( 2\n\n) 0 + 2ε (1) 0 ε (2) 0 (ε ( 1\n\n) 0 + 1)(ε ( 2\n\n) 0 + 1) (ε ( 1\n\n) 0 -1)(ε (2) 0 -1) (ε (1) 0 + ε (2) 0 ) -C 4 τ + O(τ 2 ) ,\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 13 1,2) (0) and the coefficient C 4 is given by 60\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 C 4 τ 4 + O(τ 5 ) , ( 51\n\n) S(a, T ) = k B 2πa 2 τ 2 × 3ζ(3) 32π 2 ε ( 1\n\n) 0 + ε ( 2\n\n) 0 + 2ε (1) 0 ε (2) 0 (ε ( 1\n\n) 0 + 1)(ε ( 2\n\n) 0 + 1) (ε ( 1\n\n) 0 -1)(ε ( 2\n\n) 0 -1) (ε (1) 0 + ε (2) 0 ) -C 4 τ + O(τ 2 ) . Here ε (1,2) 0 ≡ ε (\n\nC 4 = 1 720 2 + 1 ε ( 1\n\n) 0 + ε (2) 0 ε (1) 0 + ε (2) 0 2 ( 52\n\n) × -ε ( 1\n\n) 0 + ε (2) 0 2 2ε ( 1\n\n) 0 + 2ε ( 2\n\n) 0 + ε (1) 0 ε (2) 0 -ε (1) 0 ε (2) 0 + ε (1) 0 ε (2) 0 ε (1) 0 ε (2) 0 5ε (1) 0 ε (2) 0 -3ε (1) 0 -3ε ( 2\n\n) 0 + 1 + ε (1) 0 ε (2) 0 ε (1) 0 -ε (2) 0 2 ε ( 1\n\n) 0 ε (2) 0 ε (1) 0 ε ( 2\n\n) 0 -ε ( 1\n\n) 0 ε ( 2\n\n) 0 -ε ( 1\n\n) 0 -ε (2) 0 - 3 ε ( 1\n\n) 0 ε (2) 0 2 ε ( 1\n\n) 0 -1 ε ( 2\n\n) 0 -1 ε ( 1\n\n) 0 -ε (2) 0 ε ( 1\n\n) 0 + ε (2) 0 Artanh ε ( 1\n\n) 0 + ε (2) 0 ε (1) 0 -ε (2) 0 ε (1) 0 ε ( 2\n\n) 0 -ε ( 1\n\n) 0 -ε (2) 0 .\n\nIt is easily seen that in the limit ε 52) coincide with equations (29), (47)-(49) having obtained above. Note that in the application region of low-temperature asymptotic expressions the entropy of the Casimir interaction between dielectric plates is nonnegative.\n\n(1) 0 = ε ( 2\n\n) 0 = ε 0 equations (51), (\n\nThe obtained analytic behavior of the free energy, pressure and entropy at low temperatures can be compared with the results of numerical computations using the Lifshitz formula. Dielectric properties of the plates can be described by the static dielectric permittivity or more precisely using the optical tabulated data for the complex index of refraction. As an example, in Fig. 1 we present the thermal corrections to the Casimir energy (a) and pressure (b) at a separation a = 400 nm as functions of temperature in the configuration of two dissimilar plates made of high-resistivity Si and SiO 2 . The dielectric permittivities of both materials along the imaginary frequency axis were computed in Ref. 75 using the optical data of Ref. 76. The precise thermal corrections computed by taking into account these permittivities are shown by the solid lines and corrections computed by our analytical asymptotic expressions are shown by the long-dashed lines. Short-dashed lines indicate the results computed by the Lifshitz formula with constant dielectric permittivities of Si and SiO 2 equal to ε (1) 0 = 11.67 and ε (2) 0 = 3.84, respectively. As is seen in Fig. 1a ,b, at T < 60 K the results obtained using the analytical asymptotic October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 14 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 25 50 75 100 125 150 175 200 1 2 3 4 T (K) -∆F (pJ) 25 50 75 100 125 150 175 200 1 2 3 4 5 6 7 T (K) -∆P (µPa) (b) (a)\n\nFig. 1. Magnitudes of the thermal corrections to the energy (a) and pressure (b) in configuration of two plates, one made of Si and another one of SiO 2 , at a separation a = 400 nm as a function of temperature calculated by the use of different approaches: by the Lifshitz formula and tabulated optical data (solid lines), by the Lifshitz formula and static dielectric permittivities (short-dashed lines), by the asymptotic expressions in Eqs. (51) and (52) (long-dashed lines).\n\nexpressions practically coincide with the solid lines computed using the tabulated optical data for the materials of the plates.\n\nNow we return to the case of two similar dielectric plates and consider the asymptotic expressions under the condition τ ≫ 1, i.e., at high temperatures (large separations). It is well known 37,38,77 that in this case the approximation of static October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 15 dielectric permittivity works good and the main contribution is given by the zerofrequency term of the Lifshitz formula (13)\n\nF (a, T ) = cτ 64π 2 a 3 ∞ 0 ydy ln 1 -r 2 0 e -y ( 53\n\n)\n\n(the other terms being exponentially small). Performing the integration in Eq. (53) we obtain\n\nF (a, T ) = - k B T 16πa 2 Li 3 r 2 0 . ( 54\n\n)\n\nIn a similar manner for the Casimir pressure and entropy at τ ≫ 1 it follows\n\nP (a, T ) = - k B T 8πa 3 Li 3 r 2 0 , S(a, T ) = k B 16πa 2 Li 3 r 2 0 . ( 55\n\n)\n\nEquations (54) and (55) are simply generalized 60 for the case of two dissimilar dielectric plates by performing the replacement r 2 0 → r (1) 0 r (2) 0 , where r (1,2) 0 are defined by Eq. (11) with the static dielectric permittivities of dissimilar plates ε (1,2) 0 ."},{"section_type":"BACKGROUND","section_title":"IS THE DC CONDUCTIVITY RELATED TO THE CASIMIR INTERACTION BETWEEN DIELECTRICS?","text":"As was discussed in the previous section, the zero-frequency term in formula (13), i.e., the contribution with l = 0, is of prime importance and determines many of the basic properties of the Casimir interaction. In the above consideration we have described dielectric materials by Eq. ( 9 ) with finite static dielectric permittivity ε 0 . This resulted in Eq. (11) where one reflection coefficient at zero frequency is larger and the other one is smaller than the physical value for real photons at normal incidence. However, in the Lifshitz theory, the departure of both coefficients from their physical values is coordinated in such a way that the Nernst heat theorem remains valid. It is common knowledge that at nonzero temperatures dielectric materials possess a negligibly small but not equal to zero dc conductivity. From physical intuition it is reasonable to expect that the influence of this conductivity on the van der Waals and Casimir forces should be also negligible. In Ref. 57 it was shown that, on the contrary, the inclusion of small dielectric dc conductivity in the model of dielectric response leads to a large effect in dispersion forces. This raises the question if dc conductivity is related to dispersion forces or if, on the contrary, the zero-frequency contribution should be understood not literally but as an analytic continuation from the region of high frequencies determining the physical phenomenon of dispersion forces.\n\nTo illustrate this problem in more details, we consider the asymptotic behavior of the free energy and entropy at low temperature with included dc conductivity. What this means is that, instead of the dielectric permittivity ε(iξ l ) given in Eq. ( 9 ), one uses 57,58\n\nε(iξ l ) = ε(iξ l ) + 4πσ 0 ξ l = ε l (iξ) + β(T ) l , ( 56\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 16 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nwhere σ 0 is the dc conductivity of the plate material and β(T ) = 2 σ 0 /(k B T ). The conductivity of dielectrics depends on temperature as σ 0 ∼ exp(-b/T ) where b is determined by the energy gap ∆ which differs for different materials. 78 The smallness of the dc conductivity of dielectrics can be illustrated 79 by the example of SiO 2 where at T = 300 K it holds β ∼ 10 -12 . Thus, the role of the dc conductivity is really negligible for all l ≥ 1. In addition, β(T ) quickly decreases with decrease of T and, as a consequence, remains negligible at any T . In spite of this, the substitution of Eq. (56) into the reflection coefficients (16) leads to different result than in Eq. (11): r (0, y) = 1, r (0, y) = 0. (57) Equation (57) is in some analogy to Eq. ( 6 ), obtained for metals described by the Drude model, which leads to the violation of the Nernst heat theorem in the case of perfect crystal lattices. Now we substitute the dielectric permittivity εl ≡ ε(ξ l ) in Eqs. (13) -(16) instead of ε l and find the Casimir free energy F (a, T ) with included dc conductivity. For convenience, we separate the zero-frequency term, subtract and add the usual zero-frequency contribution for dielectric without dc conductivity. The result is\n\nF (a, T ) = k B T 16πa 2 ∞ 0 ydy ln 1 -e -y -ln 1 -r 2 0 e -y + k B T 16πa 2 ∞ 0 ydy ln 1 -r 2 0 e -y ( 58\n\n) + k B T 8πa 2 ∞ l=1 ∞ ζ l ydy ln 1 -r2 (ζ l , y)e -y + ln 1 -r2 ⊥ (ζ l , y)e -y ,\n\nwhere r 0 was defined in Eq. ( 11 ). Let us expand the last, third, integral on the righthand side of Eq. (58) in powers of the small parameters β/l. Then, we combine the zero-order contribution in this expansion with the second integral on the righthand side of Eq. ( 58 ) and obtain the Casimir free energy F (a, T ) calculated with the dielectric permittivities ε l . The first integral on the right-hand side of Eq. (58) is calculated explicitly. Then, Eq. ( 58 ) can be rewritten as\n\nF (a, T ) = F (a, T ) - k B T 16πa 2 ζ(3) -Li 3 r 2 0 + R(a, T ). ( 59\n\n)\n\nHere, R(a, T ) is of order O(β/l). It represents the first and higher-order contributions in the expansion of the third integral on the right-hand side of Eq. (58) in powers of β/l. Restricting its explicit form to the first order contribution we get\n\nR(a, T ) = R 1 (a, T ) + O (β/l) 2 , ( 60\n\n) R 1 (a, T ) = k B T 4πa 2 ∞ l=1 β l ∞ ζ l dy y 2 e -y y 2 + ζ 2 l (ε l -1) (2 -ε l ) ζ 2 l -2y 2 y 2 + ζ 2 l (ε l -1) + ε l y 2 × r (ζ l , y) 1 -r 2 (ζ l , y)e -y - ζ 2 l y 2 + ζ 2 l (ε l -1) + y 2 r ⊥ (ζ l , y) 1 -r 2 ⊥ (ζ l , y)e -y . ( 61\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 17\n\nCalculating the entropy by the first equality in Eq. ( 49 ), we arrive at\n\nS(a, T ) = S(a, T ) + k B 16πa 2 ζ(3) -Li 3 r 2 0 - ∂R(a, T ) ∂T , ( 62\n\n)\n\nwhere S(a, T ) is the entropy for the plates with the dielectric permittivity ε l given by Eq. ( 49 ).\n\nLet us now show that the quantity R(a, T ) exponentially goes to zero with the decrease of T . First we consider only the integral in Eq. ( 61 ), expand the integrated function in powers of τ (we recall that ζ l = τ l), restrict ourselves to the main contribution, resulting for τ = 0, and rearrange it appropriately:\n\n-2 ∞ ζ l dy ye -y (ε 0 + 1) 2 r 0 1 -r 2 0 e -y = - 2 (ε 2 0 -1) ∞ ζ l dy y r 2 0 e -y 1 -r 2 0 e -y = - 2 (ε 2 0 -1) ∞ n=1 r 2n 0 ∞ ζ l dy ye -ny = - 2 (ε 2 0 -1) ∞ n=1 r 2n 0 1 + nζ l n 2 e -nζ l . ( 63\n\n)\n\nSubstituting this in Eq. ( 61 ), we find\n\nR 1 (a, T ) = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r 2n 0 n 2 ∞ l=1 e -nτ l l + nτ ∞ l=1 e -nτ l = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r 2n 0 n 2 -ln 1 -e -nτ + nτ e nτ -1 = k B Li 2 r 2 0 2πa 2 (ε 2 0 -1) T β ln τ + T β O(τ 0 ) . ( 64\n\n)\n\nHere, the last line is obtained by using the equality\n\n-ln 1 -e -nτ + nτ e nτ -1 = -ln τ + 1 -ln n + O(τ 2 ) , ( 65\n\n)\n\nsubstituting only its leading term and observing the definition of the integral log-\n\narithm, Li 2 (z) = (1/2) ∞ n=1 z n /n 2 .\n\nTaking into account that β ∼ (1/T ) exp(-b/T ), we get the conclusion that the temperature dependence of R 1 (a, T ) is given by\n\nR 1 (a, T ) ∼ e -b/T ln T. ( 66\n\n)\n\nThus, both R 1 (a, T ) and its derivative with respect to T in Eqs. (59) and (62) go to zero. The terms of the second and higher powers in β in Eq. (60) go to zero even faster than R 1 when T → 0. Finally, in the limit T → 0 from Eq. (62) it follows\n\nS(a, 0) = k B 16πa 2 ζ(3) -Li 3 r 2 0 > 0. ( 67\n\n)\n\nThe right-hand side of this equation depends on the parameter of the system under consideration (the separation distance a) and implies a violation of the Nernst heat theorem. An analogous result was obtained 60 in the case of two dissimilar dielectrics.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 18 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThe violation of the Nernst heat theorem in the Casimir interaction for dielectrics originates from the inclusion of the dc conductivity in the model of dielectric response. This violation is, however, of different nature than the one discussed above in the case of Drude metals. In the case of dielectrics the entropy at zero temperature is positive but in the case of Drude metals it is negative. In the case of metals the violation is caused by the vanishing contribution from the transverse electric mode at zero frequency whereas the other reflection coefficient takes the physical value 1 [see Eq. ( 6 )]. For dielectrics the situation is quite opposite. In this case the transverse reflection coefficient at zero frequency is always equal to zero [compare Eqs. (11) and (57) in the absence and in the presence of the contribution from dc conductivity]. Here the violation occurs due to the unity value of the parallel reflection coefficient at zero frequency in Eqs. (57) which departs from the value r 0 = (ε 0 -1)/(ε 0 + 1) coordinated with the zero value of the transverse coefficient in Eq. (11) .\n\nOne can conclude that the dc conductivity of a dielectric is not related to the nature of the van der Waals and Casimir forces and must not be included in the model of dielectric response. Ignoring this rule results in a violation of thermodynamics. Physically it is amply clear that there is no fluctuating field of zero frequency and that for such high-frequency phenomena as the van der Waals and Casimir forces the low-frequency behavior should be obtained by analytic continuation from the region of high frequencies. This permits to conclude that the correct procedure consists in the substitution of the finite static dielectric permittivities into the zero-frequency term of the Lifshitz formula, as Lifshitz and his collaborators really did. 36,37,38"},{"section_type":"OTHER","section_title":"THERMAL CASIMIR FORCE BETWEEN DIELECTRIC AND METAL PLATES","text":"The Casimir interaction between metal and dielectric plates suggests the interesting opportunity to verify the thermodynamic consistency of the Lifshitz theory with different models of the dielectric response. This configuration was first investigated in Ref. 61 where it was proved that the Casimir entropy is in accordance with the demands of the Nernst heat theorem if the static permittivity of the dielectric plate is finite. In Ref. 61, however, only the first leading terms in the lowtemperature asymptotic expressions for the free energy and entropy were obtained and the Casimir pressure was derived only in the dilute approximation. Here we derive the more precise low-temperature behavior for the Casimir free energy, pressure and entropy in the configuration of one plate made of ideal metal and another plate made of dielectric with any finite static dielectric permittivity. For the configuration of metal and dielectric plates the Lifshitz formula takes the form 80 analogical to Eq. (13)\n\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l ydy ( 68\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 19\n\n× ln 1 -r M (ζ l , y) r D (ζ l , y) e -y + ln 1 -r M ⊥ (ζ l , y) r D ⊥ (ζ l , y) e -y .\n\nHere the reflection coefficients r M,D ,⊥ for metal and dielectric, respectively, are given by Eqs. (16) where ε l should be changed for ε M,D l\n\n= ε M,D (iξ l ).\n\nFor an ideal metal r M ,⊥ (ζ l , y) = 1 and Eq. ( 68 ) takes the more simple form\n\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l ydy ( 69\n\n) × ln 1 -r (ζ l , y)e -y + ln 1 -r ⊥ (ζ l , y)e -y\n\n(here and below we omit the index D near the reflection coefficient and permittivity of a dielectric plate). We admit that the dielectric permittivity calculated at Matsubara frequencies ε l ≡ ε 0 , i.e., is equal to its static value and find the asymptotic behavior of Eq. (69) at small τ . [In analogy with Sec. 2 it is possible to prove that the deviations of ε(iξ l ) from ε 0 at high frequencies do not influence the lowtemperature behavior of the Casimir free energy, pressure and entropy. It can be shown also that the results of this section are valid not only for ideal metal plate but for plate made of real metal as well.] The free energy (69) can be represented by Eqs. (18) -(21), with the function f (ζ, y) replaced by\n\nf (ζ, y) = y ln 1 -r (ζ, y)e -y + y ln 1 -r ⊥ (ζ, y)e -y ≡ f (ζ, y) + f⊥ (ζ, y). ( 70\n\n)\n\nIn the case of one dielectric and one metal plate both f and f⊥ contribute to F (ix) -F (-ix). The expansion of f (x, y) in powers of x takes the form\n\nf (x, y) = y ln(1 -r 0 e -y ) - ε 0 -1 4y e -y - ε 0 ε 0 + 1 ∞ n=1 r n 0 e -ny y x 2 + O(x 3 ). ( 71\n\n)\n\nNow we integrate Eq. (71) in accordance with Eq. (18) to find the function F (x). The integral of the first term on the right-hand side of Eq. (71) is evaluated using the new variable v = y -x:\n\n∞ x ydy ln(1 -r 0 e -y ) = ∞ 0 vdv ln(1 -r 0 e -v ) + O(x 2 ), ( 72\n\n)\n\nwhere the coefficient near the first-order contribution in x vanishes. As a result, this term could contribute to F (ix) -F (-ix) only starting from the third expansion order. The integrals of the second-order terms on the right-hand side of Eq. (71) are simply calculated using the formulas\n\n∞ x dy e -y y = -Ei(-x), ∞ x dy e -ny y = -Ei(-nx). ( 73\n\n)\n\nFinally, we obtain\n\nF (ix) -F (-ix) = iπ (ε 0 -1) 2 4 (ε 0 + 1) x 2 -iγ x 3 + O(x 4 ), ( 74\n\n)\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 20 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nwhere the unknown third order expansion coefficient is designated as γ. Substituting Eq. (74) in Eq. (21) and using Eq. ( 19 ), we find the free energy in the system metal-dielectric in the form\n\nF (a, T ) = E(a) - c 32π 2 a 3 ζ(3) 16π 2 (ε 0 -1) 2 ε 0 + 1 τ 3 -K 4 τ 4 + O(τ 5 ) , ( 75\n\n)\n\nwhere K 4 ≡ γ/240. The Casimir pressure in the configuration of metal and dielectric plates obtained from Eq. (75) is equal to\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 K 4 τ 4 + O(τ 5 ) . ( 76\n\n)\n\nThe direct application of the Lifshitz formula gives the expression for the pressure analogical to Eq. (31),\n\nP (a, T ) = - cτ 32π 2 a 4 ∞ l=0 1 - δ l0 2 ∞ ζ l y 2 dy r (ζ l , y) e y -r (ζ l , y) + r ⊥ (ζ l , y) e y -r ⊥ (ζ l , y) . ( 77\n\n)\n\nUsing the Abel-Plana formula (17), Eq. ( 77 ) can be represented in the form of Eqs. (32), (33) where\n\nΦ ,⊥ (x) = ∞ x dy y 2 r ,⊥ (x, y) e y -r ,⊥ (x, y) . ( 78\n\n)\n\nAgain, we deal first with Φ ⊥ (x). By adding and subtracting the asymptotic behavior of the integrated function at small x,\n\ny 2 r ⊥ (x, y) e y -r ⊥ (x, y) = 1 4 (ε 0 -1)x 2 e -y + O(x 3 ) , ( 79\n\n)\n\nand introducing the new variable v = y/x, the function Φ ⊥ (x) can be identically rearranged and expanded in powers of x as follows:\n\nΦ ⊥ (x) = 1 4 (ε 0 -1)x 2 e -x + x 3 ∞ 1 dv v 2 ∞ n=1 r n ⊥ (v)e -nvx - 1 4 (ε 0 -1)e -vx = 1 4 (ε 0 -1)x 2 (1 -x) + x 3 ∞ 1 dv v 2 r ⊥ (v) 1 -r ⊥ (v) - ε 0 -1 4 + O(x 4 ). ( 80\n\n)\n\nThe integral on the right-hand side of Eq. (80) is converging and can be simply calculated with the result\n\nΦ ⊥ (x) = ε 0 -1 4 x 2 - 1 6 (ε 0 √ ε 0 -1) x 3 + O(x 4 ). ( 81\n\n)\n\nTo deal with Φ (x) we add and subtract in Eq. ( 78 ) the two first expansion terms of the integrated function in powers of x,\n\nΦ (x) = ∞ x y 2 dy r 0 e y -r 0 - ε 0 r 0 e -y x 2 y 2 (ε 0 + 1) (1 -r 0 e -y ) 2 ( 82\n\n) + ∞ x y 2 dy r (x, y) e y -r (x, y) - r 0 e y -r 0 + ε 0 r 0 e -y x 2 y 2 (ε 0 + 1) (1 -r 0 e -y ) 2 .\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 21\n\nThe asymptotic expansion of the first integral on the right-hand side of Eq. (82) is given by\n\n2Li 3 (r 0 ) - ε 0 (ε 0 -1) 2(ε 0 + 1) x 2 + 1 12 (ε 0 -1)(3ε 0 -2)x 3 + O(x 4 ), ( 83\n\n)\n\nand of the second one by\n\n- 1 4 ε 0 (ε 0 -1) - 1 6 ε 0 (ε 0 √ ε 0 -1) + 1 2 ε 0 (ε 0 -1) √ ε 0 x 3 + O(x 4 ). ( 84\n\n)\n\nBy summing Eqs. (83) and (84) we find\n\nΦ (x) = 2Li 3 (r 0 ) - ε 0 (ε 0 -1) 2(ε 0 + 1) x 2 ( 85\n\n) - 1 6 (ε 0 -1) + (ε 0 √ ε 0 -1) -3ε 0 (ε 0 -1) √ ε 0 x 3 + O(x 4 ).\n\nFinally we add Eq. (85) to Eq. ( 81 ) and arrive at Φ(ix\n\n) -Φ(-ix) = -i 2 3 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 x 3 + O(x 4 ). ( 86\n\n)\n\nSubstituting this in Eq. (33) and using Eq. (32), the asymptotic expression for the Casimir pressure is obtained\n\nP (a, T ) = P 0 (a) - c 32π 2 a 4 1 360 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 τ 4 + O(τ 5 ) . ( 87\n\n)\n\nComparing Eqs. (87) and (76), the explicit expression for the coefficient K 4 reads\n\nK 4 = 1 360 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 . ( 88\n\n)\n\nNow we are in a position to find the asymptotic behavior of the entropy in the limit of low temperatures in the configuration of two parallel plates one of which is metallic and the other one dielectric. By calculating the negative derivative of Eq. (75) with respect to temperature, one arrives at\n\nS(a, T ) = 3k B ζ(3)(ε 0 -1) 2 128π 3 a 2 (ε 0 + 1) τ 2 ( 89\n\n) × 1 - 8π 2 (ε 0 + 1) 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 135ζ(3)(ε 0 -1) 2 τ + O(τ 2 ) .\n\nThis equation is in analogy to Eq. (49) obtained for the case of two dielectrics. As is seen from Eq. ( 89 ), the entropy of the Casimir and van der Waals interactions between metal and dielectric plates vanishes when the temperature goes to zero, i.e., the Nernst heat theorem is satisfied. Note that the first term of order τ 2 on the right-hand side of Eq. (89) was already obtained in Ref. 61. It is notable also that at low temperatures the entropy goes to zero remaining positive. At the same time, as was shown in Ref. 61, at larger temperatures entropy is nonmonotonous and may take negative values. This interesting property distinguishes the configuration of metal and dielectric plates from two dielectric plates. In the latter configuration the October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 22 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko negative entropy appears only for nonphysical dielectrics with anomalously large and frequency independent dielectric permittivities. 53 Now let us consider the case τ ≫ 1, i.e., high temperatures (large separations). In the same way as for two dielectric plates, here the main contribution to the free energy is given by the term with l = 0 in Eq. (69),\n\nF (a, T ) = cτ 64π 2 a 3 ∞ 0 ydy ln 1 -r 0 e -y . ( 90\n\n)\n\nPerforming the integration in Eq. ( 90 ), we obtain\n\nF (a, T ) = - k B T 16πa 2 Li 3 (r 0 ) . ( 91\n\n)\n\nFrom this equation, for the Casimir pressure and entropy at τ ≫ 1 it follows\n\nP (a, T ) = - k B T 8πa 3 Li 3 (r 0 ) , S(a, T ) = k B 16πa 2 Li 3 (r 0 ) . ( 92\n\n)\n\nThe results (91) and (92) are analogous to (54) and (55) for two dielectric plates."},{"section_type":"OTHER","section_title":"THE PROBLEM ORIGINATING FROM THE ACCOUNT OF DIELECTRIC DC CONDUCTIVITY","text":"In the previous section it was supposed that the static dielectric permittivity ε 0 of the dielectric plate is finite. Now we will deal with the configuration of metal and dielectric plates with included dc conductivity of the dielectric material. In doing so the permittivity of the dielectric plate is given by\n\nε(iξ l ) = ε 0 + β(T ) l , ( 93\n\n)\n\nwhere all notations were introduced in and below Eq. (56). Thus, the reflection coefficients at zero frequency satisfy Eq. ( 4 ) for a plate made of ideal metal and Eq. (57) for a plate made of dielectric with included dc conductivity. Let us find the low-temperature behavior of the Casimir entropy and verify the consistency of the Lifshitz theory with thermodynamics in this nonstandard situation. For this purpose we substitute the dielectric permittivity (93) in Eq. (69) instead of ε 0 and find the Casimir energy F (a, T ) with included dc conductivity of a dielectric plate. In the same way as in Sec. 3, it is convenient to separate the zero-frequency term of F (a, T ) and subtract and add the usual zero-frequency contribution for metal-dielectric plates computed with the dielectric permittivity ε 0 ,\n\nF (a, T ) = k B T 16πa 2 ∞ 0 ydy ln 1 -e -y -ln 1 -r 0 e -y + k B T 16πa 2 ∞ 0 ydy ln 1 -r 0 e -y ( 94\n\n) + k B T 8πa 2 ∞ l=1 ∞ ζ l ydy ln 1 -r (ζ l , y)e -y + ln 1 -r⊥ (ζ l , y)e -y .\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 23\n\nHere the reflection coefficients r ,⊥ are calculated with the permittivity (93). We expand the third integral on the right-hand side of Eq. (94) in powers of the small parameter β/l. The zero-order contribution in this expansion together with the second integral of Eq. ( 94 ) form the Casimir free energy F (a, T ) calculated with dielectric permittivity ε 0 . The first integral on the right-hand side of Eq. (94) is calculated explicitly. As a result, Eq. ( 94 ) is rearranged to\n\nF (a, T ) = F (a, T ) - k B T 16πa 2 [ζ(3) -Li 3 (r 0 )] + Q(a, T ), ( 95\n\n)\n\nwhere Q(a, T ) contains the first and higher-order contributions in the expansion of the third integral on the right-hand side of Eq. (94) in powers of β/l. The explicit form of the main first-order term in Q(a, T ) is the following:\n\nQ 1 (a, T ) = k B T 8πa 2 ∞ l=1 β l ∞ ζ l dy y 2 e -y y 2 + ζ 2 l (ε 0 -1) (2 -ε 0 )ζ 2 l -2y 2 y 2 + ζ 2 l (ε 0 -1) + ε 0 y 2 × 1 1 -r (ζ l , y)e -y - ζ 2 l y 2 + ζ 2 l (ε 0 -1) + y 2 1 1 -r ⊥ (ζ l , y)e -y . ( 96\n\n)\n\nIn the same way as in Sec. 3, we expand the integrated function in Eq. (96) in powers of τ (bearing in mind that ζ l = τ l) and preserve only the main contribution at τ = 0:\n\nQ 1 (a, T ) = - k B T 4πa 2 ∞ l=1 β l ∞ ζ l dy yr 0 e -y (ε 2 0 -1) (1 -r 0 e -y ) ( 97\n\n) = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r n 0 n 2 ∞ l=1 e -nτ l l + nτ ∞ l=1 e -nτ l .\n\nDealing with this expression in the same way as with Eq. ( 64 ), we arrive at\n\nQ 1 (a, T ) ∼ e -b/T ln T. ( 98\n\n)\n\nThe Casimir entropy in the configuration of metal and dielectric plates with included dc conductivity of the dielectric plate is obtained as minus derivative of Eq. (95) with respect to temperature,\n\nS(a, T ) = S(a, T ) + k B 16πa 2 [ζ(3) -Li 3 (r 0 )] - ∂Q(a, T ) ∂T . ( 99\n\n)\n\nUsing Eq. ( 98 ), the calculation of the limiting value at T → 0 is straightforward:\n\nS(a, 0) = k B 16πa 2 [ζ(3) -Li 3 (r 0 )] > 0. ( 100\n\n)\n\nFrom this equation it follows that the inclusion of the dc conductivity of dielectric plate in the configuration metal-dielectric results in a violation of the Nernst heat theorem. In the above this result was obtained for a metallic plate made of ideal metal. It can be shown that it remains valid for a metallic plate made of real metal with finite conductivity.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 24 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\nThus, both configurations (two dielectric plates or one metal plate and one dielectric) lead to the same conclusion that when the dc conductivity is included in the model of dielectric response of the dielectric plate, the Lifshitz theory loses its consistency with thermodynamics. This confirms the conclusion made in Sec. 3 that the actual properties of dielectric materials at very low frequencies are in fact not related to the van der Waals and Casimir forces."},{"section_type":"DISCUSSION","section_title":"QUALITATIVE DISCUSSION OF CASIMIR INTERACTION BETWEEN METAL AND SEMICONDUCTOR","text":"As was mentioned in the Introduction, semiconductors possess a wide variety of electric and optical properties ranging from metallic to dielectric. This opens the possibility to modulate the van der Waals and Casimir forces by changing the charge carrier density. Bearing in mind the discussed above problems on the consistency of the Lifshitz theory with thermodynamics, semiconductors can provide us with a test for the validity of different approaches. Thus, if for good dielectric the dc conductivity does not play any real role in the van der Waals and Casimir forces, the question arises on how much it should be increased in order to become a relevant factor in the description of dispersion forces. In Ref. 81 the Casimir force acting between two Si plates was calculated using the simple analytic expression for Si dielectric permittivity as a function of frequency. The complete tabulated optical data of Si were used in Ref. 75 to calculate the van der Waals interaction of different atoms with a Si wall. The first attempt to measure the van der Waals force between a glass lens and a Si plate and to modify it by light due to the change of carrier density was undertaken in Ref. 82. However, glass is a dielectric and therefore the electric forces due to localized point charges could not be controlled. This might explain that no force change occured on illumination at small separations where the effect should be most pronounced.\n\nThe first measurements of the Casimir force between a gold coated sphere and a single crystal Si plate were performed in Refs. 62, 63 by means of the atomic force microscope. The experiments used a p-type B doped Si plate of resistivity ρ = 0.0035 Ω cm. The chosen resistivity of the plate is in some sense intermediate between the resistivity of metals (which is usually two or three orders of magnitude lower) and the resistivity of dielectrics (it can be by about a factor of 10 5 larger; for instance, high-resistivity \"dielectric\" Si has the resistivity ρ 0 = 1000 Ω cm). Thus, the used Si plate had a relatively large absorption typical for semiconductors but it was also enough conductive to avoid the accumulation of charges. 63 In Fig. 2 , taken from Ref. 63, the differences of the theoretical and mean experimental Casimir forces acting between Au sphere and Si plate are presented as functions of separation. In Fig. 2a the theoretical force F theor is computed using the Lifshitz formula and the dielectric permittivity of a Si plate with the relatively low resistivity ρ used in experiment. This dielectric permittivity goes to infinity as ξ -1 with decreasing frequency (see the solid line in Fig. 3 ). In Fig. 2b the theo-October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 25 80 100 120 140 -10 -5 0 5 10 15 (nm) F theor -F expt (pN) 80 100 120 140 -10 -5 0 5 10 15\n\n(nm) F theor -F expt (pN) (b)\n\n(a) a a Fig. 2. Differences of the theoretical and mean experimental Casimir forces versus separation. Theoretical forces are computed (a) for the Si plate used in experiment and (b) for dielectric Si. Solid and dashed lines indicate 95 and 70% confidence intervals, respectively.\n\nretical Casimir force F theor is computed using the dielectric permittivity of the Si plate made of \"dielectric\" Si with high resistivity ρ 0 . The dielectric permittivity of high-resistivity Si is shown by the dashed line in Fig. 3 . It is characterized by the finite static value ε Si (0) = 11.67. The solid and dashed lines in Fig. 2a ,b indicate the 95% and 70% confidence intervals, respectively. As is seen from Fig. 2 , the theoretical approach using the dielectric permittivity of high-resistivity \"dielectric\" Si is excluded by experiment within the separation range from 60 to 110 nm at 70% confidence. At the same time, the theory using the dielectric permittivity of Si with a low resistivity ρ is consistent with experiment.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 26 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\n\n13.5 14 14.5 15 15.5 16 16.5 17 5 10 15 20 25 Log 10 [ξ (rad/s)] ε (iξ) D Fig. 3. Dielectric permittivity of Si plate used in experiment along the imaginary frequency axis (solid line). Dashed line shows the dielectric permittivity of dielectric Si.\n\nThe above results suggest an approach on how to correctly determine the possible role of the low-frequency conductivity properties in dispersion forces. As is seen from Fig. 3 , the dielectric permittivity of low-resistivity Si (solid line) significantly departs from the dielectric permittivity of \"dielectric\" Si in the region around the important dimensional parameter of the problem, the characteristic frequency c/(2a) ∼ 10 14 -10 15 rad/s. That is why the low-resistivity sample cannot be described at low frequencies by the static dielectric permittivity of Si equal to ε Si (0) = 11.67. To describe it, the term β(T )/l, like in Eq. ( 93 ), should be added to ε Si (0). Note that in this case β(T )/l > 1 at the first Matsubara frequencies with l = 1, 2, 3, . . . and, thus, this quantity cannot be considered as a small parameter. At the same time, for a high-resistivity sample the inclusion of the dc conductivity would lead to deviations from the dashed line in Fig. 3 only at frequencies ξ < 10 8 rad/s, which are much less than the characteristic frequency. This comparison permits to make a conclusion in what experimental situations the conductivity properties of semiconductors at low frequencies should be taken into account and when they should be omitted as being not related to dispersion forces.\n\nThe future experiments on the modification of semiconductor charge carrier density by laser light 83 will bring a more clear understanding of this problem on the connection between the low-frequency material properties and dispersion forces.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 27"},{"section_type":"OTHER","section_title":"DOES SPATIAL DISPERSION LEAD TO AN IMPORTANT IMPACT ON THERMAL CASIMIR FORCE?","text":"The presented above new analytic results on the low-temperature behavior of the Casimir free energy, pressure and entropy between dielectrics or between dielectric and metal are based on the conventional Lifshitz theory which describes dielectric materials by means of the frequency dependent dielectric permittivity. In fact, the assumption that the material of the plates possesses only the frequency dispersion means that the components of electric displacement are connected with the components of electric field by the relation\n\nD k (r, ω) = ε kl (r, ω)E l (r, ω). ( 101\n\n)\n\nThis equation is central in all different derivations of the Lifshitz formula (see, e.g., Refs. 2, 6, 36-38, 69, 70, 84). The effects of spatial nonlocality (spatial dispersion) are in fact essential only at shortest separations between the plates comparable with atomic dimensions and also for metals at sufficiently large separations (typically of about 2-3 µm) in the frequency region of the anomalous skin effect. The Casimir force in the latter region was described by the Lifshitz theory reformulated in terms of the Leontovich impedance. 85 Recently the spatial dispersion came to the attention in connection with the thermal Casimir force. 64, 65, 66, 67, 68 In particular, it was claimed 64, 65 that for real metals at any separation the account of spatial dispersion leads to practically the same result (6) for the reflection coefficients at zero frequency as was obtained earlier using the Drude model dielectric function (5). This conclusion, if it is correct, not only returns us to the contradiction with experiment 30,32,33 but also casts doubts on all results obtained by means of the conventional Lifshitz theory accounting for only the frequency dispersion. It is natural when the spatial dispersion contributes a small fraction of a percent as it is generally believed in numerous applications of the Lifshitz theory. It is, however, quite another matter when the account of the spatial dispersion results in some \"dramatic effects\", 64 i.e., in several hundred times larger thermal correction than is obtained in the local case. Below we demonstrate that the conclusions of Refs. 64 -68 are in fact not reliable because they use the Lifshitz theory of dispersion forces outside of its application range. 86 To find the electromagnetic modes associated with an empty gap between the plates, Refs. 64 -68 use the standard continuity boundary conditions,\n\nE 1t = E 2t , B 1n = B 2n , D 1n = D 2n , B 1t = B 2t , ( 102\n\n)\n\nwhich are commonly applied in the derivation of the Lifshitz formula for spatially local nonmagnetic materials. Here B is the magnetic induction, n is the normal to the boundary directed inside the medium, the subscripts n, t refer to the normal and tangential components, respectively, the subscript 1 refers to the vacuum and subscript 2 to the plate material. In Refs. 64 -68 the spatial dispersion is described by the longitudinal and transverse dielectric permittivities depending on the wave vector and frequency: ε kl = ε kl (q, ω). However, as is shown below, in the theory October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 28 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko of the Casimir effect both the boundary conditions (102) and permittivities ε kl = ε kl (q, ω) are inapplicable. We start from the boundary conditions and recall the set of Maxwell equations in a metal describing the Casimir effect, rotE + 1 c ∂B ∂t = 0, divD = 0, rotB -1 c ∂D ∂t = 0 divB = 0. (103) Equations (103) do not contain any external, i.e., independent on E, D and B, current or charge densities. The definition of the electric displacement is\n\n∂D ∂t = ∂E ∂t + 4πi, ( 104\n\n)\n\nwhere the volume current i is induced by E and B and takes into account the conduction electrons.\n\nIn electrodynamics with spatial dispersion the electric field and magnetic induction are finite at the boundary surfaces, whereas the electric displacement can tend to infinity. 87 Then, integrating Eqs. (103) over the thickness of the boundary layer as is done in Ref. 88, we reproduce the first two conditions in Eq. ( 102 ) and arrive at the modified third and fourth conditions, 87,89\n\nE 1t = E 2t , B 1n = B 2n , D 2n -D 1n = 4πσ, [n × (B 2 -B 1 )] = 4π c j, ( 105\n\n)\n\nwhere the induced surface charge and current densities are given by\n\nσ = 1 4π 2 1 div[n × [D × n]]dl, j = 1 4π 2 1 ∂D ∂t dl. ( 106\n\n)\n\nNote that the boundary conditions (105), (106) are obtained from the macroscopic Maxwell equations for physical fields. They should not be mixed with the boundary conditions arising in perturbative theories and for the fictitious fields (see below). In linear electrodynamics for a medium with time-independent properties without spatial dispersion the material equation connecting the electric displacement and electric field takes the form\n\nD k (r, t) = t -∞ εkl (r, t -t ′ )E l (r, t ′ )dt ′ . ( 107\n\n)\n\nAccording to this equation, the electric displacement at a point r and moment t is determined by the electric field at the same point r at different moments t ′ ≤ t (the spatial dispersion is absent but the temporal may be present). It is easily seen that the substitution of Eq. (107) in Eq. (106) leads to σ = 0, j= 0 and, as a result, the boundary conditions (105) coincide with the standard continuity conditions (102). It is unjustified, however, to use conditions (102) in the presence of spatial dispersion. In Refs. 87, 90 a few examples are presented illustrating that in this case neither σ nor j is equal to zero. We now turn to a discussion of the use of dielectric permittivity ε kl (q, ω) in the theory of the Casimir effect with account of spatial dispersion. In the presence of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 29 only frequency dispersion, it is possible to perform the Fourier transformation of the fields E(r, t) = ∞ -∞ E(r, ω)e -iωt dω, D(r, t) = ∞ -∞ D(r, ω)e -iωt dω, (108) in Eq. ( 107 ) and arrive at Eq. (101) where\n\nε kl (r, ω) = ∞ 0 εkl (r, τ )e iωτ dτ ( 109\n\n)\n\nis the frequency-dependent dielectric permittivity and τ ≡ t -t ′ . In fact Eqs. (101), (108) and (109) are used in parallel with the boundary conditions (102) in all derivations of the Lifshitz formula. 2,6,36,37,38,69,70, 84 If the material of the plates is characterized not only by temporal but also spatial dispersion, Eq. (107) is generalized to\n\nD k (r, t) = t -∞ dt ′ dr ′ εkl (r, r ′ , t -t ′ )E l (r ′ , t ′ ). ( 110\n\n)\n\nIf the material medium were uniform in space (i.e., all points were equivalent), the kernel ε would not depend on r and r ′ separately, as in Eq. ( 110 ), but on the difference R ≡ rr ′ . In this case, by performing the Fourier transformation, E(r, t) = ∞ -∞ dω dqE(q, ω)e i (qr-ωt) , D(r, t) = ∞ -∞ dω dqD(q, ω)e i (qr-ωt) , (111) and substituting it in Eq. ( 110 ), one could introduce the dielectric permittivities\n\nε kl (q, ω) = ∞ 0 dτ dR εkl (R, τ )e -i(qR-ωτ ) , ( 112\n\n)\n\nas Refs. 64-68 do, and rearrange Eq. (110) to the form\n\nD k (q, ω) = ε kl (q, ω)E l (q, ω). ( 113\n\n)\n\nIn the Casimir effect, however, the material medium is not uniform due to the presence of a macroscopic gap between the two plates (half spaces). Because of this, the assumption that the kernel ε depends only on R and τ is wrong. As a result, it is not possible to introduce the dielectric permittivity ε kl (q, ω) depending on the wave vector and frequency. In fact, for systems with spatial dispersion in the presence of boundaries the kernel ε depends not only on R and τ but also on the distance from the boundary. 87 In this complicated situation the following approximate phenomenological approach is sometimes applicable. 87 For electromagnetic waves with a wavelength λ the kernel ε(r, r ′ , τ ) in Eq. (110) differs essentially from zero only in a certain vicinity of the point r with characteristic dimensions l ≪ λ (for nonmetallic condensed media l is of the order of the lattice constant). Then it is reasonable to assume that ε is a function of R=r-r ′ , except for a layer of thickness l adjacent to the boundary surface. If one is mostly interested in bulk phenomena October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 30 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and neglects the role and influence of a subsurface layer, the quantity ε kl (q, ω) may be employed as a reasonable approximation. This approximate phenomenological approach is widely applied in the theoretical investigation of the anomalous skin effect (see, e.g., Ref. 91). Note that in Ref. 91 some kind of fictitious infinite system was introduced and electromagnetic fields in this system are discontinuous on the boundary surface. This discontinuity should not be confused with the discontinuity of physical fields of a real system in the presence of spatial dispersion described by Eqs. (105) and (106) . (There is also another approach to the description of the anomalous skin effect in polycrystals using the generalizations of the local Leontovich impedance which takes into account the shape of Fermi surface 92 ). The frequency and wave vector dependent dielectric permittivity in the presence of boundaries is also approximately applied in the theory of radiative heat transfer 93 or in the study of electromagnetic interaction of molecules with metal surfaces. 94 In all these applications the boundary effects are usually taken into account by the boundary conditions (105) supplemented by so called \"additional boundary conditions\".\n\nIt is unlikely, however, that the approximate phenomenological approach using such quantity as ε kl (q, ω) in the presence of boundaries would be applicable in the theory of the Casimir force where the boundary effects on the zero-point electromagnetic oscillations are of prime importance. It is notable also that this approach faces serious theoretical difficulties including the violation of the law of conservation of energy. 95 It is then not surprising that the application of this approach in Refs. 64, 65 results in the contribution to the Casimir free energy from the transverse electric mode which is in contradiction with experiment. 30,32,33 One more shortcoming of Refs. 64-68 is that they substitute the dielectric permittivity ε kl (q, ω), depending on both wave vector and frequency, into the conventional Lifshitz formula derived in the presence of only temporal dispersion. In the famous review paper 96 it has been noticed, however, that with the inclusion of spatial dispersion the free energy of a fluctuating field takes the form F = F L + ∆F , where F L is given by the conventional Lifshitz expression derived in a spatially local case and written in terms of the Fresnel reflection coefficients, and ∆F is an additional term which can be expressed in terms of the thermal Green's function of the electromagnetic field and polarization operator. Review 96 calls as not reliable the results of, e.g., Ref. 97 obtained by the substitution of dielectric permittivity ε kl (q, ω), taking account of spatial dispersion, into the conventional Lifshitz formula. It can be true that the Lifshitz formula written in terms of general reflection coefficients is applicable in both spatially local and nonlocal situations. However, as far as the exact reflection coefficients in a spatially nonlocal case are unknown, the use of some approximate phenomenological models, elaborated in literature for applications different than the Casimir effect, may lead to incorrect results for ∆F and create inconsistencies with experiment.\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 31\n\nTo conclude this section, the results of Refs. 64-68 on the influence of spatial nonlocality on the Casimir interaction are shown to be not reliable. Although at present there is no fundamental theory of the thermal Casimir force incorporating spatial dispersion, there is no reason to expect that it can play any significant role in the frequency region of infrared optics (experimental separations) or normal skin effect (i.e., at separations between plates greater than 4-5 µm)."},{"section_type":"DISCUSSION","section_title":"CONCLUSIONS AND DISCUSSION","text":"In the foregoing we have presented the derivation of analytic asymptotic expressions for the free energy, pressure and entropy of the Casimir interaction between two dielectric plates and between metal and dielectric plates at low and high temperatures. It was shown that the low-temperature behavior of the Casimir interaction between dielectrics and between dielectric and metal is determined by the static dielectric permittivities of nonpolar dielectrics. The obtained results were shown to be in agreement with thermodynamics when the static dielectric permittivities of dielectrics are finite. In particular, the entropy of the Casimir interaction goes to zero when the temperature vanishes, i.e., the Nernst heat theorem is satisfied. This demonstrates the consistency of the original Lifshitz's approach to the van der Waals forces between dielectrics which disregards the small conductivity of dielectrics at constant current.\n\nThe second important result shown above is that the inclusion of the dc conductivity of dielectrics into the model of dielectric response leads not to some small corrections to the characteristics of the Casimir interaction, as one could expect, but makes the Lifshitz theory inconsistent with thermodynamics leading to the violation of the Nernst heat theorem. This reveals that real material properties at very low, quasistatic frequencies are in fact not related to the phenomenon of the van der Waals and Casimir forces which is actually determined by sufficiently high frequencies. In this case the zero-frequency contribution to the Casimir force should be understood not literally but as analytic continuation to zero of the material physical behavior in the region around the characteristic frequency.\n\nThe presented results provide a basis for the calculation of the van der Waals and Casimir forces between real materials. Such calculations are much needed for numerous applications of the Casimir force discussed in the Introduction, in particular for the applications in nanotechnology and for constraining predictions of fundamental physical theories beyond the Standard Model. Bearing in mind that semiconductors are the main constituent materials in nanotechnological devices, it is a subject of high priority to understand the Casimir effect with semiconductor boundaries. In this connection we have discussed new experimental results and theoretical ideas on the Casimir interaction between a metal sphere and semiconductor plate. It was stressed that by changing the charge carrier density in the semiconductor it is possible to bring it in different intermediate states between metallic and dielectric. In this case the problem arises when the conductivity properties of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 32 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko semiconductor are not related to dispersion forces and when they are becoming relevant. A criterion for the resolution of this problem was formulated based on the relation between the typical frequency at which the dc conductivity properties come into play and the characteristic frequency of the Casimir effect. In fact the thermal Casimir interaction between semiconductors remains an open question and much work should be done both in experiment and theory to gain a better insight into this subject.\n\nThe last major problem discussed in this review is whether or not the spatial dispersion influences essentially the thermal Casimir force between real materials. In the present literature there is no agreement on this subject. We adduced arguments in favor of the statement that in the region of experimental separations the influence of the spatial dispersion on the Casimir force is negligible small. The statements on the opposite, contained in the literature, were shown to be not reliable because they are obtained by the application of the Lifshitz theory outside of its application range. At the same time it was ascertained that at the moment there is no consistent fundamental theory of the van der Waals and Casimir forces taking the spatial dispersion into account. This is the problem to solve in the foreseeable future. ACKNOWLEDGMENTS G.L.K. and V.M.M. are grateful to the Center of Theoretical Studies and the Institute for Theoretical Physics, Leipzig University for their kind hospitality. This work was supported by Deutsche Forschungsgemeinschaft grant 436 RUS 113/789/0-2. References 1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948). 2. P. W. Milonni, The Quantum Vacuum (Academic Press, San Diego, 1994). 3. V. M. Mostepanenko and N. N. Trunov, The Casimir Effect and its Applications (Clarendon Press, Oxford, 1997). 4. K. A. Milton, The Casimir Effect (World Scientific, Singapore, 2001). 5. M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999). 6. M. Bordag, U. Mohideen and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001). 7. S. K. Lamoreaux, Rep. Progr. Phys. 68, 201 (2005). 8. P. Candelas and S. Weinberg, Nucl. Phys. B237, 397 (1984). 9. M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D58, 075003 (1998); D60, 055004 (1999); D62, 011701(R) (2000). 10. J. C. Long, H. W. Chan and J. C. Price, Nucl. Phys. B539, 23 (1999). 11. E. Fischbach, D. E. Krause, V. M. Mostepanenko and M. Novello, Phys. Rev. D64, 075010 (2001). 12. V. M. Mostepanenko, Int. J. Mod. Phys. A17, 722 (2002); A17, 4143 (2002). 13. G. L. Klimchitskaya and U. Mohideen, Int. J. Mod. Phys. A17, 4143 (2002). 14. J. F. Babb, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A70, 042901 (2004). 15. M. Antezza, L. P. Pitaevskii and S. Stringari, Phys. Rev. A70, 053619 (2004). 16. M. Krech, The Casimir Effect in Critical Systems (World Scientific, Singapore, 1994).\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 33 17. H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D.J. Bishop and F. Capasso, Science 291, 1941 (2001); Phys. Rev. Lett. 87, 211801 (2001). 18. E. V. Blagov, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B71, 235401 (2005). 19. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer, Berlin, 1995). 20. S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997). 21. U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998). 22. G. L. Klimchitskaya, A. Roy, U. Mohideen and V. M. Mostepanenko, Phys. Rev. A60, 3487 (1999). 23. A. Roy and U. Mohideen, Phys. Rev. Lett. 82, 4380 (1999). 24. A. Roy, C.-Y. Lin and U. Mohideen, Phys. Rev. D60, 111101(R) (1999). 25. B. W. Harris, F. Chen and U. Mohideen, Phys. Rev. A62, 052109 (2000). 26. T. Ederth, Phys. Rev. A62, 062104 (2000). 27. G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, Phys. Rev. Lett. 88, 041804 (2002). 28. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. Lett. 88, 101801 (2002). 29. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A66, 032113 (2002). 30. R. S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. López and V. M. Mostepanenko, Phys. Rev. D68, 116003 (2003). 31. F. Chen, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Phys. Rev.\n\nA69, 022117 (2004). 32. R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya, D. E. Krause and V. M. Mostepanenko, Ann. Phys. (N.Y.) 318, 37 (2005). 33. G. L. Klimchitskaya, R. S. Decca, D. López, E. Fischbach, D. E. Krause and V. M. Mostepanenko, Int. J. Mod. Phys. A20, 2205 (2005). 34. Y. Srivastava, A. Widom and M. H. Friedman, Phys. Rev. Lett. 55, 2246 (1985). 35. Y. Srivastava and A. Widom, Phys. Rep. 148, 1 (1987). 36. E. M. Lifshitz, Sov. Phys. JETP 2, 73 (1956). 37. I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, Sov. Phys. Usp. 4, 153 (1961). 38. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part II (Pergamon, Oxford, 1980). 39. J. Schwinger, L. L. DeRaad and K. A. Milton, Ann. Phys. (N.Y.) 115, 1 (1978). 40. L. S. Brown and G. J. Maclay, Phys. Rev. 184, 1272 (1969). 41. M. Boström and B. E. Sernelius, Phys. Rev. Lett. 84, 4757 (2000). 42. J. Feinberg, A. Mann and M. Revzen, Ann. Phys. (N.Y.) 288, 103 (2001). 43. A. Scardiccio and R. L. Jaffe, Nucl. Phys. B743, 249 (2006). 44. C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A62, 012110 (2000). 45. M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. Lett.\n\n85, 503 (2000). 46. J. S. Høye, I. Brevik, J. B. Aarseth and K. A. Milton, Phys. Rev. E67, 056116 (2003). 47. I. Brevik, J. B. Aarseth, J. S. Høye and K. A. Milton, Phys. Rev. E71, 056101 (2005). 48. V. B. Bezerra, G. L. Klimchitskaya and C. Romero, Phys. Rev. A65, 012111 (2002). 49. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A67, 062102 (2003). 50. J. R. Torgerson and S. K. Lamoreaux, Phys. Rev. E70, 047102 (2004). 51. S. K. Lamoreaux and W. T. Buttler, Phys. Rev. E71, 036109 (2005). 52. V. B. Bezerra, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A66, 022112 (2002).\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 34 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 53. V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko and C. Romero, Phys. Rev. A69, 022119 (2004). 54. V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. López, V. M. Mostepanenko and C. Romero, Phys. Rev. E73, 028101 (2006). 55. I. Brevik, S. A. Ellingsen and K. A. Milton, New J. Phys. 8, 236 (2006). 56. V. M. Mostepanenko, V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. López and C. Romero, J. Phys. A39, 6589 (2006). 57. J. R. Zurita-Sánchez, J.-J. Greffet and L. Novotny, Phys. Rev. A69, 022902 (2004). 58. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati and J.-J. Greffet, Surf. Sci. Rep.\n\n57, 59 (2005). 59. B. S. Stipe, H. J. Mamin, T. D. Stowe, T. W. Kenny and D. Rugar, Phys. Rev. Lett.\n\n87, 096801 (2001). 60. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D72, 085009 (2005). 61. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A72, 022111 (2005). 62. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A72, 020101(R) (2005); A73, 019905(E) (2006). 63. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev.\n\nA74, 022103 (2006). 64. B. E. Sernelius, Phys. Rev. B71, 235114 (2005). 65. V. B. Svetovoy and R. Esquivel, Phys. Rev. E72, 036113 (2005). 66. R. Esquivel and V. B. Svetovoy, Phys. Rev. A69, 062102 (2004). 67. R. Esquivel, C. Villarreal and W. L. Mochán, Phys. Rev. A68, 052103 (2003). 68. A. M. Contreras-Reyes and W. L. Mochán, Phys. Rev. A72, 034102 (2005). 69. J. Mahanty and B. W. Ninham, Dispersion Forces (Academic, New York, 1976). 70. V. A. Parsegian and B. W. Ninham, Nature 224, 1197 (1969). 71. L. Bergström, Adv. Coll. Interface Sci. 70, 125 (1997). 72. H. Mitter and D. Robaschik, Eur. Phys. J. B13, 335 (2000). 73. B. W. Ninham and J. Daicic, Phys. Rev. A57, 1870 (1998). 74. G. L. Klimchitskaya, B. Geyer and V. M. Mostepanenko, J. Phys. A39, 6495 (2006). 75. A. O. Caride, G. L. Klimchitskaya, V. M. Mostepanenko and S. I. Zanette, Phys. Rev. A71, 042901 (2005). 76. Handbook of Optical Constants of Solids, ed. E. D. Palik (Academic, New York, 1985). 77. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Int. J. Mod. Phys. A16, 3291 (2001). 78. J. C. Slater, Insulators, Semiconductors and Metals. Quantum Theory of Molecules and Solids, Vol. 3, (McGraw-Hill, New York, 1967). 79. Materials Science and Engineering Handbook, eds. J. F. Shackelford and W. Alexander (CRC, Boca Raton, 2001). 80. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A65, 062109 (2002). 81. N. Inui, J. Phys. Soc. Jpn. 72, 2198 Jpn. 72, (2003);); 73, 2198 (2004); 75, 024004 (2006) . 82. W. Arnold, S. Hunklinger and K. Dransfeld, Phys. Rev. B19, 6049 (1979). 83. F. Chen and U. Mohideen, J. Phys. A39, 6233 (2006). 84. C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A67, 043811 (2003). 85. E. I. Kats, Sov. Phys. JETP 46, 109 (1977). 86. G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B75, 036101 (2007). 87. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer, Berlin, 1984).\n\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 35 88. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). 89. V. L. Ginzburg, Physics and Astrophysics (Pergamon Press, Oxford, 1985). 90. V. M. Agranovich and V. L. Ginzburg, Sov. Phys. JETP 36, 440 (1973). 91. K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968). 92. I. M. Kaganova and M. I. Kaganov, Phys. Rev. B63, 054202 (2001). 93. S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 3 (Springer, Berlin, 1989). 94. G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984). 95. J. T. Foley and A. J. Devaney, Phys. Rev. B12, 3104 (1975). 96. Yu. S. Barash and V. L. Ginzburg, Sov. Phys. Usp. 18, 305 (1975). 97. G. G. Kleinman and U. Landman, Phys. Rev. Lett. 33, 524 (1974)."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"INTRODUCTION\",\n \"text\": \"The Casimir effect 1 is the force and also the specific polarization of the vacuum arising in restricted quantization volumes and originating from the zero-point oscillations of quantized fields. This force acts between two closely spaced macrobodies, between an atom or a molecule and macrobody or between two atoms or molecules. During more than fifty years, passed after the discovery of the Casimir effect, it has attracted much theoretical attention because of numerous applications in quantum field theory, atomic physics, condensed matter physics, gravitation and cosmology, 1 October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 4 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nThese findings on the application of the Lifshitz theory to real metals have inspired a renewed interest in the Casimir force between dielectrics. As was mentioned above, at nonzero temperature dielectrics possess an although small but not equal to zero dc conductivity. In Ref. 57, 58 the van der Waals force arising from the dc conductivity of a dielectric plate was shown to lead to large effect in noncontact atomic friction, 59 a phenomenon having so far no satisfactory theoretical explanation. This brings up the question: Is it necessary or possible to take into account the dc conductivity of dielectrics in the Lifshitz theory? Recall that in the case of a positive answer the static dielectric permittivity of a dielectric material would be infinitely large. It is amply clear that the resolution of the above issue should be in accordance with the fundamentals of thermodynamics. For this reason, it is desirable to investigate the low-temperature behavior of the Casimir free energy and entropy for two dielectric plates both with neglected and included effects of the dc conductivity.\\n\\nA major breakthrough in the investigation of this problem was achieved in the year 2005. In Ref. 60 a new variant of perturbation theory was developed in a small parameter proportional to the product of the separation distance between the plates and the temperature. As a result, the behavior of the Casimir free energy, entropy and pressure at low temperatures was found analytically. If the static dielectric permittivity is finite, the thermal correction was demonstrated to be in accordance with thermodynamics. This solves positively the fundamental problem about the agreement between the Lifshitz theory and thermodynamics for the case of two dielectric plates. In Ref. 60 it was shown that, on the contrary, the formal inclusion of a small conductivity of dielectric plates at low frequencies into the model of their dielectric response leads to a violation of the Nernst heat theorem. This result gives an important guidance on how to extrapolate the tabulated optical data for the complex refractive index to low frequencies in numerous applications of the van der Waals and Casimir forces. All these problems and related ones arising for semiconductor materials are discussed in this review.\\n\\nIn Sec. 2 we derive the analytical behavior of the Casimir free energy, entropy and pressure in the configuration of two parallel dielectric plates at both low and high temperature. It is demonstrated that if the static dielectric permittivity is finite the Lifshitz theory is in agreement with thermodynamics. Sec. 3 contains the derivation of the low-temperature behavior for the Casimir free energy and entropy between two dielectric plates with included dc conductivity. In this case the Lifshitz theory is found to be in contradiction with the Nernst heat theorem. The conclusion is made that the conductivity properties of a dielectric material at a constant current are unrelated to the van der Waals and Casimir forces and must not be included into the model of dielectric response. In Sec. 4 we consider the thermal Casimir force between metal and dielectric. This problem was first investigated in Ref. 61. It was found that for dielectric plate with finite static dielectric permittivity the Nernst heat theorem is satisfied but the Casimir entropy may take negative values. Here we not only reproduce an analytical proof of the Nernst heat theorem but also October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 5 find the next perturbation orders in the expansion of the Casimir free energy and entropy in powers of a small parameter. The results obtained in high-temperature limit are also provided. Sec. 5 is devoted to the Casimir interaction between metal and dielectric plates with included dc conductivity of the dielectric material. We demonstrate that in this case the Nernst heat theorem is violated. Sec. 6 contains the discussion of semiconductors which present a wide variety of electric properties varying from metallic to dielectric. We consider the Casimir interaction between metal and semiconductor test bodies and formulate the criterion when it is appropriate to include the dc conductivity of a semiconductor into the model of dielectric response. In doing so, the results of recent experiments 62,63 on the measurement of the Casimir force between metal and semiconductor test bodies are taken into account. To this point the assumption has been made that metal, dielectric and semiconductor materials of the Casimir plates possess only temporal dispersion, i.e., can be described by the dielectric permittivity depending only on frequency. In Sec. 7 we discuss recent controversial results by different authors (see, e.g., Refs. 64 -68) attempting to take into account also spatial dispersion. As is shown in this section, the way of inclusion of spatial dispersion into the Lifshitz theory, used in Refs. 64 -68, is unjustified. We argue that the account of spatial dispersion cannot influence theoretical results obtained with the help of usual, spatially local, Lifshitz theory within presently used ranges of experimental separations. Sec. 8 contains our conclusions and discussion.\"\n },\n {\n \"section_type\": \"RESULTS\",\n \"section_title\": \"NEW ANALYTICAL RESULTS FOR THE THERMAL CASIMIR FORCE BETWEEN DIELECTRICS\",\n \"text\": \"We consider two thick dielectric plates (semispaces) described by the frequencydependent dielectric permittivity ε(ω) and restricted by the parallel planes z = ±a/2 with a separation a between them, in thermal equilibrium at temperature T . The Lifshitz formula for the free energy of the van der Waals and Casimir interaction between the plates is given by 2,3,4,5,6,36,37,\\n\\n38 F (a, T ) = k B T 2π ∞ l=0 1 - δ l0 2 ∞ 0 k ⊥ dk ⊥ ( 1\\n\\n) × ln 1 -r 2 (ξ l , k ⊥ )e -2aq l + ln 1 -r 2 ⊥ (ξ l , k ⊥ )e -2aq l ,\\n\\nwhere the reflection coefficients for two independent polarizations of electromagnetic field are defined as\\n\\nr (ξ l , k ⊥ ) = ε l q l -k l ε l q l + k l , r ⊥ (ξ l , k ⊥ ) = k l -q l k l + q l . ( 2\\n\\n)\\n\\nHere k ⊥ is the magnitude of the wave vector in the plane of plates, ξ l = 2πk B T l/ are the Matsubara frequencies, k B is the Boltzmann constant, ε l = ε(iξ l ), and q l = ξ 2 textDr 6 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nThe problems in the application of the Lifshitz theory to real materials discussed in the Introduction are closely connected with the values of the reflection coefficients at zero Matsubara frequency. For later use we discuss it for the various cases.\\n\\n• For ideal metals 40 it holds r (0, k ⊥ ) = r ⊥ (0, k ⊥ ) = 1.\\n\\n(\\n\\n) 4\\n\\n• For real metals described by the dielectric function of the Drude model,\\n\\nε(iξ l ) = 1 + ω 2 p ξ l [ξ l + ν(T )] , ( 5\\n\\n)\\n\\nwhere ω p is the plasma frequency and ν(T ) is the relaxation parameter, it holds 41 r (0, k ⊥ ) = 1, r ⊥ (0, k ⊥ ) = 0. (6) Eq. ( 6 ) results in the discontinuity between the cases of ideal and real metals and leads to the violation of the Nernst heat theorem for metallic plates having perfect crystal lattices. 52,53,54,56 • For real metals described by the dielectric function of the plasma model, ε(iξ l ) = 1 + ω 2\\n\\np ξ 2 l , ( 7\\n\\n)\\n\\nfrom Eq. (2) it follows: 44,45 r (0, k ⊥ ) = 1, r ⊥ (0, k ⊥ ) =\\n\\nc 2 k 2 ⊥ + ω 2 p -ck ⊥ c 2 k 2 ⊥ + ω 2 p + ck ⊥ . ( 8\\n\\n)\\n\\nHere, in the limit of ideal metals (ω p → ∞) the continuity is preserved because r ⊥ (0, k ⊥ ) in Eq. ( 8 ) goes to unity. The free energy (1) calculated with the permittivity (8) is also consistent with thermodynamics. • For dielectrics and semiconductors the dielectric permittivities at the imaginary Matsubara frequencies are given by the Ninham-Parsegian representation, 69,70 ε(iξ l ) = 1 + j\\n\\nC j 1 + ξ 2 l /ω 2 j , ( 9\\n\\n)\\n\\nwhere the parameters C j are the absorption strengths satisfying the condition j\\n\\nC j = ε 0 -1 ( 10\\n\\n)\\n\\nand ω j are the characteristic absorption frequencies. Here, the static dielectric permittivity ε 0 ≡ ε(0) is supposed to be finite. Although Eq. (10) is an October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 7 approximate one, it gives a very accurate description for many materials. 71 By the substitution of Eq. (9) in Eq. ( 2 ) one arrives at r (0, k ⊥ ) ≡ r 0 = ε 0 -1 ε 0 + 1 , r ⊥ (0, k ⊥ ) = 0. (11)\\n\\nNote that the vanishing of the transverse reflection coefficient for dielectrics at zero frequency in Eq. (11) has another meaning than for the Drude metals in Eq. ( 6 ). For Drude metal the parallel reflection coefficient is equal to the physical value for real photons at normal incidence, i.e., to unity, and the transverse one vanishes instead of taking unity, its physical value. This results in the violation of the Nernst heat theorem for perfect crystal lattices. In the case of dielectrics both reflection coefficients at zero frequency in Eq. (11) depart from the physical value for real photons which is equal to ( √ ε 0 -1)/( √ ε 0 + 1). In this case, however, one of them is larger and the other one is smaller than the physical value. As we will see below, this leads to the preservation of Nernst's heat theorem confirming that Eq. (9), despite being approximate, describes the material properties of dielectric and semiconductor plates in a thermodynamic consistent way.\\n\\nNow we derive the analytic representation for the Casimir free energy in Eq. (1) at low temperatures. For convenience in calculations, we introduce the dimensionless variables\\n\\nζ l = ξ l ξ c = 2aξ l c = τ l, y = 2aq l , ( 12\\n\\n)\\n\\nwhere ξ c = c/(2a) is the characteristic frequency, τ = 4πk B aT /( c), and q l was defined in Eq. ( 3 ). Then the Lifshitz formula (1) takes the form\\n\\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l dy f (ζ l , y) , ( 13\\n\\n)\\n\\nwhere f (ζ, y) = f (ζ, y) + f ⊥ (ζ, y), (14) f ,⊥ (ζ, y) = y ln 1 -r 2 ,⊥ (ζ, y)e -y , (15) and reflection coefficients (2), in terms of variables (12) , being given by\\n\\nr (ζ l , y) = ε l y -y 2 + ζ 2 l (ε l -1) ε l y + y 2 + ζ 2 l (ε l -1) , r ⊥ (ζ l , y) = y 2 + ζ 2 l (ε l -1) -y y 2 + ζ 2 l (ε l -1) + y . ( 16\\n\\n)\\n\\nTo separate the temperature independent contribution and thermal correction in Eq. ( 13 ) we apply the Abel-Plana formula, 3,6\\n\\n∞ l=0 1 - δ l0 2 F (l) = ∞ 0 F (t) dt + i ∞ 0 dt F (it) -F (-it) e 2πt -1 , ( 17\\n\\n)\\n\\nwhere F (z) is an analytic function in the right half-plane. Here, taking it as\\n\\nF (x) = ∞ x dy f (x, y) ( 18\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 8 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nand using Eq. (17), we can identically rearrange Eq. (13) to the form F (a, T ) = E(a) + ∆F (a, T ) , (19) where E(a) is the energy of the van der Waals or Casimir interaction at zero temperature,\\n\\nE(a) = c 32π 2 a 3 ∞ 0 dζ ∞ ζ dy f (ζ, y) , ( 20\\n\\n)\\n\\nand ∆F (a, T ) is the thermal correction to this energy, ∆F (a, T ) = i cτ 32π 2 a 3 ∞ 0 dt F (iτ t) -F (-iτ t) e 2πt -1 . (21) Note that, in fact, Eq. ( 21 ) describes the dependence of the free energy on the temperature arising from the dependence on temperature of the Matsubara frequencies. Thus, ∆F (a, T ) in (21) coincides with the thermal correction to the energy, defined as F (a, T ) -F (a, 0), only for plate materials with temperature independent properties. The asymptotic expressions for the energy E(a) at both short and large separations are well known. 6,37,38 Below we find the asymptotic expressions for the thermal correction (21) under the conditions τ ≪ 1 and τ ≫ 1. Taking into account the definition of τ in Eq. ( 12 ), the asymptotic expressions at τ ≪ 1 are applicable both at small and large separations if the temperature is sufficiently low.\\n\\nWe begin with condition τ ≪ 1. Let us substitute Eq. (9) in Eqs. (14) -(16), expand the function f (x, y) in powers of x = τ t, and than integrate the obtained expansion with respect to y from x to infinity in order to find F (x) in Eq. (18) and F (ix) -F (-ix) in Eq. (21) .\\n\\nIt is easy to check that f ⊥ (ζ, y) does not contribute to the leading, second, order in the expansion of F (ix) -F (-ix) in powers of x. Thus, we can restrict ourselves by the consideration of the expansion f (x, y) = y ln 1 -r 2 0 e -y + 2ε 0 r 2 0 ε 0 + 1\\n\\nx 2 y (e y -r 2 0 ) + 4ξ 2 c r 2 0 (ε 0 + 1)ω 2 1 yx 2 e y -r 2 0 + O(x 3 ), ( 22\\n\\n)\\n\\nwhere r 0 was defined in Eq. (11) . Note that for simplicity we consider here only one oscillator in Eq. ( 9 ) and put ω j = ω 1 . The case of several oscillator modes can be considered in an analogous way.\\n\\nAs a next step, we integrate Eq. ( 22 ) term by term according to Eq. ( 18 ), expand the partial results in powers of x and sum up the obtained series. Thereby we obtain the following expressions:\\n\\nZ 1 (x) ≡ ∞ x y dy ln 1 -r 2 0 e -y = - ∞ n=1 (1 + nx)e -nx n 3 r 2n 0 = -Li 3 (r 2 0 ) - x 2 2 ln(1 -r 2 0 ) + O(x 3 ) , ( 23\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 9\\n\\nZ 2 (x) ≡ 2ε 0 r 2 0 x 2 ε 0 + 1 ∞ x dy y (e y -r 2 0 ) = - 2ε 0 x 2 ε 0 + 1 ∞ n=1 r 2n 0 Ei(-nx) , ( 24\\n\\n) Z 3 (x) ≡ 4r 2 0 ξ 2 c x 2 (ε 0 + 1)ω 2 1 ∞ x ydy e y -r 2 0 = 4ξ 2 c x 2 (ε 0 + 1)ω 2 1 ∞ n=1 r 2n 0 (1 + nx)e -nx n 2 = 4ξ 2 c (ε 0 + 1)ω 2 1 x 2 Li 2 (r 2 0 ) - x 4 2 r 2 0 1 -r 2 0 + O(x 5 ) , ( 25\\n\\n)\\n\\nwhere Li n (z) is the polylogarithm function and Ei(z) is the exponential integral function. From these equations it follows\\n\\nZ 1 (ix) -Z 1 (-ix) = O(x 3 ) , Z 3 (ix) -Z 3 (-ix) = O(x 5 ) , ( 26\\n\\n) Z 2 (ix) -Z 2 (-ix) = 2iπ ε 0 ε 0 + 1 r 2 0 1 -r 2 0 x 2 + O(x 3 ) , ( 27\\n\\n)\\n\\nand, thus, Z 1 and Z 3 do not contribute to the leading order in the expansion of F (ix) -F (-ix). The latter is determined by Z 2 only. As a result, we arrive at\\n\\nF (ix) -F (-ix) = iπ (ε 0 -1) 2 2(ε 0 + 1) x 2 -iαx 3 + O(x 4 ), ( 28\\n\\n)\\n\\nwhere r 0 was substituted from Eq. ( 11 ) and α was introduced for the still unknown real coefficient of the next to leading order resulting from Z 1 and Z 2 as well as, possibly, from f ⊥ (ζ, y). At this stage it is difficult to determine the value of this coefficient because all powers in the expansion of f (x, y) contribute to it. Remarkably, the two leading orders depend only on the static dielectric permittivity ε 0 and are not influenced by the dependence of the dielectric permittivity on the frequency contained in Z 3 . Substituting Eq. (28) in Eq. (21) and using Eq. ( 19 ), we obtain\\n\\nF (a, T ) = E(a) - c 32π 2 a 3 ζ(3) 8π 2 (ε 0 -1) 2 ε 0 + 1 τ 3 -C 4 τ 4 + O(τ 5 ) , ( 29\\n\\n)\\n\\nwhere C 4 ≡ α/240 and ζ(z) is the Riemann zeta function. So far we have considered the free energy. The thermal pressure is obtained as\\n\\nP (a, T ) = - ∂F (a, T ) ∂a = P 0 (a) - c 32π 2 a 4 C 4 τ 4 + O(τ 5 ) , ( 30\\n\\n)\\n\\nwhere P 0 (a) = -∂E(a)/∂a is the Casimir pressure at zero temperature. In order to determine the value of the coefficient C 4 of the leading term, we express the pressure directly through the Lifshitz formula\\n\\nP (a, T ) = - cτ 32π 2 a 4 ∞ l=0 1 - δ l0 2 ∞ ζ l y 2 dy r 2 (ζ l , y) e y -r 2 (ζ l , y) + r 2 ⊥ (ζ l , y) e y -r 2 ⊥ (ζ l , y) . ( 31\\n\\n)\\n\\nAgain, applying the Abel-Plana formula (17), we represent the pressure as follows,\\n\\nP (a, T ) = P 0 (a) + ∆P (a, T ), ( 32\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 10 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nwhere the thermal correction to P 0 (a), the pressure at zero temperature, is\\n\\n∆P (a, T ) = - i cτ 32π 2 a 4 ∞ 0 dt Φ(iτ t) -Φ(-iτ t) e 2πt -1 ( 33\\n\\n)\\n\\nand the function Φ(x) is given by\\n\\nΦ(x) ≡ Φ (x) + Φ ⊥ (x) , Φ ,⊥ (x) = ∞ x dy y 2 r 2 ,⊥ (x, y) e y -r 2 ,⊥ (x, y) . ( 34\\n\\n)\\n\\nFirst, we determine the leading term of the expansion of Φ ⊥ (x) in powers of x. For this purpose, let us introduce the new variable v = y/x and note that the reflection coefficient r ⊥ (x, v) depends on x only through the frequency dependence of ε given by Eq. ( 9 ). Thus, we can rewrite and expand Eq. (34) as follows:\\n\\nΦ ⊥ (x) = x 3 ∞ 1 dv v 2 r 2 ⊥ (x, v) e vx -r 2 ⊥ (x, v) = x 3 ∞ 1 dv v 2 r 2 ⊥ (v) 1 -r 2 ⊥ (v) + O(x 4 ), ( 35\\n\\n)\\n\\nwhere, according to Eq. ( 16 ),\\n\\nr ⊥ (v) ≡ r ⊥ (0, v) = √ v 2 + ε 0 -1 -v √ v 2 + ε 0 -1 + v . ( 36\\n\\n)\\n\\nIntegration in Eq. ( 35 ) with account of Eq. (36) results in\\n\\nΦ ⊥ (x) = 1 - ε 0 (3 -ε 0 ) 2 √ ε 0 x 3 6 + O(x 4 ) , ( 37\\n\\n)\\n\\nfrom which it follows:\\n\\nΦ ⊥ (ix) -Φ ⊥ (-ix) = -i 1 - 1 2 √ ε 0 (3 -ε 0 ) x 3 3 + O(x 5 ). ( 38\\n\\n)\\n\\nThe expansion of Φ (x) from Eq. (34) in powers of x is somewhat more cumbersome. It can be performed in the following way. As is seen from the second equality in Eq. ( 26 ), the dependence of the dielectric permittivity on frequency contributes to F (ix) -F (-ix) starting from only the 5th power in x. Bearing in mind the connection between free energy and pressure, we can conclude that the dependence on the frequency contributes to Φ (ix) -Φ (-ix) starting from the 4th order. We are looking for the lowest (third) order expansion term of Φ (ix) -Φ (-ix). Because of this, it is permissible to disregard the frequency dependence of ε and describe the dielectric by its static dielectric permittivity.\\n\\nTo begin with, we identically rearrange Φ (x) in Eq. ( 34 ) by subtracting and adding the two first expansion terms of the function under the integral in powers of x,\\n\\nΦ (x) = ∞ x dy y 2 r 2 (x, y) e y -r 2 (x, y) -y 2 r 2 0 e y -r 2 0 + x 2 2ε 0 ε 0 + 1 r 2 0 e -y (1 -r 2 0 e -y ) 2 + ∞ x y 2 dy r 2 0 e y -r 2 0 -x 2 2ε 0 ε 0 + 1 ∞ x dy r 2 0 e -y (1 -r 2 0 e -y ) 2 , ( 39\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 11 and consider these three integrals separately. The first integral in terms of the new variable v = y/x reads\\n\\nQ 1 (x) ≡ x 3 ∞ 1 dv v 2 r 2 (v) e vx -r 2 (v) -v 2 r 2 0 e vx -r 2 0 + 2ε 0 ε 0 + 1 r 2 0 e -vx (1 -r 2 0 e -vx ) 2 , ( 40\\n\\n)\\n\\nwhere, in accordance with Eq. ( 16 ),\\n\\nr (v) ≡ r (0, v) = ε 0 v - √ v 2 + ε 0 -1 ε 0 v + √ v 2 + ε 0 -1 . ( 41\\n\\n)\\n\\nExpanding Q 1 (x) in powers of x and explicitly calculating the remaining integrals for the lowest, third, power of x results in\\n\\nQ 1 (x) = x 3 ∞ 1 dv v 2 r 2 (v) 1 -r 2 (v) -v 2 r 2 0 1 -r 2 0 + 2ε 0 ε 0 + 1 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ) ( 42\\n\\n) = x 3 6 1 - √ ε 0 2 1 + 3ε 0 -2 ε 2 0 + 2 r 2 0 1 -r 2 0 - 12 ε 0 ε 0 + 1 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ).\\n\\nThe second and third integrals on the right-hand side of Eq. ( 39 ) are simply determined with the following result:\\n\\nQ 2 (x) ≡ ∞ x y 2 dy r 2 0 e y -r 2 0 = ∞ n=1 r 2n 0 (2 + 2nx + n 2 x 2 )e -nx n 3 = 3Li 3 (r 2 0 ) - x 3 3 r 2 0 1 -r 2 0 + O(x 4 ) , ( 43\\n\\n)\\n\\nQ 3 (x) ≡ - 2ε 0 x 2 ε 0 + 1 ∞ x dy r 2 0 e -y (1 -r 2 0 e -y ) 2 = - 2ε 0 x 2 ε 0 + 1 r 2 0 e -x 1 -r 2 0 e -x = - 2ε 0 ε 0 + 1 x 2 r 2 0 1 -r 2 0 -x 3 r 2 0 (1 -r 2 0 ) 2 + O(x 4 ) . ( 44\\n\\n) Substituting Eqs. (42), (43) and (44) into Φ (x) = Q 1 (x) + Q 2 (x) + Q 3 (x), we arrive at Φ (ix) -Φ (-ix) = -i 1 + √ ε 0 2 2ε 2 0 -3ε 0 -1 x 3 3 + O(x 4 ). ( 45\\n\\n)\\n\\nThen, by summing Eqs. (38) and (45), the result is obtained\\n\\nΦ(ix) -Φ(-ix) = -i 2 + ε 5/2 0 -ε 3/2 0 -2 √ ε 0 x 3 3 + O(x 4 ). ( 46\\n\\n)\\n\\nNow we substitute Eq. (46) in Eq. ( 33 ) and perform integration. Finally, from Eq. (32) the desired expression for the Casimir pressure is derived\\n\\nP (a, T ) = P 0 (a) - c 32π 2 a 4 √ ε 0 -1 ε 2 0 + ε 3/2 0 -2 720 τ 4 + O(τ 5 ) . ( 47\\n\\n)\\n\\nBy comparison with Eq. ( 30 ) the explicit form of the coefficient C 4 is found as\\n\\nC 4 = √ ε 0 -1 ε 2 0 + ε 3/2 0 -2 /720 ( 48\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 12 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and, thus, both two first perturbation orders in the expansion for the free energy (29) are determined. Equations (29), (47) and (48) solve the fundamental problem of the thermodynamic consistency of the Lifshitz theory in the case of two dielectric plates. From Eqs. (29) and (48) the entropy of the van der Waals and Casimir interaction between plates takes the form\\n\\nS(a, T ) = - ∂F (a, T ) ∂T = 3k B ζ(3)(ε 0 -1) 2 64π 3 a 2 (ε 0 + 1) τ 2 ( 49\\n\\n) × 1 - 2π 2 (ε 0 + 1) ε 3/2 0 + 2ε 0 + 2 √ ε 0 + 2 135ζ(3)( √ ε 0 + 1) 2 τ + O(τ 2 ) .\\n\\nAs is seen from Eq. (49), in the limit τ → 0 (T → 0) the lower order contributions to the entropy are of the second and the third powers in the small parameter τ . Thus, the entropy vanishes when the temperature goes to zero as it must be in accordance with the third law of thermodynamics (the Nernst heat theorem).\\n\\nA similar behavior was obtained for ideal metals 40,72,73 and for real metals described by the plasma model. 45,52 For example, in the case of plates made of ideal metal the entropy at low temperatures is given by 40,72,73\\n\\nS(a, T ) = 3k B ζ(3) 32π 3 a 2 τ 2 1 - 2π 2 135ζ(3) τ + O(τ 2 ) . ( 50\\n\\n)\\n\\nNote, however, that the expansion coefficients in Eq. ( 50 ) cannot be obtained as a straightforward limit |ε 0 | → ∞ in Eq. ( 49 ) and the above equations for the free energy and pressure. The mathematical reason is that it is impermissible to interchange the limiting transitions τ → 0 and |ε 0 | → ∞ in the power expansions of functions depending on ε 0 as a parameter. Remarkably, the low-temperature behavior of the free energy, pressure and entropy of nonpolar dielectrics in Eqs. (29), (47) and (49) is universal, i.e., is determined only by the static dielectric permittivity. The absorption bands included in Eq. ( 9 ) do not influence the low-temperature behavior. A more simple derivation of the results (29), (47)-(49) for dielectrics with constant ε is contained in Ref. 74. As was demonstrated above, all these results remain unchanged if the dependence of dielectric permittivity on frequency is taken into account.\\n\\nIn Ref. 60 more general results were obtained related to two dissimilar dielectric plates with dielectric permittivities ε (1) (ω) and ε (2) (ω). For brevity here we present only the final expressions for the low-temperature behavior of the Casimir free energy, pressure and entropy between dissimilar plates. They are as follows: 60\\n\\nF (a, T ) = E(a) - c 32π 2 a 3 τ 3 × ζ(3) 8π 2 ε ( 1\\n\\n) 0 + ε ( 2\\n\\n) 0 + 2ε (1) 0 ε (2) 0 (ε ( 1\\n\\n) 0 + 1)(ε ( 2\\n\\n) 0 + 1) (ε ( 1\\n\\n) 0 -1)(ε (2) 0 -1) (ε (1) 0 + ε (2) 0 ) -C 4 τ + O(τ 2 ) ,\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 13 1,2) (0) and the coefficient C 4 is given by 60\\n\\nP (a, T ) = P 0 (a) - c 32π 2 a 4 C 4 τ 4 + O(τ 5 ) , ( 51\\n\\n) S(a, T ) = k B 2πa 2 τ 2 × 3ζ(3) 32π 2 ε ( 1\\n\\n) 0 + ε ( 2\\n\\n) 0 + 2ε (1) 0 ε (2) 0 (ε ( 1\\n\\n) 0 + 1)(ε ( 2\\n\\n) 0 + 1) (ε ( 1\\n\\n) 0 -1)(ε ( 2\\n\\n) 0 -1) (ε (1) 0 + ε (2) 0 ) -C 4 τ + O(τ 2 ) . Here ε (1,2) 0 ≡ ε (\\n\\nC 4 = 1 720 2 + 1 ε ( 1\\n\\n) 0 + ε (2) 0 ε (1) 0 + ε (2) 0 2 ( 52\\n\\n) × -ε ( 1\\n\\n) 0 + ε (2) 0 2 2ε ( 1\\n\\n) 0 + 2ε ( 2\\n\\n) 0 + ε (1) 0 ε (2) 0 -ε (1) 0 ε (2) 0 + ε (1) 0 ε (2) 0 ε (1) 0 ε (2) 0 5ε (1) 0 ε (2) 0 -3ε (1) 0 -3ε ( 2\\n\\n) 0 + 1 + ε (1) 0 ε (2) 0 ε (1) 0 -ε (2) 0 2 ε ( 1\\n\\n) 0 ε (2) 0 ε (1) 0 ε ( 2\\n\\n) 0 -ε ( 1\\n\\n) 0 ε ( 2\\n\\n) 0 -ε ( 1\\n\\n) 0 -ε (2) 0 - 3 ε ( 1\\n\\n) 0 ε (2) 0 2 ε ( 1\\n\\n) 0 -1 ε ( 2\\n\\n) 0 -1 ε ( 1\\n\\n) 0 -ε (2) 0 ε ( 1\\n\\n) 0 + ε (2) 0 Artanh ε ( 1\\n\\n) 0 + ε (2) 0 ε (1) 0 -ε (2) 0 ε (1) 0 ε ( 2\\n\\n) 0 -ε ( 1\\n\\n) 0 -ε (2) 0 .\\n\\nIt is easily seen that in the limit ε 52) coincide with equations (29), (47)-(49) having obtained above. Note that in the application region of low-temperature asymptotic expressions the entropy of the Casimir interaction between dielectric plates is nonnegative.\\n\\n(1) 0 = ε ( 2\\n\\n) 0 = ε 0 equations (51), (\\n\\nThe obtained analytic behavior of the free energy, pressure and entropy at low temperatures can be compared with the results of numerical computations using the Lifshitz formula. Dielectric properties of the plates can be described by the static dielectric permittivity or more precisely using the optical tabulated data for the complex index of refraction. As an example, in Fig. 1 we present the thermal corrections to the Casimir energy (a) and pressure (b) at a separation a = 400 nm as functions of temperature in the configuration of two dissimilar plates made of high-resistivity Si and SiO 2 . The dielectric permittivities of both materials along the imaginary frequency axis were computed in Ref. 75 using the optical data of Ref. 76. The precise thermal corrections computed by taking into account these permittivities are shown by the solid lines and corrections computed by our analytical asymptotic expressions are shown by the long-dashed lines. Short-dashed lines indicate the results computed by the Lifshitz formula with constant dielectric permittivities of Si and SiO 2 equal to ε (1) 0 = 11.67 and ε (2) 0 = 3.84, respectively. As is seen in Fig. 1a ,b, at T < 60 K the results obtained using the analytical asymptotic October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 14 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 25 50 75 100 125 150 175 200 1 2 3 4 T (K) -∆F (pJ) 25 50 75 100 125 150 175 200 1 2 3 4 5 6 7 T (K) -∆P (µPa) (b) (a)\\n\\nFig. 1. Magnitudes of the thermal corrections to the energy (a) and pressure (b) in configuration of two plates, one made of Si and another one of SiO 2 , at a separation a = 400 nm as a function of temperature calculated by the use of different approaches: by the Lifshitz formula and tabulated optical data (solid lines), by the Lifshitz formula and static dielectric permittivities (short-dashed lines), by the asymptotic expressions in Eqs. (51) and (52) (long-dashed lines).\\n\\nexpressions practically coincide with the solid lines computed using the tabulated optical data for the materials of the plates.\\n\\nNow we return to the case of two similar dielectric plates and consider the asymptotic expressions under the condition τ ≫ 1, i.e., at high temperatures (large separations). It is well known 37,38,77 that in this case the approximation of static October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 15 dielectric permittivity works good and the main contribution is given by the zerofrequency term of the Lifshitz formula (13)\\n\\nF (a, T ) = cτ 64π 2 a 3 ∞ 0 ydy ln 1 -r 2 0 e -y ( 53\\n\\n)\\n\\n(the other terms being exponentially small). Performing the integration in Eq. (53) we obtain\\n\\nF (a, T ) = - k B T 16πa 2 Li 3 r 2 0 . ( 54\\n\\n)\\n\\nIn a similar manner for the Casimir pressure and entropy at τ ≫ 1 it follows\\n\\nP (a, T ) = - k B T 8πa 3 Li 3 r 2 0 , S(a, T ) = k B 16πa 2 Li 3 r 2 0 . ( 55\\n\\n)\\n\\nEquations (54) and (55) are simply generalized 60 for the case of two dissimilar dielectric plates by performing the replacement r 2 0 → r (1) 0 r (2) 0 , where r (1,2) 0 are defined by Eq. (11) with the static dielectric permittivities of dissimilar plates ε (1,2) 0 .\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"IS THE DC CONDUCTIVITY RELATED TO THE CASIMIR INTERACTION BETWEEN DIELECTRICS?\",\n \"text\": \"As was discussed in the previous section, the zero-frequency term in formula (13), i.e., the contribution with l = 0, is of prime importance and determines many of the basic properties of the Casimir interaction. In the above consideration we have described dielectric materials by Eq. ( 9 ) with finite static dielectric permittivity ε 0 . This resulted in Eq. (11) where one reflection coefficient at zero frequency is larger and the other one is smaller than the physical value for real photons at normal incidence. However, in the Lifshitz theory, the departure of both coefficients from their physical values is coordinated in such a way that the Nernst heat theorem remains valid. It is common knowledge that at nonzero temperatures dielectric materials possess a negligibly small but not equal to zero dc conductivity. From physical intuition it is reasonable to expect that the influence of this conductivity on the van der Waals and Casimir forces should be also negligible. In Ref. 57 it was shown that, on the contrary, the inclusion of small dielectric dc conductivity in the model of dielectric response leads to a large effect in dispersion forces. This raises the question if dc conductivity is related to dispersion forces or if, on the contrary, the zero-frequency contribution should be understood not literally but as an analytic continuation from the region of high frequencies determining the physical phenomenon of dispersion forces.\\n\\nTo illustrate this problem in more details, we consider the asymptotic behavior of the free energy and entropy at low temperature with included dc conductivity. What this means is that, instead of the dielectric permittivity ε(iξ l ) given in Eq. ( 9 ), one uses 57,58\\n\\nε(iξ l ) = ε(iξ l ) + 4πσ 0 ξ l = ε l (iξ) + β(T ) l , ( 56\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 16 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nwhere σ 0 is the dc conductivity of the plate material and β(T ) = 2 σ 0 /(k B T ). The conductivity of dielectrics depends on temperature as σ 0 ∼ exp(-b/T ) where b is determined by the energy gap ∆ which differs for different materials. 78 The smallness of the dc conductivity of dielectrics can be illustrated 79 by the example of SiO 2 where at T = 300 K it holds β ∼ 10 -12 . Thus, the role of the dc conductivity is really negligible for all l ≥ 1. In addition, β(T ) quickly decreases with decrease of T and, as a consequence, remains negligible at any T . In spite of this, the substitution of Eq. (56) into the reflection coefficients (16) leads to different result than in Eq. (11): r (0, y) = 1, r (0, y) = 0. (57) Equation (57) is in some analogy to Eq. ( 6 ), obtained for metals described by the Drude model, which leads to the violation of the Nernst heat theorem in the case of perfect crystal lattices. Now we substitute the dielectric permittivity εl ≡ ε(ξ l ) in Eqs. (13) -(16) instead of ε l and find the Casimir free energy F (a, T ) with included dc conductivity. For convenience, we separate the zero-frequency term, subtract and add the usual zero-frequency contribution for dielectric without dc conductivity. The result is\\n\\nF (a, T ) = k B T 16πa 2 ∞ 0 ydy ln 1 -e -y -ln 1 -r 2 0 e -y + k B T 16πa 2 ∞ 0 ydy ln 1 -r 2 0 e -y ( 58\\n\\n) + k B T 8πa 2 ∞ l=1 ∞ ζ l ydy ln 1 -r2 (ζ l , y)e -y + ln 1 -r2 ⊥ (ζ l , y)e -y ,\\n\\nwhere r 0 was defined in Eq. ( 11 ). Let us expand the last, third, integral on the righthand side of Eq. (58) in powers of the small parameters β/l. Then, we combine the zero-order contribution in this expansion with the second integral on the righthand side of Eq. ( 58 ) and obtain the Casimir free energy F (a, T ) calculated with the dielectric permittivities ε l . The first integral on the right-hand side of Eq. (58) is calculated explicitly. Then, Eq. ( 58 ) can be rewritten as\\n\\nF (a, T ) = F (a, T ) - k B T 16πa 2 ζ(3) -Li 3 r 2 0 + R(a, T ). ( 59\\n\\n)\\n\\nHere, R(a, T ) is of order O(β/l). It represents the first and higher-order contributions in the expansion of the third integral on the right-hand side of Eq. (58) in powers of β/l. Restricting its explicit form to the first order contribution we get\\n\\nR(a, T ) = R 1 (a, T ) + O (β/l) 2 , ( 60\\n\\n) R 1 (a, T ) = k B T 4πa 2 ∞ l=1 β l ∞ ζ l dy y 2 e -y y 2 + ζ 2 l (ε l -1) (2 -ε l ) ζ 2 l -2y 2 y 2 + ζ 2 l (ε l -1) + ε l y 2 × r (ζ l , y) 1 -r 2 (ζ l , y)e -y - ζ 2 l y 2 + ζ 2 l (ε l -1) + y 2 r ⊥ (ζ l , y) 1 -r 2 ⊥ (ζ l , y)e -y . ( 61\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 17\\n\\nCalculating the entropy by the first equality in Eq. ( 49 ), we arrive at\\n\\nS(a, T ) = S(a, T ) + k B 16πa 2 ζ(3) -Li 3 r 2 0 - ∂R(a, T ) ∂T , ( 62\\n\\n)\\n\\nwhere S(a, T ) is the entropy for the plates with the dielectric permittivity ε l given by Eq. ( 49 ).\\n\\nLet us now show that the quantity R(a, T ) exponentially goes to zero with the decrease of T . First we consider only the integral in Eq. ( 61 ), expand the integrated function in powers of τ (we recall that ζ l = τ l), restrict ourselves to the main contribution, resulting for τ = 0, and rearrange it appropriately:\\n\\n-2 ∞ ζ l dy ye -y (ε 0 + 1) 2 r 0 1 -r 2 0 e -y = - 2 (ε 2 0 -1) ∞ ζ l dy y r 2 0 e -y 1 -r 2 0 e -y = - 2 (ε 2 0 -1) ∞ n=1 r 2n 0 ∞ ζ l dy ye -ny = - 2 (ε 2 0 -1) ∞ n=1 r 2n 0 1 + nζ l n 2 e -nζ l . ( 63\\n\\n)\\n\\nSubstituting this in Eq. ( 61 ), we find\\n\\nR 1 (a, T ) = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r 2n 0 n 2 ∞ l=1 e -nτ l l + nτ ∞ l=1 e -nτ l = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r 2n 0 n 2 -ln 1 -e -nτ + nτ e nτ -1 = k B Li 2 r 2 0 2πa 2 (ε 2 0 -1) T β ln τ + T β O(τ 0 ) . ( 64\\n\\n)\\n\\nHere, the last line is obtained by using the equality\\n\\n-ln 1 -e -nτ + nτ e nτ -1 = -ln τ + 1 -ln n + O(τ 2 ) , ( 65\\n\\n)\\n\\nsubstituting only its leading term and observing the definition of the integral log-\\n\\narithm, Li 2 (z) = (1/2) ∞ n=1 z n /n 2 .\\n\\nTaking into account that β ∼ (1/T ) exp(-b/T ), we get the conclusion that the temperature dependence of R 1 (a, T ) is given by\\n\\nR 1 (a, T ) ∼ e -b/T ln T. ( 66\\n\\n)\\n\\nThus, both R 1 (a, T ) and its derivative with respect to T in Eqs. (59) and (62) go to zero. The terms of the second and higher powers in β in Eq. (60) go to zero even faster than R 1 when T → 0. Finally, in the limit T → 0 from Eq. (62) it follows\\n\\nS(a, 0) = k B 16πa 2 ζ(3) -Li 3 r 2 0 > 0. ( 67\\n\\n)\\n\\nThe right-hand side of this equation depends on the parameter of the system under consideration (the separation distance a) and implies a violation of the Nernst heat theorem. An analogous result was obtained 60 in the case of two dissimilar dielectrics.\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 18 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nThe violation of the Nernst heat theorem in the Casimir interaction for dielectrics originates from the inclusion of the dc conductivity in the model of dielectric response. This violation is, however, of different nature than the one discussed above in the case of Drude metals. In the case of dielectrics the entropy at zero temperature is positive but in the case of Drude metals it is negative. In the case of metals the violation is caused by the vanishing contribution from the transverse electric mode at zero frequency whereas the other reflection coefficient takes the physical value 1 [see Eq. ( 6 )]. For dielectrics the situation is quite opposite. In this case the transverse reflection coefficient at zero frequency is always equal to zero [compare Eqs. (11) and (57) in the absence and in the presence of the contribution from dc conductivity]. Here the violation occurs due to the unity value of the parallel reflection coefficient at zero frequency in Eqs. (57) which departs from the value r 0 = (ε 0 -1)/(ε 0 + 1) coordinated with the zero value of the transverse coefficient in Eq. (11) .\\n\\nOne can conclude that the dc conductivity of a dielectric is not related to the nature of the van der Waals and Casimir forces and must not be included in the model of dielectric response. Ignoring this rule results in a violation of thermodynamics. Physically it is amply clear that there is no fluctuating field of zero frequency and that for such high-frequency phenomena as the van der Waals and Casimir forces the low-frequency behavior should be obtained by analytic continuation from the region of high frequencies. This permits to conclude that the correct procedure consists in the substitution of the finite static dielectric permittivities into the zero-frequency term of the Lifshitz formula, as Lifshitz and his collaborators really did. 36,37,38\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"THERMAL CASIMIR FORCE BETWEEN DIELECTRIC AND METAL PLATES\",\n \"text\": \"The Casimir interaction between metal and dielectric plates suggests the interesting opportunity to verify the thermodynamic consistency of the Lifshitz theory with different models of the dielectric response. This configuration was first investigated in Ref. 61 where it was proved that the Casimir entropy is in accordance with the demands of the Nernst heat theorem if the static permittivity of the dielectric plate is finite. In Ref. 61, however, only the first leading terms in the lowtemperature asymptotic expressions for the free energy and entropy were obtained and the Casimir pressure was derived only in the dilute approximation. Here we derive the more precise low-temperature behavior for the Casimir free energy, pressure and entropy in the configuration of one plate made of ideal metal and another plate made of dielectric with any finite static dielectric permittivity. For the configuration of metal and dielectric plates the Lifshitz formula takes the form 80 analogical to Eq. (13)\\n\\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l ydy ( 68\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 19\\n\\n× ln 1 -r M (ζ l , y) r D (ζ l , y) e -y + ln 1 -r M ⊥ (ζ l , y) r D ⊥ (ζ l , y) e -y .\\n\\nHere the reflection coefficients r M,D ,⊥ for metal and dielectric, respectively, are given by Eqs. (16) where ε l should be changed for ε M,D l\\n\\n= ε M,D (iξ l ).\\n\\nFor an ideal metal r M ,⊥ (ζ l , y) = 1 and Eq. ( 68 ) takes the more simple form\\n\\nF (a, T ) = cτ 32π 2 a 3 ∞ l=0 1 - δ l0 2 ∞ ζ l ydy ( 69\\n\\n) × ln 1 -r (ζ l , y)e -y + ln 1 -r ⊥ (ζ l , y)e -y\\n\\n(here and below we omit the index D near the reflection coefficient and permittivity of a dielectric plate). We admit that the dielectric permittivity calculated at Matsubara frequencies ε l ≡ ε 0 , i.e., is equal to its static value and find the asymptotic behavior of Eq. (69) at small τ . [In analogy with Sec. 2 it is possible to prove that the deviations of ε(iξ l ) from ε 0 at high frequencies do not influence the lowtemperature behavior of the Casimir free energy, pressure and entropy. It can be shown also that the results of this section are valid not only for ideal metal plate but for plate made of real metal as well.] The free energy (69) can be represented by Eqs. (18) -(21), with the function f (ζ, y) replaced by\\n\\nf (ζ, y) = y ln 1 -r (ζ, y)e -y + y ln 1 -r ⊥ (ζ, y)e -y ≡ f (ζ, y) + f⊥ (ζ, y). ( 70\\n\\n)\\n\\nIn the case of one dielectric and one metal plate both f and f⊥ contribute to F (ix) -F (-ix). The expansion of f (x, y) in powers of x takes the form\\n\\nf (x, y) = y ln(1 -r 0 e -y ) - ε 0 -1 4y e -y - ε 0 ε 0 + 1 ∞ n=1 r n 0 e -ny y x 2 + O(x 3 ). ( 71\\n\\n)\\n\\nNow we integrate Eq. (71) in accordance with Eq. (18) to find the function F (x). The integral of the first term on the right-hand side of Eq. (71) is evaluated using the new variable v = y -x:\\n\\n∞ x ydy ln(1 -r 0 e -y ) = ∞ 0 vdv ln(1 -r 0 e -v ) + O(x 2 ), ( 72\\n\\n)\\n\\nwhere the coefficient near the first-order contribution in x vanishes. As a result, this term could contribute to F (ix) -F (-ix) only starting from the third expansion order. The integrals of the second-order terms on the right-hand side of Eq. (71) are simply calculated using the formulas\\n\\n∞ x dy e -y y = -Ei(-x), ∞ x dy e -ny y = -Ei(-nx). ( 73\\n\\n)\\n\\nFinally, we obtain\\n\\nF (ix) -F (-ix) = iπ (ε 0 -1) 2 4 (ε 0 + 1) x 2 -iγ x 3 + O(x 4 ), ( 74\\n\\n)\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 20 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nwhere the unknown third order expansion coefficient is designated as γ. Substituting Eq. (74) in Eq. (21) and using Eq. ( 19 ), we find the free energy in the system metal-dielectric in the form\\n\\nF (a, T ) = E(a) - c 32π 2 a 3 ζ(3) 16π 2 (ε 0 -1) 2 ε 0 + 1 τ 3 -K 4 τ 4 + O(τ 5 ) , ( 75\\n\\n)\\n\\nwhere K 4 ≡ γ/240. The Casimir pressure in the configuration of metal and dielectric plates obtained from Eq. (75) is equal to\\n\\nP (a, T ) = P 0 (a) - c 32π 2 a 4 K 4 τ 4 + O(τ 5 ) . ( 76\\n\\n)\\n\\nThe direct application of the Lifshitz formula gives the expression for the pressure analogical to Eq. (31),\\n\\nP (a, T ) = - cτ 32π 2 a 4 ∞ l=0 1 - δ l0 2 ∞ ζ l y 2 dy r (ζ l , y) e y -r (ζ l , y) + r ⊥ (ζ l , y) e y -r ⊥ (ζ l , y) . ( 77\\n\\n)\\n\\nUsing the Abel-Plana formula (17), Eq. ( 77 ) can be represented in the form of Eqs. (32), (33) where\\n\\nΦ ,⊥ (x) = ∞ x dy y 2 r ,⊥ (x, y) e y -r ,⊥ (x, y) . ( 78\\n\\n)\\n\\nAgain, we deal first with Φ ⊥ (x). By adding and subtracting the asymptotic behavior of the integrated function at small x,\\n\\ny 2 r ⊥ (x, y) e y -r ⊥ (x, y) = 1 4 (ε 0 -1)x 2 e -y + O(x 3 ) , ( 79\\n\\n)\\n\\nand introducing the new variable v = y/x, the function Φ ⊥ (x) can be identically rearranged and expanded in powers of x as follows:\\n\\nΦ ⊥ (x) = 1 4 (ε 0 -1)x 2 e -x + x 3 ∞ 1 dv v 2 ∞ n=1 r n ⊥ (v)e -nvx - 1 4 (ε 0 -1)e -vx = 1 4 (ε 0 -1)x 2 (1 -x) + x 3 ∞ 1 dv v 2 r ⊥ (v) 1 -r ⊥ (v) - ε 0 -1 4 + O(x 4 ). ( 80\\n\\n)\\n\\nThe integral on the right-hand side of Eq. (80) is converging and can be simply calculated with the result\\n\\nΦ ⊥ (x) = ε 0 -1 4 x 2 - 1 6 (ε 0 √ ε 0 -1) x 3 + O(x 4 ). ( 81\\n\\n)\\n\\nTo deal with Φ (x) we add and subtract in Eq. ( 78 ) the two first expansion terms of the integrated function in powers of x,\\n\\nΦ (x) = ∞ x y 2 dy r 0 e y -r 0 - ε 0 r 0 e -y x 2 y 2 (ε 0 + 1) (1 -r 0 e -y ) 2 ( 82\\n\\n) + ∞ x y 2 dy r (x, y) e y -r (x, y) - r 0 e y -r 0 + ε 0 r 0 e -y x 2 y 2 (ε 0 + 1) (1 -r 0 e -y ) 2 .\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 21\\n\\nThe asymptotic expansion of the first integral on the right-hand side of Eq. (82) is given by\\n\\n2Li 3 (r 0 ) - ε 0 (ε 0 -1) 2(ε 0 + 1) x 2 + 1 12 (ε 0 -1)(3ε 0 -2)x 3 + O(x 4 ), ( 83\\n\\n)\\n\\nand of the second one by\\n\\n- 1 4 ε 0 (ε 0 -1) - 1 6 ε 0 (ε 0 √ ε 0 -1) + 1 2 ε 0 (ε 0 -1) √ ε 0 x 3 + O(x 4 ). ( 84\\n\\n)\\n\\nBy summing Eqs. (83) and (84) we find\\n\\nΦ (x) = 2Li 3 (r 0 ) - ε 0 (ε 0 -1) 2(ε 0 + 1) x 2 ( 85\\n\\n) - 1 6 (ε 0 -1) + (ε 0 √ ε 0 -1) -3ε 0 (ε 0 -1) √ ε 0 x 3 + O(x 4 ).\\n\\nFinally we add Eq. (85) to Eq. ( 81 ) and arrive at Φ(ix\\n\\n) -Φ(-ix) = -i 2 3 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 x 3 + O(x 4 ). ( 86\\n\\n)\\n\\nSubstituting this in Eq. (33) and using Eq. (32), the asymptotic expression for the Casimir pressure is obtained\\n\\nP (a, T ) = P 0 (a) - c 32π 2 a 4 1 360 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 τ 4 + O(τ 5 ) . ( 87\\n\\n)\\n\\nComparing Eqs. (87) and (76), the explicit expression for the coefficient K 4 reads\\n\\nK 4 = 1 360 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 . ( 88\\n\\n)\\n\\nNow we are in a position to find the asymptotic behavior of the entropy in the limit of low temperatures in the configuration of two parallel plates one of which is metallic and the other one dielectric. By calculating the negative derivative of Eq. (75) with respect to temperature, one arrives at\\n\\nS(a, T ) = 3k B ζ(3)(ε 0 -1) 2 128π 3 a 2 (ε 0 + 1) τ 2 ( 89\\n\\n) × 1 - 8π 2 (ε 0 + 1) 1 -2ε 0 √ ε 0 + ε 2 0 √ ε 0 135ζ(3)(ε 0 -1) 2 τ + O(τ 2 ) .\\n\\nThis equation is in analogy to Eq. (49) obtained for the case of two dielectrics. As is seen from Eq. ( 89 ), the entropy of the Casimir and van der Waals interactions between metal and dielectric plates vanishes when the temperature goes to zero, i.e., the Nernst heat theorem is satisfied. Note that the first term of order τ 2 on the right-hand side of Eq. (89) was already obtained in Ref. 61. It is notable also that at low temperatures the entropy goes to zero remaining positive. At the same time, as was shown in Ref. 61, at larger temperatures entropy is nonmonotonous and may take negative values. This interesting property distinguishes the configuration of metal and dielectric plates from two dielectric plates. In the latter configuration the October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 22 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko negative entropy appears only for nonphysical dielectrics with anomalously large and frequency independent dielectric permittivities. 53 Now let us consider the case τ ≫ 1, i.e., high temperatures (large separations). In the same way as for two dielectric plates, here the main contribution to the free energy is given by the term with l = 0 in Eq. (69),\\n\\nF (a, T ) = cτ 64π 2 a 3 ∞ 0 ydy ln 1 -r 0 e -y . ( 90\\n\\n)\\n\\nPerforming the integration in Eq. ( 90 ), we obtain\\n\\nF (a, T ) = - k B T 16πa 2 Li 3 (r 0 ) . ( 91\\n\\n)\\n\\nFrom this equation, for the Casimir pressure and entropy at τ ≫ 1 it follows\\n\\nP (a, T ) = - k B T 8πa 3 Li 3 (r 0 ) , S(a, T ) = k B 16πa 2 Li 3 (r 0 ) . ( 92\\n\\n)\\n\\nThe results (91) and (92) are analogous to (54) and (55) for two dielectric plates.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"THE PROBLEM ORIGINATING FROM THE ACCOUNT OF DIELECTRIC DC CONDUCTIVITY\",\n \"text\": \"In the previous section it was supposed that the static dielectric permittivity ε 0 of the dielectric plate is finite. Now we will deal with the configuration of metal and dielectric plates with included dc conductivity of the dielectric material. In doing so the permittivity of the dielectric plate is given by\\n\\nε(iξ l ) = ε 0 + β(T ) l , ( 93\\n\\n)\\n\\nwhere all notations were introduced in and below Eq. (56). Thus, the reflection coefficients at zero frequency satisfy Eq. ( 4 ) for a plate made of ideal metal and Eq. (57) for a plate made of dielectric with included dc conductivity. Let us find the low-temperature behavior of the Casimir entropy and verify the consistency of the Lifshitz theory with thermodynamics in this nonstandard situation. For this purpose we substitute the dielectric permittivity (93) in Eq. (69) instead of ε 0 and find the Casimir energy F (a, T ) with included dc conductivity of a dielectric plate. In the same way as in Sec. 3, it is convenient to separate the zero-frequency term of F (a, T ) and subtract and add the usual zero-frequency contribution for metal-dielectric plates computed with the dielectric permittivity ε 0 ,\\n\\nF (a, T ) = k B T 16πa 2 ∞ 0 ydy ln 1 -e -y -ln 1 -r 0 e -y + k B T 16πa 2 ∞ 0 ydy ln 1 -r 0 e -y ( 94\\n\\n) + k B T 8πa 2 ∞ l=1 ∞ ζ l ydy ln 1 -r (ζ l , y)e -y + ln 1 -r⊥ (ζ l , y)e -y .\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 23\\n\\nHere the reflection coefficients r ,⊥ are calculated with the permittivity (93). We expand the third integral on the right-hand side of Eq. (94) in powers of the small parameter β/l. The zero-order contribution in this expansion together with the second integral of Eq. ( 94 ) form the Casimir free energy F (a, T ) calculated with dielectric permittivity ε 0 . The first integral on the right-hand side of Eq. (94) is calculated explicitly. As a result, Eq. ( 94 ) is rearranged to\\n\\nF (a, T ) = F (a, T ) - k B T 16πa 2 [ζ(3) -Li 3 (r 0 )] + Q(a, T ), ( 95\\n\\n)\\n\\nwhere Q(a, T ) contains the first and higher-order contributions in the expansion of the third integral on the right-hand side of Eq. (94) in powers of β/l. The explicit form of the main first-order term in Q(a, T ) is the following:\\n\\nQ 1 (a, T ) = k B T 8πa 2 ∞ l=1 β l ∞ ζ l dy y 2 e -y y 2 + ζ 2 l (ε 0 -1) (2 -ε 0 )ζ 2 l -2y 2 y 2 + ζ 2 l (ε 0 -1) + ε 0 y 2 × 1 1 -r (ζ l , y)e -y - ζ 2 l y 2 + ζ 2 l (ε 0 -1) + y 2 1 1 -r ⊥ (ζ l , y)e -y . ( 96\\n\\n)\\n\\nIn the same way as in Sec. 3, we expand the integrated function in Eq. (96) in powers of τ (bearing in mind that ζ l = τ l) and preserve only the main contribution at τ = 0:\\n\\nQ 1 (a, T ) = - k B T 4πa 2 ∞ l=1 β l ∞ ζ l dy yr 0 e -y (ε 2 0 -1) (1 -r 0 e -y ) ( 97\\n\\n) = - k B T β 4πa 2 (ε 2 0 -1) ∞ n=1 r n 0 n 2 ∞ l=1 e -nτ l l + nτ ∞ l=1 e -nτ l .\\n\\nDealing with this expression in the same way as with Eq. ( 64 ), we arrive at\\n\\nQ 1 (a, T ) ∼ e -b/T ln T. ( 98\\n\\n)\\n\\nThe Casimir entropy in the configuration of metal and dielectric plates with included dc conductivity of the dielectric plate is obtained as minus derivative of Eq. (95) with respect to temperature,\\n\\nS(a, T ) = S(a, T ) + k B 16πa 2 [ζ(3) -Li 3 (r 0 )] - ∂Q(a, T ) ∂T . ( 99\\n\\n)\\n\\nUsing Eq. ( 98 ), the calculation of the limiting value at T → 0 is straightforward:\\n\\nS(a, 0) = k B 16πa 2 [ζ(3) -Li 3 (r 0 )] > 0. ( 100\\n\\n)\\n\\nFrom this equation it follows that the inclusion of the dc conductivity of dielectric plate in the configuration metal-dielectric results in a violation of the Nernst heat theorem. In the above this result was obtained for a metallic plate made of ideal metal. It can be shown that it remains valid for a metallic plate made of real metal with finite conductivity.\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 24 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\nThus, both configurations (two dielectric plates or one metal plate and one dielectric) lead to the same conclusion that when the dc conductivity is included in the model of dielectric response of the dielectric plate, the Lifshitz theory loses its consistency with thermodynamics. This confirms the conclusion made in Sec. 3 that the actual properties of dielectric materials at very low frequencies are in fact not related to the van der Waals and Casimir forces.\"\n },\n {\n \"section_type\": \"DISCUSSION\",\n \"section_title\": \"QUALITATIVE DISCUSSION OF CASIMIR INTERACTION BETWEEN METAL AND SEMICONDUCTOR\",\n \"text\": \"As was mentioned in the Introduction, semiconductors possess a wide variety of electric and optical properties ranging from metallic to dielectric. This opens the possibility to modulate the van der Waals and Casimir forces by changing the charge carrier density. Bearing in mind the discussed above problems on the consistency of the Lifshitz theory with thermodynamics, semiconductors can provide us with a test for the validity of different approaches. Thus, if for good dielectric the dc conductivity does not play any real role in the van der Waals and Casimir forces, the question arises on how much it should be increased in order to become a relevant factor in the description of dispersion forces. In Ref. 81 the Casimir force acting between two Si plates was calculated using the simple analytic expression for Si dielectric permittivity as a function of frequency. The complete tabulated optical data of Si were used in Ref. 75 to calculate the van der Waals interaction of different atoms with a Si wall. The first attempt to measure the van der Waals force between a glass lens and a Si plate and to modify it by light due to the change of carrier density was undertaken in Ref. 82. However, glass is a dielectric and therefore the electric forces due to localized point charges could not be controlled. This might explain that no force change occured on illumination at small separations where the effect should be most pronounced.\\n\\nThe first measurements of the Casimir force between a gold coated sphere and a single crystal Si plate were performed in Refs. 62, 63 by means of the atomic force microscope. The experiments used a p-type B doped Si plate of resistivity ρ = 0.0035 Ω cm. The chosen resistivity of the plate is in some sense intermediate between the resistivity of metals (which is usually two or three orders of magnitude lower) and the resistivity of dielectrics (it can be by about a factor of 10 5 larger; for instance, high-resistivity \\\"dielectric\\\" Si has the resistivity ρ 0 = 1000 Ω cm). Thus, the used Si plate had a relatively large absorption typical for semiconductors but it was also enough conductive to avoid the accumulation of charges. 63 In Fig. 2 , taken from Ref. 63, the differences of the theoretical and mean experimental Casimir forces acting between Au sphere and Si plate are presented as functions of separation. In Fig. 2a the theoretical force F theor is computed using the Lifshitz formula and the dielectric permittivity of a Si plate with the relatively low resistivity ρ used in experiment. This dielectric permittivity goes to infinity as ξ -1 with decreasing frequency (see the solid line in Fig. 3 ). In Fig. 2b the theo-October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 25 80 100 120 140 -10 -5 0 5 10 15 (nm) F theor -F expt (pN) 80 100 120 140 -10 -5 0 5 10 15\\n\\n(nm) F theor -F expt (pN) (b)\\n\\n(a) a a Fig. 2. Differences of the theoretical and mean experimental Casimir forces versus separation. Theoretical forces are computed (a) for the Si plate used in experiment and (b) for dielectric Si. Solid and dashed lines indicate 95 and 70% confidence intervals, respectively.\\n\\nretical Casimir force F theor is computed using the dielectric permittivity of the Si plate made of \\\"dielectric\\\" Si with high resistivity ρ 0 . The dielectric permittivity of high-resistivity Si is shown by the dashed line in Fig. 3 . It is characterized by the finite static value ε Si (0) = 11.67. The solid and dashed lines in Fig. 2a ,b indicate the 95% and 70% confidence intervals, respectively. As is seen from Fig. 2 , the theoretical approach using the dielectric permittivity of high-resistivity \\\"dielectric\\\" Si is excluded by experiment within the separation range from 60 to 110 nm at 70% confidence. At the same time, the theory using the dielectric permittivity of Si with a low resistivity ρ is consistent with experiment.\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 26 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko\\n\\n13.5 14 14.5 15 15.5 16 16.5 17 5 10 15 20 25 Log 10 [ξ (rad/s)] ε (iξ) D Fig. 3. Dielectric permittivity of Si plate used in experiment along the imaginary frequency axis (solid line). Dashed line shows the dielectric permittivity of dielectric Si.\\n\\nThe above results suggest an approach on how to correctly determine the possible role of the low-frequency conductivity properties in dispersion forces. As is seen from Fig. 3 , the dielectric permittivity of low-resistivity Si (solid line) significantly departs from the dielectric permittivity of \\\"dielectric\\\" Si in the region around the important dimensional parameter of the problem, the characteristic frequency c/(2a) ∼ 10 14 -10 15 rad/s. That is why the low-resistivity sample cannot be described at low frequencies by the static dielectric permittivity of Si equal to ε Si (0) = 11.67. To describe it, the term β(T )/l, like in Eq. ( 93 ), should be added to ε Si (0). Note that in this case β(T )/l > 1 at the first Matsubara frequencies with l = 1, 2, 3, . . . and, thus, this quantity cannot be considered as a small parameter. At the same time, for a high-resistivity sample the inclusion of the dc conductivity would lead to deviations from the dashed line in Fig. 3 only at frequencies ξ < 10 8 rad/s, which are much less than the characteristic frequency. This comparison permits to make a conclusion in what experimental situations the conductivity properties of semiconductors at low frequencies should be taken into account and when they should be omitted as being not related to dispersion forces.\\n\\nThe future experiments on the modification of semiconductor charge carrier density by laser light 83 will bring a more clear understanding of this problem on the connection between the low-frequency material properties and dispersion forces.\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 27\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"DOES SPATIAL DISPERSION LEAD TO AN IMPORTANT IMPACT ON THERMAL CASIMIR FORCE?\",\n \"text\": \"The presented above new analytic results on the low-temperature behavior of the Casimir free energy, pressure and entropy between dielectrics or between dielectric and metal are based on the conventional Lifshitz theory which describes dielectric materials by means of the frequency dependent dielectric permittivity. In fact, the assumption that the material of the plates possesses only the frequency dispersion means that the components of electric displacement are connected with the components of electric field by the relation\\n\\nD k (r, ω) = ε kl (r, ω)E l (r, ω). ( 101\\n\\n)\\n\\nThis equation is central in all different derivations of the Lifshitz formula (see, e.g., Refs. 2, 6, 36-38, 69, 70, 84). The effects of spatial nonlocality (spatial dispersion) are in fact essential only at shortest separations between the plates comparable with atomic dimensions and also for metals at sufficiently large separations (typically of about 2-3 µm) in the frequency region of the anomalous skin effect. The Casimir force in the latter region was described by the Lifshitz theory reformulated in terms of the Leontovich impedance. 85 Recently the spatial dispersion came to the attention in connection with the thermal Casimir force. 64, 65, 66, 67, 68 In particular, it was claimed 64, 65 that for real metals at any separation the account of spatial dispersion leads to practically the same result (6) for the reflection coefficients at zero frequency as was obtained earlier using the Drude model dielectric function (5). This conclusion, if it is correct, not only returns us to the contradiction with experiment 30,32,33 but also casts doubts on all results obtained by means of the conventional Lifshitz theory accounting for only the frequency dispersion. It is natural when the spatial dispersion contributes a small fraction of a percent as it is generally believed in numerous applications of the Lifshitz theory. It is, however, quite another matter when the account of the spatial dispersion results in some \\\"dramatic effects\\\", 64 i.e., in several hundred times larger thermal correction than is obtained in the local case. Below we demonstrate that the conclusions of Refs. 64 -68 are in fact not reliable because they use the Lifshitz theory of dispersion forces outside of its application range. 86 To find the electromagnetic modes associated with an empty gap between the plates, Refs. 64 -68 use the standard continuity boundary conditions,\\n\\nE 1t = E 2t , B 1n = B 2n , D 1n = D 2n , B 1t = B 2t , ( 102\\n\\n)\\n\\nwhich are commonly applied in the derivation of the Lifshitz formula for spatially local nonmagnetic materials. Here B is the magnetic induction, n is the normal to the boundary directed inside the medium, the subscripts n, t refer to the normal and tangential components, respectively, the subscript 1 refers to the vacuum and subscript 2 to the plate material. In Refs. 64 -68 the spatial dispersion is described by the longitudinal and transverse dielectric permittivities depending on the wave vector and frequency: ε kl = ε kl (q, ω). However, as is shown below, in the theory October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 28 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko of the Casimir effect both the boundary conditions (102) and permittivities ε kl = ε kl (q, ω) are inapplicable. We start from the boundary conditions and recall the set of Maxwell equations in a metal describing the Casimir effect, rotE + 1 c ∂B ∂t = 0, divD = 0, rotB -1 c ∂D ∂t = 0 divB = 0. (103) Equations (103) do not contain any external, i.e., independent on E, D and B, current or charge densities. The definition of the electric displacement is\\n\\n∂D ∂t = ∂E ∂t + 4πi, ( 104\\n\\n)\\n\\nwhere the volume current i is induced by E and B and takes into account the conduction electrons.\\n\\nIn electrodynamics with spatial dispersion the electric field and magnetic induction are finite at the boundary surfaces, whereas the electric displacement can tend to infinity. 87 Then, integrating Eqs. (103) over the thickness of the boundary layer as is done in Ref. 88, we reproduce the first two conditions in Eq. ( 102 ) and arrive at the modified third and fourth conditions, 87,89\\n\\nE 1t = E 2t , B 1n = B 2n , D 2n -D 1n = 4πσ, [n × (B 2 -B 1 )] = 4π c j, ( 105\\n\\n)\\n\\nwhere the induced surface charge and current densities are given by\\n\\nσ = 1 4π 2 1 div[n × [D × n]]dl, j = 1 4π 2 1 ∂D ∂t dl. ( 106\\n\\n)\\n\\nNote that the boundary conditions (105), (106) are obtained from the macroscopic Maxwell equations for physical fields. They should not be mixed with the boundary conditions arising in perturbative theories and for the fictitious fields (see below). In linear electrodynamics for a medium with time-independent properties without spatial dispersion the material equation connecting the electric displacement and electric field takes the form\\n\\nD k (r, t) = t -∞ εkl (r, t -t ′ )E l (r, t ′ )dt ′ . ( 107\\n\\n)\\n\\nAccording to this equation, the electric displacement at a point r and moment t is determined by the electric field at the same point r at different moments t ′ ≤ t (the spatial dispersion is absent but the temporal may be present). It is easily seen that the substitution of Eq. (107) in Eq. (106) leads to σ = 0, j= 0 and, as a result, the boundary conditions (105) coincide with the standard continuity conditions (102). It is unjustified, however, to use conditions (102) in the presence of spatial dispersion. In Refs. 87, 90 a few examples are presented illustrating that in this case neither σ nor j is equal to zero. We now turn to a discussion of the use of dielectric permittivity ε kl (q, ω) in the theory of the Casimir effect with account of spatial dispersion. In the presence of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 29 only frequency dispersion, it is possible to perform the Fourier transformation of the fields E(r, t) = ∞ -∞ E(r, ω)e -iωt dω, D(r, t) = ∞ -∞ D(r, ω)e -iωt dω, (108) in Eq. ( 107 ) and arrive at Eq. (101) where\\n\\nε kl (r, ω) = ∞ 0 εkl (r, τ )e iωτ dτ ( 109\\n\\n)\\n\\nis the frequency-dependent dielectric permittivity and τ ≡ t -t ′ . In fact Eqs. (101), (108) and (109) are used in parallel with the boundary conditions (102) in all derivations of the Lifshitz formula. 2,6,36,37,38,69,70, 84 If the material of the plates is characterized not only by temporal but also spatial dispersion, Eq. (107) is generalized to\\n\\nD k (r, t) = t -∞ dt ′ dr ′ εkl (r, r ′ , t -t ′ )E l (r ′ , t ′ ). ( 110\\n\\n)\\n\\nIf the material medium were uniform in space (i.e., all points were equivalent), the kernel ε would not depend on r and r ′ separately, as in Eq. ( 110 ), but on the difference R ≡ rr ′ . In this case, by performing the Fourier transformation, E(r, t) = ∞ -∞ dω dqE(q, ω)e i (qr-ωt) , D(r, t) = ∞ -∞ dω dqD(q, ω)e i (qr-ωt) , (111) and substituting it in Eq. ( 110 ), one could introduce the dielectric permittivities\\n\\nε kl (q, ω) = ∞ 0 dτ dR εkl (R, τ )e -i(qR-ωτ ) , ( 112\\n\\n)\\n\\nas Refs. 64-68 do, and rearrange Eq. (110) to the form\\n\\nD k (q, ω) = ε kl (q, ω)E l (q, ω). ( 113\\n\\n)\\n\\nIn the Casimir effect, however, the material medium is not uniform due to the presence of a macroscopic gap between the two plates (half spaces). Because of this, the assumption that the kernel ε depends only on R and τ is wrong. As a result, it is not possible to introduce the dielectric permittivity ε kl (q, ω) depending on the wave vector and frequency. In fact, for systems with spatial dispersion in the presence of boundaries the kernel ε depends not only on R and τ but also on the distance from the boundary. 87 In this complicated situation the following approximate phenomenological approach is sometimes applicable. 87 For electromagnetic waves with a wavelength λ the kernel ε(r, r ′ , τ ) in Eq. (110) differs essentially from zero only in a certain vicinity of the point r with characteristic dimensions l ≪ λ (for nonmetallic condensed media l is of the order of the lattice constant). Then it is reasonable to assume that ε is a function of R=r-r ′ , except for a layer of thickness l adjacent to the boundary surface. If one is mostly interested in bulk phenomena October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 30 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko and neglects the role and influence of a subsurface layer, the quantity ε kl (q, ω) may be employed as a reasonable approximation. This approximate phenomenological approach is widely applied in the theoretical investigation of the anomalous skin effect (see, e.g., Ref. 91). Note that in Ref. 91 some kind of fictitious infinite system was introduced and electromagnetic fields in this system are discontinuous on the boundary surface. This discontinuity should not be confused with the discontinuity of physical fields of a real system in the presence of spatial dispersion described by Eqs. (105) and (106) . (There is also another approach to the description of the anomalous skin effect in polycrystals using the generalizations of the local Leontovich impedance which takes into account the shape of Fermi surface 92 ). The frequency and wave vector dependent dielectric permittivity in the presence of boundaries is also approximately applied in the theory of radiative heat transfer 93 or in the study of electromagnetic interaction of molecules with metal surfaces. 94 In all these applications the boundary effects are usually taken into account by the boundary conditions (105) supplemented by so called \\\"additional boundary conditions\\\".\\n\\nIt is unlikely, however, that the approximate phenomenological approach using such quantity as ε kl (q, ω) in the presence of boundaries would be applicable in the theory of the Casimir force where the boundary effects on the zero-point electromagnetic oscillations are of prime importance. It is notable also that this approach faces serious theoretical difficulties including the violation of the law of conservation of energy. 95 It is then not surprising that the application of this approach in Refs. 64, 65 results in the contribution to the Casimir free energy from the transverse electric mode which is in contradiction with experiment. 30,32,33 One more shortcoming of Refs. 64-68 is that they substitute the dielectric permittivity ε kl (q, ω), depending on both wave vector and frequency, into the conventional Lifshitz formula derived in the presence of only temporal dispersion. In the famous review paper 96 it has been noticed, however, that with the inclusion of spatial dispersion the free energy of a fluctuating field takes the form F = F L + ∆F , where F L is given by the conventional Lifshitz expression derived in a spatially local case and written in terms of the Fresnel reflection coefficients, and ∆F is an additional term which can be expressed in terms of the thermal Green's function of the electromagnetic field and polarization operator. Review 96 calls as not reliable the results of, e.g., Ref. 97 obtained by the substitution of dielectric permittivity ε kl (q, ω), taking account of spatial dispersion, into the conventional Lifshitz formula. It can be true that the Lifshitz formula written in terms of general reflection coefficients is applicable in both spatially local and nonlocal situations. However, as far as the exact reflection coefficients in a spatially nonlocal case are unknown, the use of some approximate phenomenological models, elaborated in literature for applications different than the Casimir effect, may lead to incorrect results for ∆F and create inconsistencies with experiment.\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 31\\n\\nTo conclude this section, the results of Refs. 64-68 on the influence of spatial nonlocality on the Casimir interaction are shown to be not reliable. Although at present there is no fundamental theory of the thermal Casimir force incorporating spatial dispersion, there is no reason to expect that it can play any significant role in the frequency region of infrared optics (experimental separations) or normal skin effect (i.e., at separations between plates greater than 4-5 µm).\"\n },\n {\n \"section_type\": \"DISCUSSION\",\n \"section_title\": \"CONCLUSIONS AND DISCUSSION\",\n \"text\": \"In the foregoing we have presented the derivation of analytic asymptotic expressions for the free energy, pressure and entropy of the Casimir interaction between two dielectric plates and between metal and dielectric plates at low and high temperatures. It was shown that the low-temperature behavior of the Casimir interaction between dielectrics and between dielectric and metal is determined by the static dielectric permittivities of nonpolar dielectrics. The obtained results were shown to be in agreement with thermodynamics when the static dielectric permittivities of dielectrics are finite. In particular, the entropy of the Casimir interaction goes to zero when the temperature vanishes, i.e., the Nernst heat theorem is satisfied. This demonstrates the consistency of the original Lifshitz's approach to the van der Waals forces between dielectrics which disregards the small conductivity of dielectrics at constant current.\\n\\nThe second important result shown above is that the inclusion of the dc conductivity of dielectrics into the model of dielectric response leads not to some small corrections to the characteristics of the Casimir interaction, as one could expect, but makes the Lifshitz theory inconsistent with thermodynamics leading to the violation of the Nernst heat theorem. This reveals that real material properties at very low, quasistatic frequencies are in fact not related to the phenomenon of the van der Waals and Casimir forces which is actually determined by sufficiently high frequencies. In this case the zero-frequency contribution to the Casimir force should be understood not literally but as analytic continuation to zero of the material physical behavior in the region around the characteristic frequency.\\n\\nThe presented results provide a basis for the calculation of the van der Waals and Casimir forces between real materials. Such calculations are much needed for numerous applications of the Casimir force discussed in the Introduction, in particular for the applications in nanotechnology and for constraining predictions of fundamental physical theories beyond the Standard Model. Bearing in mind that semiconductors are the main constituent materials in nanotechnological devices, it is a subject of high priority to understand the Casimir effect with semiconductor boundaries. In this connection we have discussed new experimental results and theoretical ideas on the Casimir interaction between a metal sphere and semiconductor plate. It was stressed that by changing the charge carrier density in the semiconductor it is possible to bring it in different intermediate states between metallic and dielectric. In this case the problem arises when the conductivity properties of October 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 32 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko semiconductor are not related to dispersion forces and when they are becoming relevant. A criterion for the resolution of this problem was formulated based on the relation between the typical frequency at which the dc conductivity properties come into play and the characteristic frequency of the Casimir effect. In fact the thermal Casimir interaction between semiconductors remains an open question and much work should be done both in experiment and theory to gain a better insight into this subject.\\n\\nThe last major problem discussed in this review is whether or not the spatial dispersion influences essentially the thermal Casimir force between real materials. In the present literature there is no agreement on this subject. We adduced arguments in favor of the statement that in the region of experimental separations the influence of the spatial dispersion on the Casimir force is negligible small. The statements on the opposite, contained in the literature, were shown to be not reliable because they are obtained by the application of the Lifshitz theory outside of its application range. At the same time it was ascertained that at the moment there is no consistent fundamental theory of the van der Waals and Casimir forces taking the spatial dispersion into account. This is the problem to solve in the foreseeable future. ACKNOWLEDGMENTS G.L.K. and V.M.M. are grateful to the Center of Theoretical Studies and the Institute for Theoretical Physics, Leipzig University for their kind hospitality. This work was supported by Deutsche Forschungsgemeinschaft grant 436 RUS 113/789/0-2. References 1. H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793 (1948). 2. P. W. Milonni, The Quantum Vacuum (Academic Press, San Diego, 1994). 3. V. M. Mostepanenko and N. N. Trunov, The Casimir Effect and its Applications (Clarendon Press, Oxford, 1997). 4. K. A. Milton, The Casimir Effect (World Scientific, Singapore, 2001). 5. M. Kardar and R. Golestanian, Rev. Mod. Phys. 71, 1233 (1999). 6. M. Bordag, U. Mohideen and V. M. Mostepanenko, Phys. Rep. 353, 1 (2001). 7. S. K. Lamoreaux, Rep. Progr. Phys. 68, 201 (2005). 8. P. Candelas and S. Weinberg, Nucl. Phys. B237, 397 (1984). 9. M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D58, 075003 (1998); D60, 055004 (1999); D62, 011701(R) (2000). 10. J. C. Long, H. W. Chan and J. C. Price, Nucl. Phys. B539, 23 (1999). 11. E. Fischbach, D. E. Krause, V. M. Mostepanenko and M. Novello, Phys. Rev. D64, 075010 (2001). 12. V. M. Mostepanenko, Int. J. Mod. Phys. A17, 722 (2002); A17, 4143 (2002). 13. G. L. Klimchitskaya and U. Mohideen, Int. J. Mod. Phys. A17, 4143 (2002). 14. J. F. Babb, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A70, 042901 (2004). 15. M. Antezza, L. P. Pitaevskii and S. Stringari, Phys. Rev. A70, 053619 (2004). 16. M. Krech, The Casimir Effect in Critical Systems (World Scientific, Singapore, 1994).\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 33 17. H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D.J. Bishop and F. Capasso, Science 291, 1941 (2001); Phys. Rev. Lett. 87, 211801 (2001). 18. E. V. Blagov, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B71, 235401 (2005). 19. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions (Springer, Berlin, 1995). 20. S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1997). 21. U. Mohideen and A. Roy, Phys. Rev. Lett. 81, 4549 (1998). 22. G. L. Klimchitskaya, A. Roy, U. Mohideen and V. M. Mostepanenko, Phys. Rev. A60, 3487 (1999). 23. A. Roy and U. Mohideen, Phys. Rev. Lett. 82, 4380 (1999). 24. A. Roy, C.-Y. Lin and U. Mohideen, Phys. Rev. D60, 111101(R) (1999). 25. B. W. Harris, F. Chen and U. Mohideen, Phys. Rev. A62, 052109 (2000). 26. T. Ederth, Phys. Rev. A62, 062104 (2000). 27. G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, Phys. Rev. Lett. 88, 041804 (2002). 28. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. Lett. 88, 101801 (2002). 29. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A66, 032113 (2002). 30. R. S. Decca, E. Fischbach, G. L. Klimchitskaya, D. E. Krause, D. López and V. M. Mostepanenko, Phys. Rev. D68, 116003 (2003). 31. F. Chen, G. L. Klimchitskaya, U. Mohideen and V. M. Mostepanenko, Phys. Rev.\\n\\nA69, 022117 (2004). 32. R. S. Decca, D. López, E. Fischbach, G. L. Klimchitskaya, D. E. Krause and V. M. Mostepanenko, Ann. Phys. (N.Y.) 318, 37 (2005). 33. G. L. Klimchitskaya, R. S. Decca, D. López, E. Fischbach, D. E. Krause and V. M. Mostepanenko, Int. J. Mod. Phys. A20, 2205 (2005). 34. Y. Srivastava, A. Widom and M. H. Friedman, Phys. Rev. Lett. 55, 2246 (1985). 35. Y. Srivastava and A. Widom, Phys. Rep. 148, 1 (1987). 36. E. M. Lifshitz, Sov. Phys. JETP 2, 73 (1956). 37. I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, Sov. Phys. Usp. 4, 153 (1961). 38. E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part II (Pergamon, Oxford, 1980). 39. J. Schwinger, L. L. DeRaad and K. A. Milton, Ann. Phys. (N.Y.) 115, 1 (1978). 40. L. S. Brown and G. J. Maclay, Phys. Rev. 184, 1272 (1969). 41. M. Boström and B. E. Sernelius, Phys. Rev. Lett. 84, 4757 (2000). 42. J. Feinberg, A. Mann and M. Revzen, Ann. Phys. (N.Y.) 288, 103 (2001). 43. A. Scardiccio and R. L. Jaffe, Nucl. Phys. B743, 249 (2006). 44. C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A62, 012110 (2000). 45. M. Bordag, B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. Lett.\\n\\n85, 503 (2000). 46. J. S. Høye, I. Brevik, J. B. Aarseth and K. A. Milton, Phys. Rev. E67, 056116 (2003). 47. I. Brevik, J. B. Aarseth, J. S. Høye and K. A. Milton, Phys. Rev. E71, 056101 (2005). 48. V. B. Bezerra, G. L. Klimchitskaya and C. Romero, Phys. Rev. A65, 012111 (2002). 49. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A67, 062102 (2003). 50. J. R. Torgerson and S. K. Lamoreaux, Phys. Rev. E70, 047102 (2004). 51. S. K. Lamoreaux and W. T. Buttler, Phys. Rev. E71, 036109 (2005). 52. V. B. Bezerra, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A66, 022112 (2002).\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr 34 B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko 53. V. B. Bezerra, G. L. Klimchitskaya, V. M. Mostepanenko and C. Romero, Phys. Rev. A69, 022119 (2004). 54. V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. López, V. M. Mostepanenko and C. Romero, Phys. Rev. E73, 028101 (2006). 55. I. Brevik, S. A. Ellingsen and K. A. Milton, New J. Phys. 8, 236 (2006). 56. V. M. Mostepanenko, V. B. Bezerra, R. S. Decca, E. Fischbach, B. Geyer, G. L. Klimchitskaya, D. E. Krause, D. López and C. Romero, J. Phys. A39, 6589 (2006). 57. J. R. Zurita-Sánchez, J.-J. Greffet and L. Novotny, Phys. Rev. A69, 022902 (2004). 58. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati and J.-J. Greffet, Surf. Sci. Rep.\\n\\n57, 59 (2005). 59. B. S. Stipe, H. J. Mamin, T. D. Stowe, T. W. Kenny and D. Rugar, Phys. Rev. Lett.\\n\\n87, 096801 (2001). 60. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. D72, 085009 (2005). 61. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A72, 022111 (2005). 62. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A72, 020101(R) (2005); A73, 019905(E) (2006). 63. F. Chen, U. Mohideen, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev.\\n\\nA74, 022103 (2006). 64. B. E. Sernelius, Phys. Rev. B71, 235114 (2005). 65. V. B. Svetovoy and R. Esquivel, Phys. Rev. E72, 036113 (2005). 66. R. Esquivel and V. B. Svetovoy, Phys. Rev. A69, 062102 (2004). 67. R. Esquivel, C. Villarreal and W. L. Mochán, Phys. Rev. A68, 052103 (2003). 68. A. M. Contreras-Reyes and W. L. Mochán, Phys. Rev. A72, 034102 (2005). 69. J. Mahanty and B. W. Ninham, Dispersion Forces (Academic, New York, 1976). 70. V. A. Parsegian and B. W. Ninham, Nature 224, 1197 (1969). 71. L. Bergström, Adv. Coll. Interface Sci. 70, 125 (1997). 72. H. Mitter and D. Robaschik, Eur. Phys. J. B13, 335 (2000). 73. B. W. Ninham and J. Daicic, Phys. Rev. A57, 1870 (1998). 74. G. L. Klimchitskaya, B. Geyer and V. M. Mostepanenko, J. Phys. A39, 6495 (2006). 75. A. O. Caride, G. L. Klimchitskaya, V. M. Mostepanenko and S. I. Zanette, Phys. Rev. A71, 042901 (2005). 76. Handbook of Optical Constants of Solids, ed. E. D. Palik (Academic, New York, 1985). 77. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Int. J. Mod. Phys. A16, 3291 (2001). 78. J. C. Slater, Insulators, Semiconductors and Metals. Quantum Theory of Molecules and Solids, Vol. 3, (McGraw-Hill, New York, 1967). 79. Materials Science and Engineering Handbook, eds. J. F. Shackelford and W. Alexander (CRC, Boca Raton, 2001). 80. B. Geyer, G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. A65, 062109 (2002). 81. N. Inui, J. Phys. Soc. Jpn. 72, 2198 Jpn. 72, (2003);); 73, 2198 (2004); 75, 024004 (2006) . 82. W. Arnold, S. Hunklinger and K. Dransfeld, Phys. Rev. B19, 6049 (1979). 83. F. Chen and U. Mohideen, J. Phys. A39, 6233 (2006). 84. C. Genet, A. Lambrecht and S. Reynaud, Phys. Rev. A67, 043811 (2003). 85. E. I. Kats, Sov. Phys. JETP 46, 109 (1977). 86. G. L. Klimchitskaya and V. M. Mostepanenko, Phys. Rev. B75, 036101 (2007). 87. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer, Berlin, 1984).\\n\\nOctober 22, 2018 0:31 WSPC/INSTRUCTION FILE textDr Thermal Casimir force between dielectrics 35 88. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941). 89. V. L. Ginzburg, Physics and Astrophysics (Pergamon Press, Oxford, 1985). 90. V. M. Agranovich and V. L. Ginzburg, Sov. Phys. JETP 36, 440 (1973). 91. K. L. Kliewer and R. Fuchs, Phys. Rev. 172, 607 (1968). 92. I. M. Kaganova and M. I. Kaganov, Phys. Rev. B63, 054202 (2001). 93. S. M. Rytov, Yu. A. Kravtsov and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 3 (Springer, Berlin, 1989). 94. G. W. Ford and W. H. Weber, Phys. Rep. 113, 195 (1984). 95. J. T. Foley and A. J. Devaney, Phys. Rev. B12, 3104 (1975). 96. Yu. S. Barash and V. L. Ginzburg, Sov. Phys. Usp. 18, 305 (1975). 97. G. G. Kleinman and U. Landman, Phys. Rev. Lett. 33, 524 (1974).\"\n }\n]","isConversational":false}}},{"rowIdx":184,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1042"},"arxiv_id":{"kind":"string","value":"0704.1042"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"9f451056a440032725ead173c36ec150fe683509dc3e530d946da326af93a5e0"},"title":{"kind":"string","value":"Entangling and disentangling capacities of nonlocal maps"},"abstract":{"kind":"string","value":"Entangling and disentangling capacities are the key manifestation of the\nnonlocal content of a quantum operation. A lot of effort has been put recently\ninto investigating (dis)entangling capacities of unitary operations, but very\nlittle is known about capacities of non-unitary operations. Here we investigate\n(dis)entangling capacities of unital CPTP maps acting on two qubits."},"authors":{"kind":"list like","value":["Berry Groisman"],"string":"[\n \"Berry Groisman\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-08"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Entanglement content is one of the fundamental ways to characterize nonlocal quantum resources (nonlocal states and operations). For pure bipartite states the ultimate measure of entanglement, the von Neumann entropy of entanglement, had been recently discovered [1] . A universal measure of entanglement for mixed states had not been found yet and different measures are used depending on the operational context. Nevertheless, the important feature of all entanglement measures of states is that their values are directly inferred using the parameters of a state itself.\n\nSimilarly to mixed states, the entanglement content of quantum operations can be characterized in different ways, e.g. via the amount of entanglement necessary to generate that operation or via the amount of entanglement the operation is able to produce/destroy (the so called entangling/disentangling capacities). This article is concerned with the two latter measures.\n\nUnlike the amount of entanglement in a state, the (dis)entangling capacities of an operation do not have an operational interpretation on their own. They manifest themselves via the change of the entanglement of a particular state that the operation acts upon. And the operation has to act on a specific state (the \"optimal\" state) in order to realize its (dis)entangling capacity in full. Thus, the straightforward way to calculate these quantities is to maximize the change of entanglement over all possible initial states. Substantial progress have been made recently in investigating (dis)entangling capacities of unitary operations. The capacities of two-qubit unitary operations were explicitly calculated [2, 3, 4] . It was also shown that singleshot capacities are equal to asymptotic capacities [3, 5, 6] . Some results for higher dimensions were also obtained [7] . However, extending these techniques to systems of higher dimensionality seems to be a very difficult task. Even in the two-qubit case the capacities of a general unitary operation have been calculated numerically, no analytical technique is known.\n\nIn all real situations an experimentalist never deals with a perfect unitary in the laboratory. And it is needless to say that calculating capacities of non-unitary operations, i.e. nonlocal quantum maps, is even a bigger challenge.\n\nIn this article we consider nonlocal completely positive trace preserving (CPTP) maps of the form\n\nτ (ρ) → k p k U k ρU † k , ( 1\n\n)\n\nwhere U k are unitary transformations. Maps of this type are often called random unitary processes, and they are doubly stochastic. The map (1) may arise, for example, if the desired unitary transformation can be implemented successfully only with certain probability, while another unitary is realized in the case of failure. A continuous version of the map (1) may arise naturally in experiment if parameters of a desired unitary transformation are subject to a noise (the case of Gaussian noise will be analyzed in detail in Section IV C). The scope of this article covers the case of τ that act on two qubits. We calculate singleshot (dis)entangling capacities of τ in some particular cases. The structure of the article is as follows. In Sec. II the definition(s) of (dis)entangling capacities of unitaries are presented and some recent results concerning twoqubit unitaries. Sec. III generalizes the definition of (dis)entangling capacities for non-unitaries. Some numerical results for (lower bounds on) (dis)entangling capacities for discreet and continuous mixtures of unitaries are presented in Sec. IV.\n\nConsider an unitary operation U AB that acts on a tensor product Hilbert space H A ⊗ H B of two spatially separated particles A and B. If U AB can not be decomposed into a tensor product of local unitaries, i.e. U AB = V A ⊗ W B , then we say that U AB is nonlocal. Unlike local unitaries, nonlocal unitaries have an ability to produce or destroy entanglement. This ability is usually characterized by the entangling, E ↑ (U ), and the disentangling, E ↓ (U ), capacities, i.e. by the maximal increase (decrease) of entanglement that can be achieved when U acts on quantum states. To quantify these capacities we have to choose appropriate measures of entanglement. The most sensible choice is to use the entanglement of formation [11] as a measure of entanglement of the initial state ρ, and the distillable entanglement [12] as a measure 2 of entanglement of the final state U ρU † . The reason for this asymmetric choice is purely operational one. What counts is the amount of resources (pure maximally entangled states) needed to create ρ (asymptotically) and the amount of pure-state entanglement one will be able to extract from U ρU † , again asymptotically. Thus the most general definition is\n\nE ↑ (U ) = max ρ [D(U ρU † ) -E F (ρ)], E ↓ (U ) = max ρ [E F (ρ) -D(U ρU † )], ( 2\n\n)\n\nwhere the maximization is over all possible states ρ (mixed and pure) accessible to U . The Hilbert space of an accessible ρ is not necessarily restricted to H A ⊗ H B . It turns out to be the case that some U create more entanglement if the original particles are entangled with local ancillary particles [2, 3]. It also appears to be the case that the maximization in Eq. ( 2 ) can be restricted to pure-states only [3], therefore, the definition (2) can be simplified as\n\nE ↑ (U ) = max ψ [E(U |ψ ) -E(|ψ )] E ↓ (U ) = max ψ [E(|ψ ) -E(U |ψ )], ( 3\n\n)\n\nwhere E is an entanglement measure for pure state (Throughout this paper we will use the von Neumann entropy of entanglement as the most appropriate measure). This obviously simplifies the job significantly.\n\nLet us briefly recall the main results for A and B being two-level particles, qubits.\n\nAny U AB acting on qubits can be decomposed as [2, 10]\n\nU AB = [V A ⊗ V B ] e i α=x,y,z ξασ A α σ B α [W A ⊗ W B ] , ( 4\n\n) where π/4 ≥ ξ x ≥ ξ y ≥ |ξ z | ≥ 0.\n\nThe middle term sandwiched by local unitaries is called the canonical decomposition of U . Any U can be transformed to its canonical form by sandwiching it with Hermitian conjugates of corresponding local unitaries. That means that the canonical form is genuinely nonlocal part of U -everything else is local. The beauty of this results is that out of 15 real parameters that parameterize a general two-qubit unitary only three are necessary to describe its nonlocal nature. It simplifies considerably the classification of nonlocal unitaries. For the purpose of our discussion three classes can be identified; namely, the Controlled-NOT(CNOT)-class (\n\nξ x = 0, ξ y = ξ z = 0), the DoubleCNOT-class (ξ x = 0, ξ y = 0, ξ z = 0)\n\n, and the SWAP-class (all three ξ α are not equal zero) [15, 16] . The names reflect the fact that the corresponding \"mother\" unitary transformation (i.e. with ξ α = π/4 for α = 0) belongs to that class. The main results for qubits are [2, 3]:\n\n(a) E ↑ (U ) = E ↓ (U ).\n\n(b) For CNOT-class the optimal state, i.e. the state that satisfies definition (3), lives solely in the Hilbert space of particles A and B (no ancillas are needed) and takes the form\n\n|ψ opt = cos α|0 A |0 B ± i sin α|1 A |1 B , ( 5\n\n)\n\nwhere ± correspond to E ↑ and E ↓ respectively. Thus all U from that class achieve their capacity by acting on pure states with the same Schmidt basis (only values of Schmidt coefficients differ depending on the value of ξ x ). The values α = f (ξ x ) can be obtained by straightforward numerical optimization. (c) If ξ α < π/4 the maximal capacity is achieved when |ψ opt is already entangled.\n\n(d) Unitaries of the CNOT-class achieve their capacities by acting on optimal states that lie in H A ⊗H B . However, unitaries of the DCNOT and SWAP-classes achieve their capacities only if the original particles are entangled with local ancillas. It was conjectured that it is sufficient to take the size of ancillas equal to the size of original particles. This conjecture was supported by numerical simulations for qubits [3].\n\nFor non-unitaries we will use a definition similar to Eq. ( 2 ), i.e. we define\n\nE ↑ (τ ) = max ρ [D(τ (ρ)) -E F (ρ)] E ↓ (τ ) = max ρ [E F (ρ) -D(τ (ρ))]. ( 6\n\n)\n\nHowever, in general here we cannot justify reducing the search to pure states. This is due to the fact that distillable entanglement is not necessary a convex measure.\n\nWe can argue, nevertheless, that in the case of mixtures of unitaries acting on two qubits without ancillas the distillable entanglement can be regarded as a convex measure. Indeed, a mixture of optimal states (5) forms a Bell-diagonal state for which the lower and upper bounds on distillable entanglement [17]\n\nS(ρ A ) -S(ρ AB ) ≤ D(ρ) ≤ E RE (ρ AB ) ( 7\n\n)\n\ncoincide. Here we recall that the relative entropy of entanglement E RE (x) is a convex measure. If ancillas are used then the situation is more complicated. We leave the question of whether the capacities are attained on pure states as an open question and calculate the lower bounds on these capacities using pure states.\n\nThe properties of two-qubit unitaries described above in Section II can help us to generalize that approach to 3 -0.75 -0.5 -0.25 0.25 0.5 0.75 0.2 0.4 0.6 0.8 1 E FIG. 1: (color online) E ↑ (τ ) (solid line) and E ↓ (τ ) (dashed line) as functions of ∆ for different values of ξ, where p = 0.5. The highest curve corresponds to ξ = π/4. The lowest curve corresponds to ξ = 0. Here ∆ is measured in radians.\n\nmixtures of unitaries as in Eq. ( 1 ) [18] . Here we use two methods for calculating E ↑ (τ ) and E ↓ (τ ). Method I: We make an assumption about the particular form of the optimal input state, and subsequently find the optimal values of its parameters.\n\nMethod II: We perform a direct numerical optimization without making any a priori assumption about the optimal state (except of its purity).\n\nConsider a mixture of unitaries of the CNOT-class,\n\nU k = e iξ k x σ A x σ B x . ( 8\n\n)\n\nHere we use Method I. From continuity it follows that the optimal state is expected to lie on the 2-dimensional manifold of (superpositions of) states of the form (5) or their convex mixtures. Moreover, in this special case we can adopt the argument of [3] (see Sec. II) and claim that the search can be restricted to pure states only. Thus, the state optimal for τ is again of the type (5). As a simplest case let us consider only two unitaries U 1 and U 2 :\n\nτ (ρ) → pU 1 ρU † 1 + (1 -p)U 2 ρU † 2 ( 9\n\n)\n\nFor convenience let us define ∆ = ξ 2 x -ξ 1 x , and denote ξ 1 x simply by ξ, so ξ 2 x = ξ + ∆. We will fix ξ and analyze E ↑ and E ↓ for various ∆. For ∆ = 0 the map reduces to a unitary (with an appropriate capacity). As smaller angle means smaller E ↑ (U ), we would expect that if U 1 is mixed with U 2 , where ∆ < 0, then the entangling capacity of the resulting map will decrease relative to E ↑ (U ). This intuition is consistent with the results presented on Fig. 1. Similarly, we might expect that the entangling capacity of the map will increase with ∆ > 0, and that -0.4 -0.2 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 E FIG. 2: (color online) DCNOT: E ↑ (τ ) (empty diamonds with solid line fit) and E ↓ (τ ) (empty triangles with dashed line fit) as functions of ∆ for ξ 1 x = ξ 1 y = π/8. SWAP: E ↑ (τ ) (filled diamonds with solid line fit) and E ↓ (τ ) (filled triangles with dashed line fit) as functions of ∆ for ξ 1 x = ξ 1 y = ξ 1 z = π/8. In both cases p = 0.5. Here ∆ is measured in radians.\n\nthis capacity will reach its maximum for maximal ∆, i.e. maximal E ↑ (U 2 ). However, Fig. 1 shows that this is not the case. E ↑ (τ ) indeed grows while ∆ is positive and relatively small, reaching its maximum for certain intermediate positive value of ∆ and then starting to decrease. In other words, if U 1 and p are fixed, then maximal E ↑ (τ ) is achieved for some intermediate U 2 with ξ 2 x > ξ 1 x , but not for ξ 2 x = π/4. This result might seem counterintuitive from the first sight, but it has a clear explanation. Let α 1 , α 2 be the corresponding optimal values of α in Eq. ( 5 ) for U 1 and U 2 respectively, then the optimal value of α τ will lie somewhere in between, i.e. satisfy α 1 > α τ > α 2 . For ξ 2 x = π/4, U 2 can realize its entangling capacity of 1 ebit if α 2 = 0. However, when U 2 acts on a state with α τ > α 2 , then it creates less than 1 -H[(cos α τ ) 2 ] ebit.\n\nDisentangling capacity, E ↓ (τ ), behaves differently. It is monotonic with ∆. It equals E ↑ for ∆ = 0 as expected, and it is strictly larger than E ↑ for all other values of ∆. The last observation shows behavior completely opposite to that of unitaries. In a sense, it is easier for non-local map to destroy entanglement rather than create it, while for unitary operations the opposite holds [7] .\n\nThis approach can be similarly applied to any finite number of unitaries and to continuous distribution of unitaries, which will be discussed in Section IV C.\n\nFor mixtures of unitaries of DCNOT and SWAP-class Method II was used. The details of numerical calculations are presented in Appendix. We conjectured that it is sufficient if local ancillas are qubits. Selected results are shown in Fig. 2 . We can see that for DCNOTmixture the behavior of E ↑ (τ ) and E ↓ (τ ) is qualitatively 4 similar to CNOT-mixture (Fig. 1 ). However, for SWAPmixture slightly different behavior is obtained. In particular E ↓ (τ ) exhibits maximum at an intermediate value of ∆. It is also noticeable that for D > 0.357, E ↑ (τ ) is smaller for SWAP-mixture than for DCNOT-mixture. It is a counterintuitive result that SWAP-mixture which is naturally considered as \"stronger\" that DCNOT-mixture has lower entangling capacity. However, again similar to the line of thought in Example I we can argue that for (relatively) large ∆ the second unitary, U 2 is too strong, and therefore when it acts on the optimal state (optimal for the mixture, not for itself) it causes more destruction that corresponding U 2 of DCNOT-class would have caused.\n\nSo far we analyzed discrete mixtures of unitaries. In this section we analyze a continuous distribution, which is usually what experimentalists deal with. These distributions arise due to uncertainty in one or more parameters. Such uncertainties may be caused by the limits of calibration precision of the devices and by high sensitivity of systems used to generate desired interactions. For example, the strength of exchange coupling between donors in silicon based solid-state architectures for quantum computing exhibit significant uncertainty resulting in error in gate operation [19] .\n\nIn particular, we consider the case when a non-unitary map arises if a unitary from CNOT-class is subject to a Gaussian noise.\n\nRecent work [20] analyzed the capability of noisy Hamiltonians to create entanglement. In particular, interactions of the form Eq. ( 8 ), where ξ is Gaussian distributed with the mean ξ = π/4 and standard deviation Ω, were considered. Without noise this operation (CNOT operation) is able to create a maximally entangled state if it acts on a disentangled pure state. The authors analyzed the situation when the noisy operation acts on initially disentangled state, which is by itself subject to a Gaussian noise. Its capability to create entanglement was characterized in terms of the condition for inseparability of the resulting mixed state (via Peres-Horodecci separability criterion).\n\nThe aim of our analysis is different. We consider noisy interactions with ξ ∈ [0, π/4] and calculate their entangling and disentangling capacities in terms of ξ and Ω. Thus, we give a comprehensive quantitative characterization of the non-local content of these noisy maps in terms of their entangling and disentangling capacities. Unlike [20] we do not test the resulting state on inseparability, rather calculate its distillable entanglement explicitly.\n\nThe action of a unitary U = exp[iξσ A x σ B x ], where ξ is Gaussian distributed with the mean ξ and the standard deviation Ω, on the state ρ AB can be seen as a non-0.2 0.4 0.6 0.8 Ξ 0.2 0.4 0.6 0.8 1 FIG. 3: (color online) E ↑ (τG) (solid line) and E ↓ (τG) (dashed line) as a function of ξ for several values of Ω: 0.01, 0.18, 0.35, 0.52, 0.69, and 0.86. The dotted line corresponds to Ω = 0, i.e. a unitary transformation. ξ and Ω are measured in radians.\n\nunitary CPTP map\n\nτ G (ρ) → 1 √ 2πΩ +∞ -∞ e -(ξ-ξ) 2 2Ω 2 U ρU † dξ, ( 10\n\n)\n\nwhich is a continuous mixture of unitaries of the CNOTclass. Similarly to the Sec. IV A we consider pure initial state, i.e. ρ = |ψ ψ|, where ψ takes the form (5), calculate the distillable entanglement of the output mixed state, and maximize it over α. Figure 3 presents numerical results for E ↑ (τ ) and E ↓ (τ ) as functions of ξ for several values of Ω. We can see that already for Ω ≈ 0.01 rad we obtain only very small deviation from the (dis)entangling capacity of the unitary, i.e. τ G without noise -Ω = 0. As Ω increases the disentangling capacity increases and the entangling capacity decreases. The former fact should not be surprizing as it is known that entanglement can be destroyed even by local CPTP unital maps [21] . Thus, the more dispersed the distribution of ξ becomes, the easier for τ G to destroy entanglement and the harder to create it. Nevertheless, we see that even when Ω is relatively large τ G is still able to create considerable amount of entanglement.\n\nWe have discussed the entangling and disentangled capacities of nonlocal CPTP unital maps, i.e. maps that can be represented as probabilistic mixtures of unitaries, and have calculated these capacities in some particular cases for two qubits. Three classes of unitaries were considered, namely the CNOT, DCNOT, and SWAP classes.\n\nWe have observed that the disentangling capacity was always larger than corresponding entangling capacity, which contrasts with the unitary case where the both capacities are equal for qubits and for higher dimensions disentangling capacity cannot be greater than entangling capacity [7].\n\n5 In the case of the CNOT-class our results were obtained via straightforward generalization of the method for CNOT-class unitaries. We argue that the (dis)entangling capacity is achieved when a map acts on the optimal pure state from the same family as in the unitary case. Both discrete and continuous mixtures were analyzed. In the case of the DCNOT and SWAP-class direct numerical optimization was performed. We have conjectured that dimensions of the local ancillas are equal to the dimensions of the original particles, i.e. the ancillas were taken to be qubits.\n\nA number of open question can be addressed in a future research.\n\nIt will be interesting and useful to prove (or disprove) the general conjecture that the sizes of local ancillas can be taken equal to the sizes of original particles.\n\nHere we have calculated single-shot capacities. In the case of unitaries it had been shown that in the asymptotic regime one cannot do better [5, 6] . It is important to check whether this result holds in the non-unitary case.\n\nIn the case of DCNOT and SWAP-mixtures we performed maximization over pure states only thereby obtaining lover bounds on E ↑ (τ ) and E ↓ (τ ), but not their actual values.\n\nIn our future research we will address the question of whether these bounds are tight. It might be the case that optimal states for these operations are mixed and, consequently, the capacities are higher than we have calculated.\n\nThis work was funded by the U.K. Engineering and Physical Sciences Research Council, Grant No. EP/C528042/1, and supported by the European Union through the Integrated Project QAP (IST-3-015848) and SECOQC.\n\nWe have used two-dimensional ancilla on each side. Consider a general state of four qubits in the tensor-product of the computational bases of the original particles A, B and the ancillary particles A ′ , B ′\n\n|Ψ AA ′ BB ′ = i,j,k,l c i,j,k,l |i A |j A ′ |k B |l B ′ . ( A1\n\n)\n\nThere are 16 terms in the above superposition with 16 complex amplitudes c i,j,k,l , therefore |Ψ can be parameterized using 30 real numbers (if we take into account the global phase and normalization). We will parameterize it in the following way [22] . First, to facilitate our analysis it is easier to incorporate four indexes i, j, k and l, each of which runs from 0 to 1, into a single index, x, that runs from 1 to 16. This can be done by using the formula x = 8i + 4j + 2k + l + 1, which is essentially a formula for converting a number from the Boolean representation to the decimal. Second, we present amplitudes c x in the form\n\nc x = |c x |e iθx , ( A2\n\n) where 16 x=1 |c x | 2 = 1\n\nand θ 1 = 0. Third, we introduce new parameters φ x such that\n\nc x = sin φ x-1 15 y=x cos φ y , ( A3\n\n)\n\nwhere φ 0 = π/2. Thus the state |Ψ is parameterized by 30 angles. The advantage of this parametrization is that we restrict their values only to the interval [0, 2π] that simplifies numerics. We proceed as follows. A program generates a vector of 30 random numbers in the interval [0, 2π] . This is the initial state. We then apply the non-local map and obtain a final state. We calculate the value of the gain in entanglement ∆S = S(τ\n\n(Ψ) BB ′ ) -S(τ (Ψ) AA ′ BB ′ ) - S(T r AA ′ |Ψ Ψ|).\n\nAfter that we vary the values of the random vector by a small amount and repeat these calculations again, thereby obtaining a gradient of the change in entanglement in that point. We move along the gradient to obtain the next |Ψ , and the procedure is repeated. Eventually, the program reaches the maximum where it stops.\n\n[1] C.H.Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A , 53 2046 (1996). [2] W. Dür, G. Vidal, J. I. Cirac, N. Linden, and S. Popescu, Phys. Rev. Lett. , 87, 137901 (2001). [3] M.S. Leifer, L. Henderson, and N. Linden, Phys. Rev. A 67, 012306 (2003). [4] D.W.Berry and B.C.Sanders, Phys. Rev. A 71, 022304 (2005); P.Zanardi, C. Zalka, and Lara Faoro, Phys. Rev. A 62, 030301(R) (2000); L. Clarisse, S. Ghosh, S. Severini, A. Sudbery, e-print arXiv:quant-ph/0611075v2. [5] C.H. Bennett, A. Harrow, D.W. Leung, and J.A. Smolin, IEEE Tran. Inf. Theory, 49, 8, 1895 Theory, 49, 8, (2003)) . [6] A.M. Childs, D.W. Leung, F. Verstraete, and G. Vidal, Quant. Inf. Comp. 3, 97 (2003). [7] N. Linden, J.A. Smolin, and A. Winter, e-print quant-ph/0511217. [8] M.A. Nielsen and I.L.Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2004). [9] It was explicitly shown [3] that the optimization can be restricted to pure states. [10] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001).\n\n[11] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [12] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54, 3824 (1996). [13] J. I. Cirac, W. Dür, B. Kraus and M. Lewenstein, Phys.\n\nRev. Lett. , 86, 544 (2001); W. Dür and J. I. Cirac, Phys. Rev. A , 64, 012317 (2001). [14] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001). [15] Strictly speaking DCNOT-class and SWAP-class should be unified under a single class if analyzed according to the criteria of interconvertability under LOCC [16] . In the framework of (dis)entangling capacity it is useful to identify them as separate classes, because their behavior differs qualitatively. [16] W. Dür, G. Vidal, J. I. Cirac, Phys. Rev. Lett. , 89, 057901 (2002).\n\n[17] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998). [18] Here we assume that all unitaries in Eq. (1) belong to the same class, which is typical for real situations where only parameters of the coupling are subject to variations. [19] M. J. Testolin, C. D. Hill, C. J. Wellard, L. C. L. Hollenberg, e-print quant-ph/0701165. [20] S. Bandyopadhyay and D.A. Lidar, Phys. Rev. A 70, 010301(R) (2004). [21] B.Groisman, S. Popescu, and A. Winter, Phys. Rev. A 72, 032317 (2005). [22] This method is a partial adaptation of the method used in Ref. [17] for calculation the relative entropy of entanglement."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"Entanglement content is one of the fundamental ways to characterize nonlocal quantum resources (nonlocal states and operations). For pure bipartite states the ultimate measure of entanglement, the von Neumann entropy of entanglement, had been recently discovered [1] . A universal measure of entanglement for mixed states had not been found yet and different measures are used depending on the operational context. Nevertheless, the important feature of all entanglement measures of states is that their values are directly inferred using the parameters of a state itself.\n\nSimilarly to mixed states, the entanglement content of quantum operations can be characterized in different ways, e.g. via the amount of entanglement necessary to generate that operation or via the amount of entanglement the operation is able to produce/destroy (the so called entangling/disentangling capacities). This article is concerned with the two latter measures.\n\nUnlike the amount of entanglement in a state, the (dis)entangling capacities of an operation do not have an operational interpretation on their own. They manifest themselves via the change of the entanglement of a particular state that the operation acts upon. And the operation has to act on a specific state (the \"optimal\" state) in order to realize its (dis)entangling capacity in full. Thus, the straightforward way to calculate these quantities is to maximize the change of entanglement over all possible initial states. Substantial progress have been made recently in investigating (dis)entangling capacities of unitary operations. The capacities of two-qubit unitary operations were explicitly calculated [2, 3, 4] . It was also shown that singleshot capacities are equal to asymptotic capacities [3, 5, 6] . Some results for higher dimensions were also obtained [7] . However, extending these techniques to systems of higher dimensionality seems to be a very difficult task. Even in the two-qubit case the capacities of a general unitary operation have been calculated numerically, no analytical technique is known.\n\nIn all real situations an experimentalist never deals with a perfect unitary in the laboratory. And it is needless to say that calculating capacities of non-unitary operations, i.e. nonlocal quantum maps, is even a bigger challenge.\n\nIn this article we consider nonlocal completely positive trace preserving (CPTP) maps of the form\n\nτ (ρ) → k p k U k ρU † k , ( 1\n\n)\n\nwhere U k are unitary transformations. Maps of this type are often called random unitary processes, and they are doubly stochastic. The map (1) may arise, for example, if the desired unitary transformation can be implemented successfully only with certain probability, while another unitary is realized in the case of failure. A continuous version of the map (1) may arise naturally in experiment if parameters of a desired unitary transformation are subject to a noise (the case of Gaussian noise will be analyzed in detail in Section IV C). The scope of this article covers the case of τ that act on two qubits. We calculate singleshot (dis)entangling capacities of τ in some particular cases. The structure of the article is as follows. In Sec. II the definition(s) of (dis)entangling capacities of unitaries are presented and some recent results concerning twoqubit unitaries. Sec. III generalizes the definition of (dis)entangling capacities for non-unitaries. Some numerical results for (lower bounds on) (dis)entangling capacities for discreet and continuous mixtures of unitaries are presented in Sec. IV."},{"section_type":"RESULTS","section_title":"II. (DIS)ENTANGLING CAPACITY OF A UNITARY: DEFINITIONS AND SOME RELATED RESULTS","text":"Consider an unitary operation U AB that acts on a tensor product Hilbert space H A ⊗ H B of two spatially separated particles A and B. If U AB can not be decomposed into a tensor product of local unitaries, i.e. U AB = V A ⊗ W B , then we say that U AB is nonlocal. Unlike local unitaries, nonlocal unitaries have an ability to produce or destroy entanglement. This ability is usually characterized by the entangling, E ↑ (U ), and the disentangling, E ↓ (U ), capacities, i.e. by the maximal increase (decrease) of entanglement that can be achieved when U acts on quantum states. To quantify these capacities we have to choose appropriate measures of entanglement. The most sensible choice is to use the entanglement of formation [11] as a measure of entanglement of the initial state ρ, and the distillable entanglement [12] as a measure 2 of entanglement of the final state U ρU † . The reason for this asymmetric choice is purely operational one. What counts is the amount of resources (pure maximally entangled states) needed to create ρ (asymptotically) and the amount of pure-state entanglement one will be able to extract from U ρU † , again asymptotically. Thus the most general definition is\n\nE ↑ (U ) = max ρ [D(U ρU † ) -E F (ρ)], E ↓ (U ) = max ρ [E F (ρ) -D(U ρU † )], ( 2\n\n)\n\nwhere the maximization is over all possible states ρ (mixed and pure) accessible to U . The Hilbert space of an accessible ρ is not necessarily restricted to H A ⊗ H B . It turns out to be the case that some U create more entanglement if the original particles are entangled with local ancillary particles [2, 3]. It also appears to be the case that the maximization in Eq. ( 2 ) can be restricted to pure-states only [3], therefore, the definition (2) can be simplified as\n\nE ↑ (U ) = max ψ [E(U |ψ ) -E(|ψ )] E ↓ (U ) = max ψ [E(|ψ ) -E(U |ψ )], ( 3\n\n)\n\nwhere E is an entanglement measure for pure state (Throughout this paper we will use the von Neumann entropy of entanglement as the most appropriate measure). This obviously simplifies the job significantly.\n\nLet us briefly recall the main results for A and B being two-level particles, qubits.\n\nAny U AB acting on qubits can be decomposed as [2, 10]\n\nU AB = [V A ⊗ V B ] e i α=x,y,z ξασ A α σ B α [W A ⊗ W B ] , ( 4\n\n) where π/4 ≥ ξ x ≥ ξ y ≥ |ξ z | ≥ 0.\n\nThe middle term sandwiched by local unitaries is called the canonical decomposition of U . Any U can be transformed to its canonical form by sandwiching it with Hermitian conjugates of corresponding local unitaries. That means that the canonical form is genuinely nonlocal part of U -everything else is local. The beauty of this results is that out of 15 real parameters that parameterize a general two-qubit unitary only three are necessary to describe its nonlocal nature. It simplifies considerably the classification of nonlocal unitaries. For the purpose of our discussion three classes can be identified; namely, the Controlled-NOT(CNOT)-class (\n\nξ x = 0, ξ y = ξ z = 0), the DoubleCNOT-class (ξ x = 0, ξ y = 0, ξ z = 0)\n\n, and the SWAP-class (all three ξ α are not equal zero) [15, 16] . The names reflect the fact that the corresponding \"mother\" unitary transformation (i.e. with ξ α = π/4 for α = 0) belongs to that class. The main results for qubits are [2, 3]:\n\n(a) E ↑ (U ) = E ↓ (U ).\n\n(b) For CNOT-class the optimal state, i.e. the state that satisfies definition (3), lives solely in the Hilbert space of particles A and B (no ancillas are needed) and takes the form\n\n|ψ opt = cos α|0 A |0 B ± i sin α|1 A |1 B , ( 5\n\n)\n\nwhere ± correspond to E ↑ and E ↓ respectively. Thus all U from that class achieve their capacity by acting on pure states with the same Schmidt basis (only values of Schmidt coefficients differ depending on the value of ξ x ). The values α = f (ξ x ) can be obtained by straightforward numerical optimization. (c) If ξ α < π/4 the maximal capacity is achieved when |ψ opt is already entangled.\n\n(d) Unitaries of the CNOT-class achieve their capacities by acting on optimal states that lie in H A ⊗H B . However, unitaries of the DCNOT and SWAP-classes achieve their capacities only if the original particles are entangled with local ancillas. It was conjectured that it is sufficient to take the size of ancillas equal to the size of original particles. This conjecture was supported by numerical simulations for qubits [3]."},{"section_type":"OTHER","section_title":"III. ENTANGLING AND DISENTANGLING CAPACITIES OF A NON-UNITARY","text":"For non-unitaries we will use a definition similar to Eq. ( 2 ), i.e. we define\n\nE ↑ (τ ) = max ρ [D(τ (ρ)) -E F (ρ)] E ↓ (τ ) = max ρ [E F (ρ) -D(τ (ρ))]. ( 6\n\n)\n\nHowever, in general here we cannot justify reducing the search to pure states. This is due to the fact that distillable entanglement is not necessary a convex measure.\n\nWe can argue, nevertheless, that in the case of mixtures of unitaries acting on two qubits without ancillas the distillable entanglement can be regarded as a convex measure. Indeed, a mixture of optimal states (5) forms a Bell-diagonal state for which the lower and upper bounds on distillable entanglement [17]\n\nS(ρ A ) -S(ρ AB ) ≤ D(ρ) ≤ E RE (ρ AB ) ( 7\n\n)\n\ncoincide. Here we recall that the relative entropy of entanglement E RE (x) is a convex measure. If ancillas are used then the situation is more complicated. We leave the question of whether the capacities are attained on pure states as an open question and calculate the lower bounds on these capacities using pure states."},{"section_type":"OTHER","section_title":"IV. MIXTURES OF UNITARIES ACTING ON TWO QUBITS","text":"The properties of two-qubit unitaries described above in Section II can help us to generalize that approach to 3 -0.75 -0.5 -0.25 0.25 0.5 0.75 0.2 0.4 0.6 0.8 1 E FIG. 1: (color online) E ↑ (τ ) (solid line) and E ↓ (τ ) (dashed line) as functions of ∆ for different values of ξ, where p = 0.5. The highest curve corresponds to ξ = π/4. The lowest curve corresponds to ξ = 0. Here ∆ is measured in radians.\n\nmixtures of unitaries as in Eq. ( 1 ) [18] . Here we use two methods for calculating E ↑ (τ ) and E ↓ (τ ). Method I: We make an assumption about the particular form of the optimal input state, and subsequently find the optimal values of its parameters.\n\nMethod II: We perform a direct numerical optimization without making any a priori assumption about the optimal state (except of its purity)."},{"section_type":"OTHER","section_title":"A. Example I: Discreet CNOT-mixtures","text":"Consider a mixture of unitaries of the CNOT-class,\n\nU k = e iξ k x σ A x σ B x . ( 8\n\n)\n\nHere we use Method I. From continuity it follows that the optimal state is expected to lie on the 2-dimensional manifold of (superpositions of) states of the form (5) or their convex mixtures. Moreover, in this special case we can adopt the argument of [3] (see Sec. II) and claim that the search can be restricted to pure states only. Thus, the state optimal for τ is again of the type (5). As a simplest case let us consider only two unitaries U 1 and U 2 :\n\nτ (ρ) → pU 1 ρU † 1 + (1 -p)U 2 ρU † 2 ( 9\n\n)\n\nFor convenience let us define ∆ = ξ 2 x -ξ 1 x , and denote ξ 1 x simply by ξ, so ξ 2 x = ξ + ∆. We will fix ξ and analyze E ↑ and E ↓ for various ∆. For ∆ = 0 the map reduces to a unitary (with an appropriate capacity). As smaller angle means smaller E ↑ (U ), we would expect that if U 1 is mixed with U 2 , where ∆ < 0, then the entangling capacity of the resulting map will decrease relative to E ↑ (U ). This intuition is consistent with the results presented on Fig. 1. Similarly, we might expect that the entangling capacity of the map will increase with ∆ > 0, and that -0.4 -0.2 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 E FIG. 2: (color online) DCNOT: E ↑ (τ ) (empty diamonds with solid line fit) and E ↓ (τ ) (empty triangles with dashed line fit) as functions of ∆ for ξ 1 x = ξ 1 y = π/8. SWAP: E ↑ (τ ) (filled diamonds with solid line fit) and E ↓ (τ ) (filled triangles with dashed line fit) as functions of ∆ for ξ 1 x = ξ 1 y = ξ 1 z = π/8. In both cases p = 0.5. Here ∆ is measured in radians.\n\nthis capacity will reach its maximum for maximal ∆, i.e. maximal E ↑ (U 2 ). However, Fig. 1 shows that this is not the case. E ↑ (τ ) indeed grows while ∆ is positive and relatively small, reaching its maximum for certain intermediate positive value of ∆ and then starting to decrease. In other words, if U 1 and p are fixed, then maximal E ↑ (τ ) is achieved for some intermediate U 2 with ξ 2 x > ξ 1 x , but not for ξ 2 x = π/4. This result might seem counterintuitive from the first sight, but it has a clear explanation. Let α 1 , α 2 be the corresponding optimal values of α in Eq. ( 5 ) for U 1 and U 2 respectively, then the optimal value of α τ will lie somewhere in between, i.e. satisfy α 1 > α τ > α 2 . For ξ 2 x = π/4, U 2 can realize its entangling capacity of 1 ebit if α 2 = 0. However, when U 2 acts on a state with α τ > α 2 , then it creates less than 1 -H[(cos α τ ) 2 ] ebit.\n\nDisentangling capacity, E ↓ (τ ), behaves differently. It is monotonic with ∆. It equals E ↑ for ∆ = 0 as expected, and it is strictly larger than E ↑ for all other values of ∆. The last observation shows behavior completely opposite to that of unitaries. In a sense, it is easier for non-local map to destroy entanglement rather than create it, while for unitary operations the opposite holds [7] .\n\nThis approach can be similarly applied to any finite number of unitaries and to continuous distribution of unitaries, which will be discussed in Section IV C."},{"section_type":"OTHER","section_title":"B. Example II: Discreet DCNOT and SWAP-mixtures","text":"For mixtures of unitaries of DCNOT and SWAP-class Method II was used. The details of numerical calculations are presented in Appendix. We conjectured that it is sufficient if local ancillas are qubits. Selected results are shown in Fig. 2 . We can see that for DCNOTmixture the behavior of E ↑ (τ ) and E ↓ (τ ) is qualitatively 4 similar to CNOT-mixture (Fig. 1 ). However, for SWAPmixture slightly different behavior is obtained. In particular E ↓ (τ ) exhibits maximum at an intermediate value of ∆. It is also noticeable that for D > 0.357, E ↑ (τ ) is smaller for SWAP-mixture than for DCNOT-mixture. It is a counterintuitive result that SWAP-mixture which is naturally considered as \"stronger\" that DCNOT-mixture has lower entangling capacity. However, again similar to the line of thought in Example I we can argue that for (relatively) large ∆ the second unitary, U 2 is too strong, and therefore when it acts on the optimal state (optimal for the mixture, not for itself) it causes more destruction that corresponding U 2 of DCNOT-class would have caused."},{"section_type":"OTHER","section_title":"C. Example III: Entangling capacity of noisy unitary with Gaussian fluctuations","text":"So far we analyzed discrete mixtures of unitaries. In this section we analyze a continuous distribution, which is usually what experimentalists deal with. These distributions arise due to uncertainty in one or more parameters. Such uncertainties may be caused by the limits of calibration precision of the devices and by high sensitivity of systems used to generate desired interactions. For example, the strength of exchange coupling between donors in silicon based solid-state architectures for quantum computing exhibit significant uncertainty resulting in error in gate operation [19] .\n\nIn particular, we consider the case when a non-unitary map arises if a unitary from CNOT-class is subject to a Gaussian noise.\n\nRecent work [20] analyzed the capability of noisy Hamiltonians to create entanglement. In particular, interactions of the form Eq. ( 8 ), where ξ is Gaussian distributed with the mean ξ = π/4 and standard deviation Ω, were considered. Without noise this operation (CNOT operation) is able to create a maximally entangled state if it acts on a disentangled pure state. The authors analyzed the situation when the noisy operation acts on initially disentangled state, which is by itself subject to a Gaussian noise. Its capability to create entanglement was characterized in terms of the condition for inseparability of the resulting mixed state (via Peres-Horodecci separability criterion).\n\nThe aim of our analysis is different. We consider noisy interactions with ξ ∈ [0, π/4] and calculate their entangling and disentangling capacities in terms of ξ and Ω. Thus, we give a comprehensive quantitative characterization of the non-local content of these noisy maps in terms of their entangling and disentangling capacities. Unlike [20] we do not test the resulting state on inseparability, rather calculate its distillable entanglement explicitly.\n\nThe action of a unitary U = exp[iξσ A x σ B x ], where ξ is Gaussian distributed with the mean ξ and the standard deviation Ω, on the state ρ AB can be seen as a non-0.2 0.4 0.6 0.8 Ξ 0.2 0.4 0.6 0.8 1 FIG. 3: (color online) E ↑ (τG) (solid line) and E ↓ (τG) (dashed line) as a function of ξ for several values of Ω: 0.01, 0.18, 0.35, 0.52, 0.69, and 0.86. The dotted line corresponds to Ω = 0, i.e. a unitary transformation. ξ and Ω are measured in radians.\n\nunitary CPTP map\n\nτ G (ρ) → 1 √ 2πΩ +∞ -∞ e -(ξ-ξ) 2 2Ω 2 U ρU † dξ, ( 10\n\n)\n\nwhich is a continuous mixture of unitaries of the CNOTclass. Similarly to the Sec. IV A we consider pure initial state, i.e. ρ = |ψ ψ|, where ψ takes the form (5), calculate the distillable entanglement of the output mixed state, and maximize it over α. Figure 3 presents numerical results for E ↑ (τ ) and E ↓ (τ ) as functions of ξ for several values of Ω. We can see that already for Ω ≈ 0.01 rad we obtain only very small deviation from the (dis)entangling capacity of the unitary, i.e. τ G without noise -Ω = 0. As Ω increases the disentangling capacity increases and the entangling capacity decreases. The former fact should not be surprizing as it is known that entanglement can be destroyed even by local CPTP unital maps [21] . Thus, the more dispersed the distribution of ξ becomes, the easier for τ G to destroy entanglement and the harder to create it. Nevertheless, we see that even when Ω is relatively large τ G is still able to create considerable amount of entanglement."},{"section_type":"DISCUSSION","section_title":"V. DISCUSSION AND CONCLUSION","text":"We have discussed the entangling and disentangled capacities of nonlocal CPTP unital maps, i.e. maps that can be represented as probabilistic mixtures of unitaries, and have calculated these capacities in some particular cases for two qubits. Three classes of unitaries were considered, namely the CNOT, DCNOT, and SWAP classes.\n\nWe have observed that the disentangling capacity was always larger than corresponding entangling capacity, which contrasts with the unitary case where the both capacities are equal for qubits and for higher dimensions disentangling capacity cannot be greater than entangling capacity [7].\n\n5 In the case of the CNOT-class our results were obtained via straightforward generalization of the method for CNOT-class unitaries. We argue that the (dis)entangling capacity is achieved when a map acts on the optimal pure state from the same family as in the unitary case. Both discrete and continuous mixtures were analyzed. In the case of the DCNOT and SWAP-class direct numerical optimization was performed. We have conjectured that dimensions of the local ancillas are equal to the dimensions of the original particles, i.e. the ancillas were taken to be qubits.\n\nA number of open question can be addressed in a future research.\n\nIt will be interesting and useful to prove (or disprove) the general conjecture that the sizes of local ancillas can be taken equal to the sizes of original particles.\n\nHere we have calculated single-shot capacities. In the case of unitaries it had been shown that in the asymptotic regime one cannot do better [5, 6] . It is important to check whether this result holds in the non-unitary case.\n\nIn the case of DCNOT and SWAP-mixtures we performed maximization over pure states only thereby obtaining lover bounds on E ↑ (τ ) and E ↓ (τ ), but not their actual values.\n\nIn our future research we will address the question of whether these bounds are tight. It might be the case that optimal states for these operations are mixed and, consequently, the capacities are higher than we have calculated."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"This work was funded by the U.K. Engineering and Physical Sciences Research Council, Grant No. EP/C528042/1, and supported by the European Union through the Integrated Project QAP (IST-3-015848) and SECOQC."},{"section_type":"OTHER","section_title":"APPENDIX A","text":"We have used two-dimensional ancilla on each side. Consider a general state of four qubits in the tensor-product of the computational bases of the original particles A, B and the ancillary particles A ′ , B ′\n\n|Ψ AA ′ BB ′ = i,j,k,l c i,j,k,l |i A |j A ′ |k B |l B ′ . ( A1\n\n)\n\nThere are 16 terms in the above superposition with 16 complex amplitudes c i,j,k,l , therefore |Ψ can be parameterized using 30 real numbers (if we take into account the global phase and normalization). We will parameterize it in the following way [22] . First, to facilitate our analysis it is easier to incorporate four indexes i, j, k and l, each of which runs from 0 to 1, into a single index, x, that runs from 1 to 16. This can be done by using the formula x = 8i + 4j + 2k + l + 1, which is essentially a formula for converting a number from the Boolean representation to the decimal. Second, we present amplitudes c x in the form\n\nc x = |c x |e iθx , ( A2\n\n) where 16 x=1 |c x | 2 = 1\n\nand θ 1 = 0. Third, we introduce new parameters φ x such that\n\nc x = sin φ x-1 15 y=x cos φ y , ( A3\n\n)\n\nwhere φ 0 = π/2. Thus the state |Ψ is parameterized by 30 angles. The advantage of this parametrization is that we restrict their values only to the interval [0, 2π] that simplifies numerics. We proceed as follows. A program generates a vector of 30 random numbers in the interval [0, 2π] . This is the initial state. We then apply the non-local map and obtain a final state. We calculate the value of the gain in entanglement ∆S = S(τ\n\n(Ψ) BB ′ ) -S(τ (Ψ) AA ′ BB ′ ) - S(T r AA ′ |Ψ Ψ|).\n\nAfter that we vary the values of the random vector by a small amount and repeat these calculations again, thereby obtaining a gradient of the change in entanglement in that point. We move along the gradient to obtain the next |Ψ , and the procedure is repeated. Eventually, the program reaches the maximum where it stops.\n\n[1] C.H.Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A , 53 2046 (1996). [2] W. Dür, G. Vidal, J. I. Cirac, N. Linden, and S. Popescu, Phys. Rev. Lett. , 87, 137901 (2001). [3] M.S. Leifer, L. Henderson, and N. Linden, Phys. Rev. A 67, 012306 (2003). [4] D.W.Berry and B.C.Sanders, Phys. Rev. A 71, 022304 (2005); P.Zanardi, C. Zalka, and Lara Faoro, Phys. Rev. A 62, 030301(R) (2000); L. Clarisse, S. Ghosh, S. Severini, A. Sudbery, e-print arXiv:quant-ph/0611075v2. [5] C.H. Bennett, A. Harrow, D.W. Leung, and J.A. Smolin, IEEE Tran. Inf. Theory, 49, 8, 1895 Theory, 49, 8, (2003)) . [6] A.M. Childs, D.W. Leung, F. Verstraete, and G. Vidal, Quant. Inf. Comp. 3, 97 (2003). [7] N. Linden, J.A. Smolin, and A. Winter, e-print quant-ph/0511217. [8] M.A. Nielsen and I.L.Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2004). [9] It was explicitly shown [3] that the optimization can be restricted to pure states. [10] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001).\n\n[11] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [12] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54, 3824 (1996). [13] J. I. Cirac, W. Dür, B. Kraus and M. Lewenstein, Phys.\n\nRev. Lett. , 86, 544 (2001); W. Dür and J. I. Cirac, Phys. Rev. A , 64, 012317 (2001). [14] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001). [15] Strictly speaking DCNOT-class and SWAP-class should be unified under a single class if analyzed according to the criteria of interconvertability under LOCC [16] . In the framework of (dis)entangling capacity it is useful to identify them as separate classes, because their behavior differs qualitatively. [16] W. Dür, G. Vidal, J. I. Cirac, Phys. Rev. Lett. , 89, 057901 (2002).\n\n[17] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998). [18] Here we assume that all unitaries in Eq. (1) belong to the same class, which is typical for real situations where only parameters of the coupling are subject to variations. [19] M. J. Testolin, C. D. Hill, C. J. Wellard, L. C. L. Hollenberg, e-print quant-ph/0701165. [20] S. Bandyopadhyay and D.A. Lidar, Phys. Rev. A 70, 010301(R) (2004). [21] B.Groisman, S. Popescu, and A. Winter, Phys. Rev. A 72, 032317 (2005). [22] This method is a partial adaptation of the method used in Ref. [17] for calculation the relative entropy of entanglement."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"Entanglement content is one of the fundamental ways to characterize nonlocal quantum resources (nonlocal states and operations). For pure bipartite states the ultimate measure of entanglement, the von Neumann entropy of entanglement, had been recently discovered [1] . A universal measure of entanglement for mixed states had not been found yet and different measures are used depending on the operational context. Nevertheless, the important feature of all entanglement measures of states is that their values are directly inferred using the parameters of a state itself.\\n\\nSimilarly to mixed states, the entanglement content of quantum operations can be characterized in different ways, e.g. via the amount of entanglement necessary to generate that operation or via the amount of entanglement the operation is able to produce/destroy (the so called entangling/disentangling capacities). This article is concerned with the two latter measures.\\n\\nUnlike the amount of entanglement in a state, the (dis)entangling capacities of an operation do not have an operational interpretation on their own. They manifest themselves via the change of the entanglement of a particular state that the operation acts upon. And the operation has to act on a specific state (the \\\"optimal\\\" state) in order to realize its (dis)entangling capacity in full. Thus, the straightforward way to calculate these quantities is to maximize the change of entanglement over all possible initial states. Substantial progress have been made recently in investigating (dis)entangling capacities of unitary operations. The capacities of two-qubit unitary operations were explicitly calculated [2, 3, 4] . It was also shown that singleshot capacities are equal to asymptotic capacities [3, 5, 6] . Some results for higher dimensions were also obtained [7] . However, extending these techniques to systems of higher dimensionality seems to be a very difficult task. Even in the two-qubit case the capacities of a general unitary operation have been calculated numerically, no analytical technique is known.\\n\\nIn all real situations an experimentalist never deals with a perfect unitary in the laboratory. And it is needless to say that calculating capacities of non-unitary operations, i.e. nonlocal quantum maps, is even a bigger challenge.\\n\\nIn this article we consider nonlocal completely positive trace preserving (CPTP) maps of the form\\n\\nτ (ρ) → k p k U k ρU † k , ( 1\\n\\n)\\n\\nwhere U k are unitary transformations. Maps of this type are often called random unitary processes, and they are doubly stochastic. The map (1) may arise, for example, if the desired unitary transformation can be implemented successfully only with certain probability, while another unitary is realized in the case of failure. A continuous version of the map (1) may arise naturally in experiment if parameters of a desired unitary transformation are subject to a noise (the case of Gaussian noise will be analyzed in detail in Section IV C). The scope of this article covers the case of τ that act on two qubits. We calculate singleshot (dis)entangling capacities of τ in some particular cases. The structure of the article is as follows. In Sec. II the definition(s) of (dis)entangling capacities of unitaries are presented and some recent results concerning twoqubit unitaries. Sec. III generalizes the definition of (dis)entangling capacities for non-unitaries. Some numerical results for (lower bounds on) (dis)entangling capacities for discreet and continuous mixtures of unitaries are presented in Sec. IV.\"\n },\n {\n \"section_type\": \"RESULTS\",\n \"section_title\": \"II. (DIS)ENTANGLING CAPACITY OF A UNITARY: DEFINITIONS AND SOME RELATED RESULTS\",\n \"text\": \"Consider an unitary operation U AB that acts on a tensor product Hilbert space H A ⊗ H B of two spatially separated particles A and B. If U AB can not be decomposed into a tensor product of local unitaries, i.e. U AB = V A ⊗ W B , then we say that U AB is nonlocal. Unlike local unitaries, nonlocal unitaries have an ability to produce or destroy entanglement. This ability is usually characterized by the entangling, E ↑ (U ), and the disentangling, E ↓ (U ), capacities, i.e. by the maximal increase (decrease) of entanglement that can be achieved when U acts on quantum states. To quantify these capacities we have to choose appropriate measures of entanglement. The most sensible choice is to use the entanglement of formation [11] as a measure of entanglement of the initial state ρ, and the distillable entanglement [12] as a measure 2 of entanglement of the final state U ρU † . The reason for this asymmetric choice is purely operational one. What counts is the amount of resources (pure maximally entangled states) needed to create ρ (asymptotically) and the amount of pure-state entanglement one will be able to extract from U ρU † , again asymptotically. Thus the most general definition is\\n\\nE ↑ (U ) = max ρ [D(U ρU † ) -E F (ρ)], E ↓ (U ) = max ρ [E F (ρ) -D(U ρU † )], ( 2\\n\\n)\\n\\nwhere the maximization is over all possible states ρ (mixed and pure) accessible to U . The Hilbert space of an accessible ρ is not necessarily restricted to H A ⊗ H B . It turns out to be the case that some U create more entanglement if the original particles are entangled with local ancillary particles [2, 3]. It also appears to be the case that the maximization in Eq. ( 2 ) can be restricted to pure-states only [3], therefore, the definition (2) can be simplified as\\n\\nE ↑ (U ) = max ψ [E(U |ψ ) -E(|ψ )] E ↓ (U ) = max ψ [E(|ψ ) -E(U |ψ )], ( 3\\n\\n)\\n\\nwhere E is an entanglement measure for pure state (Throughout this paper we will use the von Neumann entropy of entanglement as the most appropriate measure). This obviously simplifies the job significantly.\\n\\nLet us briefly recall the main results for A and B being two-level particles, qubits.\\n\\nAny U AB acting on qubits can be decomposed as [2, 10]\\n\\nU AB = [V A ⊗ V B ] e i α=x,y,z ξασ A α σ B α [W A ⊗ W B ] , ( 4\\n\\n) where π/4 ≥ ξ x ≥ ξ y ≥ |ξ z | ≥ 0.\\n\\nThe middle term sandwiched by local unitaries is called the canonical decomposition of U . Any U can be transformed to its canonical form by sandwiching it with Hermitian conjugates of corresponding local unitaries. That means that the canonical form is genuinely nonlocal part of U -everything else is local. The beauty of this results is that out of 15 real parameters that parameterize a general two-qubit unitary only three are necessary to describe its nonlocal nature. It simplifies considerably the classification of nonlocal unitaries. For the purpose of our discussion three classes can be identified; namely, the Controlled-NOT(CNOT)-class (\\n\\nξ x = 0, ξ y = ξ z = 0), the DoubleCNOT-class (ξ x = 0, ξ y = 0, ξ z = 0)\\n\\n, and the SWAP-class (all three ξ α are not equal zero) [15, 16] . The names reflect the fact that the corresponding \\\"mother\\\" unitary transformation (i.e. with ξ α = π/4 for α = 0) belongs to that class. The main results for qubits are [2, 3]:\\n\\n(a) E ↑ (U ) = E ↓ (U ).\\n\\n(b) For CNOT-class the optimal state, i.e. the state that satisfies definition (3), lives solely in the Hilbert space of particles A and B (no ancillas are needed) and takes the form\\n\\n|ψ opt = cos α|0 A |0 B ± i sin α|1 A |1 B , ( 5\\n\\n)\\n\\nwhere ± correspond to E ↑ and E ↓ respectively. Thus all U from that class achieve their capacity by acting on pure states with the same Schmidt basis (only values of Schmidt coefficients differ depending on the value of ξ x ). The values α = f (ξ x ) can be obtained by straightforward numerical optimization. (c) If ξ α < π/4 the maximal capacity is achieved when |ψ opt is already entangled.\\n\\n(d) Unitaries of the CNOT-class achieve their capacities by acting on optimal states that lie in H A ⊗H B . However, unitaries of the DCNOT and SWAP-classes achieve their capacities only if the original particles are entangled with local ancillas. It was conjectured that it is sufficient to take the size of ancillas equal to the size of original particles. This conjecture was supported by numerical simulations for qubits [3].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. ENTANGLING AND DISENTANGLING CAPACITIES OF A NON-UNITARY\",\n \"text\": \"For non-unitaries we will use a definition similar to Eq. ( 2 ), i.e. we define\\n\\nE ↑ (τ ) = max ρ [D(τ (ρ)) -E F (ρ)] E ↓ (τ ) = max ρ [E F (ρ) -D(τ (ρ))]. ( 6\\n\\n)\\n\\nHowever, in general here we cannot justify reducing the search to pure states. This is due to the fact that distillable entanglement is not necessary a convex measure.\\n\\nWe can argue, nevertheless, that in the case of mixtures of unitaries acting on two qubits without ancillas the distillable entanglement can be regarded as a convex measure. Indeed, a mixture of optimal states (5) forms a Bell-diagonal state for which the lower and upper bounds on distillable entanglement [17]\\n\\nS(ρ A ) -S(ρ AB ) ≤ D(ρ) ≤ E RE (ρ AB ) ( 7\\n\\n)\\n\\ncoincide. Here we recall that the relative entropy of entanglement E RE (x) is a convex measure. If ancillas are used then the situation is more complicated. We leave the question of whether the capacities are attained on pure states as an open question and calculate the lower bounds on these capacities using pure states.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. MIXTURES OF UNITARIES ACTING ON TWO QUBITS\",\n \"text\": \"The properties of two-qubit unitaries described above in Section II can help us to generalize that approach to 3 -0.75 -0.5 -0.25 0.25 0.5 0.75 0.2 0.4 0.6 0.8 1 E FIG. 1: (color online) E ↑ (τ ) (solid line) and E ↓ (τ ) (dashed line) as functions of ∆ for different values of ξ, where p = 0.5. The highest curve corresponds to ξ = π/4. The lowest curve corresponds to ξ = 0. Here ∆ is measured in radians.\\n\\nmixtures of unitaries as in Eq. ( 1 ) [18] . Here we use two methods for calculating E ↑ (τ ) and E ↓ (τ ). Method I: We make an assumption about the particular form of the optimal input state, and subsequently find the optimal values of its parameters.\\n\\nMethod II: We perform a direct numerical optimization without making any a priori assumption about the optimal state (except of its purity).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Example I: Discreet CNOT-mixtures\",\n \"text\": \"Consider a mixture of unitaries of the CNOT-class,\\n\\nU k = e iξ k x σ A x σ B x . ( 8\\n\\n)\\n\\nHere we use Method I. From continuity it follows that the optimal state is expected to lie on the 2-dimensional manifold of (superpositions of) states of the form (5) or their convex mixtures. Moreover, in this special case we can adopt the argument of [3] (see Sec. II) and claim that the search can be restricted to pure states only. Thus, the state optimal for τ is again of the type (5). As a simplest case let us consider only two unitaries U 1 and U 2 :\\n\\nτ (ρ) → pU 1 ρU † 1 + (1 -p)U 2 ρU † 2 ( 9\\n\\n)\\n\\nFor convenience let us define ∆ = ξ 2 x -ξ 1 x , and denote ξ 1 x simply by ξ, so ξ 2 x = ξ + ∆. We will fix ξ and analyze E ↑ and E ↓ for various ∆. For ∆ = 0 the map reduces to a unitary (with an appropriate capacity). As smaller angle means smaller E ↑ (U ), we would expect that if U 1 is mixed with U 2 , where ∆ < 0, then the entangling capacity of the resulting map will decrease relative to E ↑ (U ). This intuition is consistent with the results presented on Fig. 1. Similarly, we might expect that the entangling capacity of the map will increase with ∆ > 0, and that -0.4 -0.2 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 E FIG. 2: (color online) DCNOT: E ↑ (τ ) (empty diamonds with solid line fit) and E ↓ (τ ) (empty triangles with dashed line fit) as functions of ∆ for ξ 1 x = ξ 1 y = π/8. SWAP: E ↑ (τ ) (filled diamonds with solid line fit) and E ↓ (τ ) (filled triangles with dashed line fit) as functions of ∆ for ξ 1 x = ξ 1 y = ξ 1 z = π/8. In both cases p = 0.5. Here ∆ is measured in radians.\\n\\nthis capacity will reach its maximum for maximal ∆, i.e. maximal E ↑ (U 2 ). However, Fig. 1 shows that this is not the case. E ↑ (τ ) indeed grows while ∆ is positive and relatively small, reaching its maximum for certain intermediate positive value of ∆ and then starting to decrease. In other words, if U 1 and p are fixed, then maximal E ↑ (τ ) is achieved for some intermediate U 2 with ξ 2 x > ξ 1 x , but not for ξ 2 x = π/4. This result might seem counterintuitive from the first sight, but it has a clear explanation. Let α 1 , α 2 be the corresponding optimal values of α in Eq. ( 5 ) for U 1 and U 2 respectively, then the optimal value of α τ will lie somewhere in between, i.e. satisfy α 1 > α τ > α 2 . For ξ 2 x = π/4, U 2 can realize its entangling capacity of 1 ebit if α 2 = 0. However, when U 2 acts on a state with α τ > α 2 , then it creates less than 1 -H[(cos α τ ) 2 ] ebit.\\n\\nDisentangling capacity, E ↓ (τ ), behaves differently. It is monotonic with ∆. It equals E ↑ for ∆ = 0 as expected, and it is strictly larger than E ↑ for all other values of ∆. The last observation shows behavior completely opposite to that of unitaries. In a sense, it is easier for non-local map to destroy entanglement rather than create it, while for unitary operations the opposite holds [7] .\\n\\nThis approach can be similarly applied to any finite number of unitaries and to continuous distribution of unitaries, which will be discussed in Section IV C.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Example II: Discreet DCNOT and SWAP-mixtures\",\n \"text\": \"For mixtures of unitaries of DCNOT and SWAP-class Method II was used. The details of numerical calculations are presented in Appendix. We conjectured that it is sufficient if local ancillas are qubits. Selected results are shown in Fig. 2 . We can see that for DCNOTmixture the behavior of E ↑ (τ ) and E ↓ (τ ) is qualitatively 4 similar to CNOT-mixture (Fig. 1 ). However, for SWAPmixture slightly different behavior is obtained. In particular E ↓ (τ ) exhibits maximum at an intermediate value of ∆. It is also noticeable that for D > 0.357, E ↑ (τ ) is smaller for SWAP-mixture than for DCNOT-mixture. It is a counterintuitive result that SWAP-mixture which is naturally considered as \\\"stronger\\\" that DCNOT-mixture has lower entangling capacity. However, again similar to the line of thought in Example I we can argue that for (relatively) large ∆ the second unitary, U 2 is too strong, and therefore when it acts on the optimal state (optimal for the mixture, not for itself) it causes more destruction that corresponding U 2 of DCNOT-class would have caused.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Example III: Entangling capacity of noisy unitary with Gaussian fluctuations\",\n \"text\": \"So far we analyzed discrete mixtures of unitaries. In this section we analyze a continuous distribution, which is usually what experimentalists deal with. These distributions arise due to uncertainty in one or more parameters. Such uncertainties may be caused by the limits of calibration precision of the devices and by high sensitivity of systems used to generate desired interactions. For example, the strength of exchange coupling between donors in silicon based solid-state architectures for quantum computing exhibit significant uncertainty resulting in error in gate operation [19] .\\n\\nIn particular, we consider the case when a non-unitary map arises if a unitary from CNOT-class is subject to a Gaussian noise.\\n\\nRecent work [20] analyzed the capability of noisy Hamiltonians to create entanglement. In particular, interactions of the form Eq. ( 8 ), where ξ is Gaussian distributed with the mean ξ = π/4 and standard deviation Ω, were considered. Without noise this operation (CNOT operation) is able to create a maximally entangled state if it acts on a disentangled pure state. The authors analyzed the situation when the noisy operation acts on initially disentangled state, which is by itself subject to a Gaussian noise. Its capability to create entanglement was characterized in terms of the condition for inseparability of the resulting mixed state (via Peres-Horodecci separability criterion).\\n\\nThe aim of our analysis is different. We consider noisy interactions with ξ ∈ [0, π/4] and calculate their entangling and disentangling capacities in terms of ξ and Ω. Thus, we give a comprehensive quantitative characterization of the non-local content of these noisy maps in terms of their entangling and disentangling capacities. Unlike [20] we do not test the resulting state on inseparability, rather calculate its distillable entanglement explicitly.\\n\\nThe action of a unitary U = exp[iξσ A x σ B x ], where ξ is Gaussian distributed with the mean ξ and the standard deviation Ω, on the state ρ AB can be seen as a non-0.2 0.4 0.6 0.8 Ξ 0.2 0.4 0.6 0.8 1 FIG. 3: (color online) E ↑ (τG) (solid line) and E ↓ (τG) (dashed line) as a function of ξ for several values of Ω: 0.01, 0.18, 0.35, 0.52, 0.69, and 0.86. The dotted line corresponds to Ω = 0, i.e. a unitary transformation. ξ and Ω are measured in radians.\\n\\nunitary CPTP map\\n\\nτ G (ρ) → 1 √ 2πΩ +∞ -∞ e -(ξ-ξ) 2 2Ω 2 U ρU † dξ, ( 10\\n\\n)\\n\\nwhich is a continuous mixture of unitaries of the CNOTclass. Similarly to the Sec. IV A we consider pure initial state, i.e. ρ = |ψ ψ|, where ψ takes the form (5), calculate the distillable entanglement of the output mixed state, and maximize it over α. Figure 3 presents numerical results for E ↑ (τ ) and E ↓ (τ ) as functions of ξ for several values of Ω. We can see that already for Ω ≈ 0.01 rad we obtain only very small deviation from the (dis)entangling capacity of the unitary, i.e. τ G without noise -Ω = 0. As Ω increases the disentangling capacity increases and the entangling capacity decreases. The former fact should not be surprizing as it is known that entanglement can be destroyed even by local CPTP unital maps [21] . Thus, the more dispersed the distribution of ξ becomes, the easier for τ G to destroy entanglement and the harder to create it. Nevertheless, we see that even when Ω is relatively large τ G is still able to create considerable amount of entanglement.\"\n },\n {\n \"section_type\": \"DISCUSSION\",\n \"section_title\": \"V. DISCUSSION AND CONCLUSION\",\n \"text\": \"We have discussed the entangling and disentangled capacities of nonlocal CPTP unital maps, i.e. maps that can be represented as probabilistic mixtures of unitaries, and have calculated these capacities in some particular cases for two qubits. Three classes of unitaries were considered, namely the CNOT, DCNOT, and SWAP classes.\\n\\nWe have observed that the disentangling capacity was always larger than corresponding entangling capacity, which contrasts with the unitary case where the both capacities are equal for qubits and for higher dimensions disentangling capacity cannot be greater than entangling capacity [7].\\n\\n5 In the case of the CNOT-class our results were obtained via straightforward generalization of the method for CNOT-class unitaries. We argue that the (dis)entangling capacity is achieved when a map acts on the optimal pure state from the same family as in the unitary case. Both discrete and continuous mixtures were analyzed. In the case of the DCNOT and SWAP-class direct numerical optimization was performed. We have conjectured that dimensions of the local ancillas are equal to the dimensions of the original particles, i.e. the ancillas were taken to be qubits.\\n\\nA number of open question can be addressed in a future research.\\n\\nIt will be interesting and useful to prove (or disprove) the general conjecture that the sizes of local ancillas can be taken equal to the sizes of original particles.\\n\\nHere we have calculated single-shot capacities. In the case of unitaries it had been shown that in the asymptotic regime one cannot do better [5, 6] . It is important to check whether this result holds in the non-unitary case.\\n\\nIn the case of DCNOT and SWAP-mixtures we performed maximization over pure states only thereby obtaining lover bounds on E ↑ (τ ) and E ↓ (τ ), but not their actual values.\\n\\nIn our future research we will address the question of whether these bounds are tight. It might be the case that optimal states for these operations are mixed and, consequently, the capacities are higher than we have calculated.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"This work was funded by the U.K. Engineering and Physical Sciences Research Council, Grant No. EP/C528042/1, and supported by the European Union through the Integrated Project QAP (IST-3-015848) and SECOQC.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"APPENDIX A\",\n \"text\": \"We have used two-dimensional ancilla on each side. Consider a general state of four qubits in the tensor-product of the computational bases of the original particles A, B and the ancillary particles A ′ , B ′\\n\\n|Ψ AA ′ BB ′ = i,j,k,l c i,j,k,l |i A |j A ′ |k B |l B ′ . ( A1\\n\\n)\\n\\nThere are 16 terms in the above superposition with 16 complex amplitudes c i,j,k,l , therefore |Ψ can be parameterized using 30 real numbers (if we take into account the global phase and normalization). We will parameterize it in the following way [22] . First, to facilitate our analysis it is easier to incorporate four indexes i, j, k and l, each of which runs from 0 to 1, into a single index, x, that runs from 1 to 16. This can be done by using the formula x = 8i + 4j + 2k + l + 1, which is essentially a formula for converting a number from the Boolean representation to the decimal. Second, we present amplitudes c x in the form\\n\\nc x = |c x |e iθx , ( A2\\n\\n) where 16 x=1 |c x | 2 = 1\\n\\nand θ 1 = 0. Third, we introduce new parameters φ x such that\\n\\nc x = sin φ x-1 15 y=x cos φ y , ( A3\\n\\n)\\n\\nwhere φ 0 = π/2. Thus the state |Ψ is parameterized by 30 angles. The advantage of this parametrization is that we restrict their values only to the interval [0, 2π] that simplifies numerics. We proceed as follows. A program generates a vector of 30 random numbers in the interval [0, 2π] . This is the initial state. We then apply the non-local map and obtain a final state. We calculate the value of the gain in entanglement ∆S = S(τ\\n\\n(Ψ) BB ′ ) -S(τ (Ψ) AA ′ BB ′ ) - S(T r AA ′ |Ψ Ψ|).\\n\\nAfter that we vary the values of the random vector by a small amount and repeat these calculations again, thereby obtaining a gradient of the change in entanglement in that point. We move along the gradient to obtain the next |Ψ , and the procedure is repeated. Eventually, the program reaches the maximum where it stops.\\n\\n[1] C.H.Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A , 53 2046 (1996). [2] W. Dür, G. Vidal, J. I. Cirac, N. Linden, and S. Popescu, Phys. Rev. Lett. , 87, 137901 (2001). [3] M.S. Leifer, L. Henderson, and N. Linden, Phys. Rev. A 67, 012306 (2003). [4] D.W.Berry and B.C.Sanders, Phys. Rev. A 71, 022304 (2005); P.Zanardi, C. Zalka, and Lara Faoro, Phys. Rev. A 62, 030301(R) (2000); L. Clarisse, S. Ghosh, S. Severini, A. Sudbery, e-print arXiv:quant-ph/0611075v2. [5] C.H. Bennett, A. Harrow, D.W. Leung, and J.A. Smolin, IEEE Tran. Inf. Theory, 49, 8, 1895 Theory, 49, 8, (2003)) . [6] A.M. Childs, D.W. Leung, F. Verstraete, and G. Vidal, Quant. Inf. Comp. 3, 97 (2003). [7] N. Linden, J.A. Smolin, and A. Winter, e-print quant-ph/0511217. [8] M.A. Nielsen and I.L.Chuang, Quantum Computation and Quantum Information, Cambridge University Press (2004). [9] It was explicitly shown [3] that the optimization can be restricted to pure states. [10] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001).\\n\\n[11] W.K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [12] C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters, Phys. Rev. A 54, 3824 (1996). [13] J. I. Cirac, W. Dür, B. Kraus and M. Lewenstein, Phys.\\n\\nRev. Lett. , 86, 544 (2001); W. Dür and J. I. Cirac, Phys. Rev. A , 64, 012317 (2001). [14] B. Kraus and J.I. Cirac, Phys. Rev. A 63, 062309 (2001). [15] Strictly speaking DCNOT-class and SWAP-class should be unified under a single class if analyzed according to the criteria of interconvertability under LOCC [16] . In the framework of (dis)entangling capacity it is useful to identify them as separate classes, because their behavior differs qualitatively. [16] W. Dür, G. Vidal, J. I. Cirac, Phys. Rev. Lett. , 89, 057901 (2002).\\n\\n[17] V. Vedral and M. B. Plenio, Phys. Rev. A 57, 1619 (1998). [18] Here we assume that all unitaries in Eq. (1) belong to the same class, which is typical for real situations where only parameters of the coupling are subject to variations. [19] M. J. Testolin, C. D. Hill, C. J. Wellard, L. C. L. Hollenberg, e-print quant-ph/0701165. [20] S. Bandyopadhyay and D.A. Lidar, Phys. Rev. A 70, 010301(R) (2004). [21] B.Groisman, S. Popescu, and A. Winter, Phys. Rev. A 72, 032317 (2005). [22] This method is a partial adaptation of the method used in Ref. [17] for calculation the relative entropy of entanglement.\"\n }\n]","isConversational":false}}},{"rowIdx":185,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1058"},"arxiv_id":{"kind":"string","value":"0704.1058"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/0264-9381/24/11/R01"},"content_hash":{"kind":"string","value":"418b473054216467def87b4ea566874d20859da100de8f3f0f5e2c48ba15e9c9"},"title":{"kind":"string","value":"Dark energy, gravitation and supernovae"},"abstract":{"kind":"string","value":"The discovery of the acceleration of the rate of expansion of the Universe\nfosters new explorations of the behavior of gravitation theories in the\ncosmological context. Either the GR framework is valid but a cosmic component\nwith a negative equation of state is dominating the energy--matter contents or\nthe Universe is better described at large by a theory that departs from GR. In\nthis review we address theoretical alternatives that have been explored through\nsupernovae."},"authors":{"kind":"list like","value":["Pilar Ruiz-Lapuente"],"string":"[\n \"Pilar Ruiz-Lapuente\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["gr-qc","astro-ph","hep-ph"],"string":"[\n \"gr-qc\",\n \"astro-ph\",\n \"hep-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"gr-qc"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"2 Contents 1. Introduction 2. Gravity and extra dimensions 3. The dark energy field and scalar-tensor gravity 4. The cosmological constant? 5. The d L test from high-z SNe Ia 6. Testing the adequacy of the FRW metric 7. Next decade experiments 8. Complementarity 9. Conclusion 3 1. Introduction\n\nThe discovery of the acceleration of the expansion of the universe [1, 2] has revived questions concerning the domain of validity of Einstein's General Relativity. At present, the data gathered on the expansion rate do not disclose whether the acceleration is due to a component formally equivalent to the cosmological constant introduced by Einstein [3], or whether we are finding the effective behavior of a theory of a wider scope whose low energy limit slightly departs from GR. The bare Einstein's equations (without cosmological constant) for a Universe with a FRW metric and dust-like matter imply a continuous deceleration of the expansion rate. However, a Universe containing a fluid with an equation of state p = wρ with index w < -1/3 overcomes the deceleration when the density of this fluid dominates over that of the dust-like matter. In this context, the cosmological constant, if it is positive and added to the equations, balances the deceleration by acting as a fluid with an equation of state p = -ρ.\n\nThe cosmological constant was originally in the metric part of Einstein's equations, and alternative metric theories of gravity provide a wealth of terms in that side. When describing a homogeneous and isotropic universe within a FRW model, it is frequent to come as close as possible to the standard Friedmann-Einstein equation for the expansion rate by bringing terms to the right-hand side within the so-called \"effective index\" w eff of the cosmic fluid whose equation of state is p = w eff ρ.\n\nIn the field of gravitation, ideas that appeared in the classical domain several decades ago, are now being brought back within the different context of quantum gravity theories. Scalar-tensor theories of gravitation are being reconsidered. Scalar fields coupled to gravity were proposed as a way to incorporate Mach inertial induction. They gave rise to the Brans-Dicke theory of gravitation [4] , which is well restricted by solar system experiments [5] . Superstring theory and M-theory introduce a number of quantum scalar fields, the dilaton, the radion, the moduli [6] . They can play a role in the late acceleration of the Universe. On the other hand, brane worlds inspired by M-theory provide modifications to gravity and new equations of cosmic expansion [7] . In the brane world context, gravity lives in the bulk of 3+1+d dimensions while the rest of the interactions are confined to the 3+1 brane. Some braneworld proposals do not explain the observed cosmic expansion. However, it was realized [8, 9] that it is possible to obtain late-time acceleration with one extra dimension. This possibility is interesting as it accounts for the observed cosmic expansion without the need of a cosmological constant term. Acceleration results from the induced gravity term in the brane. The number and size of additional dimensions has been explored in a wide range of proposals. The existence of additional dimensions and their size are constrained by tests of the gravity law [10, 11, 12] . At the time when gravity theories of the scalar-tensor type and others were examined, the solar system tests were the most restrictive ones. Nowadays, the theories have to confront the cosmological observations. The theories have to allow Big Bang 4 nucleosynthesis [13] . They have to fullfil the constraints from the growth of density perturbations and give correct predictions for the evolution of the growth factor. Such growth function of density perturbations requires careful evaluation for several modified gravity scenarios [14, 15] . Ultimately, gravity theories have to be consistent with the CMB data and with the expansion rate of the Universe provided by supernovae. Supernovae enable to measure the rate of the expansion of the Universe along redshift z. They do so by tracing the luminosity distance along z for a standard candle. If the Universe is flat, as inferred from WMAP [16] , the d L relation is simply:\n\nd L (z) = cH -1 0 (1 + z) z 0 dz ′ H(z ′ ) ( 1\n\n)\n\nwhile in the general case,\n\nd L (z) = cH -1 0 (1 + z)| Ω K | -1/2 sinn | Ω K | 1/2 z 0 H 0 dz ′ H(z ′ ) ( 2\n\n)\n\nwhere sinn(x) = sin(x), x, or sinh(x) for closed, flat, and open models respectively, where the curvature parameter Ω K , is defined as Ω K = Ω T -1. H(z) is the Hubble parameter. For a FRW universe containing matter and a general form of matter-energy X with index of the equation of state w X , H(z) is given by:\n\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω M (1 + z) 3 + Ω X (1 + z) 3(1+w X ) ( 3\n\n)\n\nwhere H 0 is the present value of the Hubble parameter and\n\nΩ K ≡ -k H 2 0 a 2 0 ( 4\n\n)\n\nwhere k accounts for the geometry and a 0 is the present value of the scale factor.\n\nΩ M = 8πG 3H 2 0 ρ M ( 5\n\n) Ω X is the density parameter for dark energy.\n\nWhen w(z) varies along z, H(z) can be expressed as:\n\nH(z) 2 = H 2 0 Ω K (1 + z) 2 + Ω M (1 + z) 3 + Ω X exp 3 z 0 1 + w(z ′ ) 1 + z ′ dz ′ ( 6\n\n)\n\nOne sees from (1-2), that d L contains w(z) through a double integral. This leads to limitations in recovering w(z) if d L data along z are scarce and with large errorbars. Data are often given as m(z) which relates to d L (z) as:\n\nm(z) = 5 logd L (z) + [M + 25 -5 log(H 0 )] ( 7\n\n)\n\n5 The quantity within brackets is M ≡ M -5 log H 0 + 25, the \"Hubble-constant-free\" peak absolute magnitude of a supernova. The data are given sometimes in the form of distance moduli:\n\nµ = 5 log d L + 25 ( 8\n\n)\n\nThe measurement of w(z) through d L (z) is the standard approach, though possibilities of measuring directly H(z) have been investigated (for a review see [17] ). This Hubble law in (3) is derived from the Einstein-Hilbert action. We will review other possibilities related to expansion histories arising from modified gravity theories and effective theories which contain scalar fields accelerating the cosmic expansion. In the present overview we concentrate on the empirical evaluation of those theories. Some of the models to be explored here might contain a ghost problem or have a subclass amongst them that present quantum instabilities. We refer for this topic to recent updates in [18, 19, 20, 21, 22] .\n\nThe idea of allowing gravity to live in extra dimensions arose to explain the hierarchy problem and constitutes one of the major challenges to the classical picture of a 4dimensional description of gravity [7] . The Planck scale of 10 19 GeV, relevant for gravitation, is much larger than the electroweak scale, of about 1 TeV of the standard model of particle physics. According to the proposal by Arkani-Hamed, Dimopoulos and Dvali [7], the Planck scale is not a fundamental scale, but its enormity is simply a consequence of the large size of new extra dimensions. While gravitons can propagate in the new extra dimensions, the standard model fields are confined to our submanifold of 3 + 1 dimensions, the brane, which is embedded in a higher dimensional bulk. In this scenario, the Planck scale in the bulk can be much smaller than in the 4dimensional brane as\n\nM 2 P,4 = R n M n+2 P,n+4 ( 9\n\n)\n\nThe existence of extra dimensions in which gravity propagates produces a transition in the gravitation law at the compactification radius, proposed to be of the submillimetre scale.\n\nIn the cosmological context, this idea has some interesting consequences since it predicts a departure from GR. As soon as the cosmological model was implemented for n = 1 extra dimensions [23, 24, 9, 8] , it was noted that the brane cosmology leads to Friedmann-like equations very different from the standard ones [24] . In the 5dimensional model by Dvali, Gabadadze and Porrati (DGP) whose cosmology is explored by Deffayet and collaborators in [9] , the competition between the five-dimensional term of the Einstein-Hilbert action and the four-dimensional term induced in the brane leads to a modified Friedmann equation which simulates an acceleration term without the need to include a cosmological constant. The transition from the gravity law r -2 to 6 the modified one is characterized by a scale r c , the so-called cross-over scale defined as [9]:\n\nr c ≡ M 2 P 2M 3 ( 5\n\n) ( 10\n\n)\n\nAt large distances, the five dimensional term takes over and gravity gets weaker [9] . If the parameter r c is chosen to be of the order of H 0 -1 ∼ 10 29 mm, this is a choice for M (5) ∼ 10-100 MeV. At short distances r << r c , Newton's law is modified by a logarithmic repulsion. At large distances r >> r c , the modification involves a factor r rc . The Hubble expansion law shows significant departures from the standard Friedmann expansion law:\n\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω rc + Ω rc + α Ω α (1 + z) 3(1+wα) 2 ( 11\n\n)\n\nwhere the sum over α stands for a sum over each component with equation of state w α , Ω α being the density parameter:\n\nΩ α ≡ ρ 0 α 3M 2 Pl H 2 0 a 3(1+wα) 0 ( 12\n\n)\n\nand\n\nΩ rc ≡ 1 4r 2 c H 2 0 ( 13\n\n)\n\nIn the matter-dominated epoch, the expansion equation is:\n\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω rc + Ω rc + Ω M (1 + z) 3 2 ( 14\n\n)\n\nIn the original 5D model in [8] , M 5 was estimated to be ∼ 1 TeV, which corresponds to a cross-over scale r c of 10 15 cm. Such r c led to incompatibilities with the solar system data. When M 5 is taken to be M 5 < ∼ 1 GeV, i.e. r c > ∼ 10 25 cm, the conflict disappears. High-z supernovae give useful constraints to the parameter Ω rc in (10) [25, 26] . In [26] , two samples of high-z supernovae are used in conjunction with measurements from baryon oscillations [27] and the CMB shift parameter [28] to estimate the favored values for Ω rc and Ω M . Ω rc = 0.125 for the supernova data in the Riess sample from 2004 [29] and Ω rc = 0.13 for the SNLS data [30] . It is found in [26] that, for a flat Universe, this 5D modified gravity model gives a worse fit than GR plus cosmological constant. Further supernova samples would reexamine the empirical compatibility of the DGP model and other modified gravity models involving more dimensions. Those models can be explored through a phenomenological description. The Hubble law arising from gravity with extra dimensions can be expressed by adding H α terms to the standard Friedmann equation [31] :\n\nH 2 - H α r 2-α c = 8πGρ M 3 ( 15\n\n)\n\n7 where r c is the cross-over radius, i.e. the transition radius from a regime to another in the gravity law, and α is the parameter to be empirically determined. From redshift z= 0 to z= 2, the above Friedmann equation gives an effective index of the equation of state [31]: w eff ≈ -1 + 0.3α (16) Such effective equation of state evolves too fast to be compatible with present SNe Ia data, unless α is small. Present data favor α close to 0 [25]. Though the most simple implementations lack empirical confirmation, room is left for several models. Cosmological SNe Ia are useful tests in restricting the cross-over radius that relates to departures from GR.\n\nA different kind of model in which the brane is embedded in a 5D bulk has been proposed by Randall and Sundrum [32, 33] . Contrary to the previous approach, here the bulk has a AdS 5 geometry as opposed to the flat geometry of the DGP model. In the original Randall-Sundrum (R-S) model, the brane has tension σ, i.e. cosmological constant, as well as the bulk Λ 5 . The bulk cosmological constant is negative. The resulting Hubble expansion in the brane is the result of the compensating effect of both terms. To make zero the effective cosmological constant in the brane, Λ 4 , a fine tuning between the brane tension and the bulk cosmological constant is required. The original proposal sets the effective cosmological constant, Λ 4 , to zero. Generally, a tiny deviation from exact compensation [34] would lead to an effective cosmological constant at late times. Modifications of the original R-S proposal include in the brane, in addition to tension (cosmological constant), a matter-energy density ρ. The total brane density ρ B is:\n\nρ B = σ + ρ ( 17\n\n)\n\nThis gives the modified Friedmann equation [35] :\n\nH 2 = κ 4 36 σ 2 - 1 l 2 + κ 4 18 σρ + κ 4 36 ρ 2 + C a 4 ( 18\n\n)\n\nwhere κ is the inverse Planck mass M -1 P and l is related to Λ 5 . Generalizations of the Randall-Sundrum scenario in cosmology consider the AdS 5 bulk geometry containing the fifth-dimensional cosmological constant Λ 5 , but explore arbitrary densities in the brane and in the bulk.\n\nExplorations of braneworld ideas that allow a late acceleration of the Universe can be found in [36, 37, 38] .\n\n3. The dark energy scalar field and scalar-tensor gravity\n\nBefore the development of braneworld scenarios, dark energy as associated to a scalar field was considered in analogous way to research done within the context of inflation 8 [40] . This theoretical possibility was entertained a decade before there was evidence of a non-zero cosmological constant from supernovae and that the CMB results indicated a flat Universe. The favored Ω T = 1 value by inflation and the evidence that Ω M < 1 led to think about a possible Λ term. In the recent expansion history a scalar field could play a role in analogy with inflation, where a scalar field dubbed the inflaton gives rise to the exponential phase of early acceleration. The scalar field responsible for the late acceleration, later named quintessence, is proposed as a new massless scalar field which induces a cosmological constant behavior at late times. In this original setting, the field is minimally coupled to gravity, i.e., thus the inclusion of the field remains within the framework of GR. The interaction of the field with the matter fields is set to be negligible. The scalar field responsible for dark energy can be identified by recovering its potential, or equivalently its effective equation of state. The presence of such scalar field φ, would induce an action:\n\nS = d 4 x √ -g 1 2 g µν ∂ µ φ∂ ν φ -V (φ) ( 19\n\n)\n\nThe field contributes to the stress-energy momentum tensor with an effective mass density and pressure:\n\nρ φ = 1 2 φ2 + V (φ) p φ = 1 2 φ2 -V (φ) ( 20\n\n)\n\nIn this case, the effective value of the equation of state w eff depends on the form of the potential V (φ) and it can evolve with time [39] .\n\nw eff = p φ ρ φ ( 21\n\n)\n\nA search for general properties of this potential, in analogy to the search for the inflation potential, leads to suggestions for potentials with tracking behavior [40, 41, 42, 43] . Several potentials have been proposed, from the original V = κ/φ α [40] to other recent proposals [44] . The dark energy scalar field that does not couple with matter has the feature of avoiding effects on variations of constants and other effects subject to experimental evidence [45, 46] . There is already a reconstruction of what could be the potential of the minimally coupled field V (φ) obtained from the SNLS SNe Ia [47] following reconstruction ideas given in [48, 49] . Another reconstruction of V (φ) using SNe Ia from various collaborations and considering Padé approximant expansions of the potential is found in [50] . By now, many proposals have been discarded, since the equation of state between z= 0 and z= 1.5 evolves very slowly as shown by high-z SNe Ia data [2, 51, 29, 52] . 9 3.2. Scalar-tensor gravity One can still consider the case of a scalar field that is coupled to other matter fields or to gravity. Some of these proposals come from the ballpark of superstring theory. The most well-known scalar-tensor gravity proposal, i.e. Brans-Dicke theory, incorporated the coupling of a field to gravity to account for the Mach induction on Newton's constant. In Brans-Dicke gravity [53, 4] , Newton's constant is replaced by a dynamical field G leading to the generalized Friedmann equation:\n\nH 2 = 8π 3 ω 2πφ 2 ρ + 1 2 φ2 + V (φ) - 3 2ω Hφ φ ( 22\n\n)\n\nThe Brans-Dicke action contains a φ 2 R term:\n\nS = √ -g d 4 x ± 1 8ω φ 2 R ∓ 1 2 g µν φ ,µ φ ,ν -V (φ) + L matter ( 23\n\n)\n\nThe theory has been well tested [5]. Observationally, for other scalar-tensor theories, we aim at retrieving the self-interaction potential of the scalar V (φ). and the coupling function to matter and gravity A(φ) in the matter term of the action [54]:\n\nS matter (ψ, A 2 (φ)g µν ) ( 24\n\n)\n\nIn the case of quintessence (minimally coupled field) A(φ) = 1. The present supernova data have explored the potential for a minimally coupled field [47] . In the non-minimally coupled case, also referred to as coupled quintessence or coupled dark energy [55, 56] , the CMB data can put constraints on the strength of scalar gravity as compared to ordinary tensorial gravity [55] . However, there are limitations to the recovery of the form of the coupling and the coupled dark energy potential from CMB observations [55] . Supernovae together with observational constraints on the growth of density perturbations would allow to retrieve A(φ) [57, 58] . We expand A(φ) in its derivatives α 0 , β 0 and higher orders:\n\nln A(φ) ≡ α 0 (φ -φ 0 ) + 1 2 β 0 (φ -φ 0 ) 2 + O(φ -φ 0 ) 3 ( 25\n\n)\n\nBy exploring the empirical evidence it is found [54] that solar system tests and binary pulsar tests allow to impose precise bounds on the first and second derivatives of the matter-scalar coupling function, while SNe Ia could \"a priori\" allow to reconstruct the full shape of the function A(φ) or the higher order derivatives. It is argued in [57, 58, 59] that the knowledge of the luminosity distance and the density fluctuations δρ/ρ as functions of redshift z is sufficient to reconstruct the potential V (φ) and the coupling function A(φ).\n\nThese results can be generalized to proposed actions that can be re-expressed as a theory of interacting matter fields in general relativity [60] . Actions containing e -Φ R are motivated from the lowest order effective action including the dilaton Φ [61] . They can be brought by redefinition of the field to an Einstein frame in which the field is minimally coupled to the metric.\n\n10 A number of interesting cases have been examined and have been expressed in terms of a scalar field which is minimally coupled to the metric, but nonminimally coupled to the other fields [62, 63, 64] . The self-interaction potential shows an explicit energy transfer between the scalar field and the matter fields. Related to the previous exploration is that of f(R) gravity or extended curvature gravity [65], also known as \"c-essence\" when including an inverse power of R [67] . Those modified gravity theories with a Lagrangian containing an arbitrary function of R are equivalent to a particular class of scalar-tensor theories of gravity. For instance, the action\n\nS = 1 2κ 2 d 4 x √ -g R - µ 4 R + S matter (g µν ) ( 26\n\n)\n\nis found to be equivalent to [67]:\n\nS = 1 2κ 2 d 4 x √ -g 1 + µ 4 φ 2 R - 2µ 4 φ + S matter (g µν ) ( 27\n\n)\n\nThe equivalence is generalized to an arbitrary function of R [60, 65]:\n\nS = 1 2κ 2 d 4 x √ -g F (R) + S matter (g µν ) ( 28\n\n)\n\nwith an equivalent action [60, 65]:\n\nS = 1 2κ 2 d 4 x √ -g (F (φ) + F ′ (φ)(R -φ)) + S matter (g µν ) ( 29\n\n)\n\nwhere F ′ (φ) = dF/dφ. The introduction of a canonical scalar field ϕ such that F ′ (φ) = exp( 2/3 κϕ), allows to rewrite the action with F(R) as:\n\nS = d 4 x √ -g E 1 2κ 2 R E - 1 2 (∇ϕ) 2 -V (ϕ) + S matter (g E µν /F ′ (φ(ϕ)), V (ϕ) = (φ(ϕ)F ′ (φ(ϕ)) -F (φ(ϕ)))/2κ 2 F ′ (φ(ϕ)) 2 ( 30\n\n)\n\nThe gravity described by g E µν is the Einstein-scalar system and g µν (= g E µν /F ′ (φ)). g µν is a metric of scalar-tensor gravity which is, in this case, subject to constraints coming from solar system experiments. Again, a classification of when one recovers an action equivalent to the Einstein-Hilbert case and when one recovers a Brans-Dicketype action in f(R) gravity is given in [60] . When considering a general function of R, it is not always possible to obtain a description equivalent to Einstein GR plus additional fields minimally coupled to the metric but coupled to other fields. When this is possible, we have in the matter term the coupling between the matter fields ψ and the scalar field φ. Examples of f(R) casted in terms of fields can be found in [62, 64, 66, 67] . For the Starobinsky model [66] with F (R) = R + R 2 /M 2 with M ∼ 10 12 GeV, for instance, the effective potential can be rewritten, in terms of the scalar field ϕ, as\n\nV (ϕ) = M 2 e -2 √ 2/3 κϕ 8κ 2 (e √ 2/3 κϕ -1) 2 ( 31\n\n)\n\n11 Actions with generalized inverse powers of R have been investigate as well [68, 69, 70, 71]. The supernovae gathered up to now are compatible with some f(R) gravity models and incompatible with others [68, 70, 72]. More general scalar-tensor theories than the ones here defined, such as those where the potential depends on several scalar fields V (φ n ) and matter-scalar coupling A(φ n ), give rise to a wider phenomenology. In the superstring scenario, nonminimal coupling of the scalar fields to the spacetime curvature leads to effective actions with similarities to scalar-tensor proposals. Some scalars present in the effective action arise from compactification to four dimensions.\n\nThose are structure moduli which do not couple directly with the spacetime curvature tensor. One modulus field σ associated with the overall size of the internal compactificaton couples to Einstein gravity via Riemann curvature invariants, such as the Gauss-Bonnet invariant [74] . If one considers the Gauss-Bonnet term, at least an extra function, the coupling to the Gauss-Bonnet term, W (σ) needs to be determined.\n\nIn the following, we review attempts to include such enlarged action and their test with cosmological d L data.\n\nThe Gauss-Bonnet term can be found in the effective low energy string Lagrangian and in brane theories [73, 74, 75, 35, 76] . The case of the simplest action containing the Gauss-Bonnet term can be written as:\n\nS = √ -g R 2κ 2 - 1 2 (∂ µ φ) 2 - √ -g W (σ) R 2 GB + S matter [matter ; g µν ] ( 32\n\n)\n\nThis Lagrangian includes the Gauss-Bonnet term coupled to a field φ ≡ σ, for instance the modulus field in [74] where W (σ) is the coupling of the modulus field to the Gauss-Bonnet term, R GB .\n\nR 2 GB ≡ R µνρσ R µνρσ -4R µν R µν + R 2 ( 33\n\n)\n\nThe possible coupling of a scalar field to the Gauss-Bonnet topological invariant can be constrained if one takes into account cosmological supernovae and solar system tests.\n\nFrom supernova data gathered in [29] and with a parameterized analysis of the evolution, it is found that the present value of the equation of state w 0 is below -1. Though for the measurement of the index of the equation of state at z = 0 one awaits a large enough nearby supernova sample, the results presented in [29, 52] move to examine the possibility of w eff < -1. In the original exploration of the idea [77], a Universe which evolves towards w eff < -1 would suffer a super-aceleration phase, that has come to be called the Big Rip. From an action which includes the Gauss-Bonnet term, one can obtain w eff < -1 for a canonical scalar field (positive kinetic energy) [78, 35] . It avoids the Big Rip for w eff < -1. To add a word on this, this is not the main motivation for 12 Gauss-Bonnet gravity, since variations of the braneworld scenario can also get w eff < -1 and avoid the Big Rip [79, 80, 81] . The interest on the GB term is justified by its presence in a variety of gravity proposals from string and M-Theory. Recently some tests using supernovae have already been done [82] . The extra GB term opens naturally many possibilities for the evolution of the scale factor. The supernova test succeeds for a range of effective low energy actions including a Gauss-Bonnet term [82].\n\nThe fields that interact with the dark energy scalar field are constrained by a large number of tests [45] . The possibility that the dark energy scalar couples to nucleons and photons has been investigated in relation to implications for a fifth force [83, 84] . Within current scenarios of particle physics such as MSSM and MSUGRA the role of this dark energy field shows some observational signatures of the fifth force type as well [85] . For instance, the interaction of the dark energy scalar field φ with the Higgs doublet H u and H d leads to an action where the field couples differently to each of the Higgs ψ u and ψ d and one expects a violation of the weak equivalence principle. In the string scenario, where radion, dilaton and moduli can be responsible for the early and late acceleration, one has as well some possible strong couplings of the fields with matter [86, 75, 87]. Thus, the cosmological context has to be considered together with classical gravitational tests on Earth. Thus far, the torsion balance experiments set strict limits on the variation of Newton's law using different substances and locations. There is a wide range of tests being done to examine the predictions from scalar-tensor proposals and modified gravity ideas [12, 10, 11, 88, 89] . The empirical bounds on some types of couplings are tight and worth considering together with the cosmological probes.\n\nFinding that the negative pressure component gives w = -1, would likely be the biggest challenge to gravitation theories. At present, SNe Ia have not provided such evidence, though averaged values of w are close to -1 at redshifts up to z∼ 1. To probe that w stays -1 up to the surface of last scattering, involves refining several other methods (growth of density perturbations, integrated Sachs-Wolfe effect) [90] . From the view point of string theorists [91] , there is no natural explanation for the vanishing or extreme smallness of the vacuum energy. A way to address the small value of the cosmological constant in the context of a stringy landscape is to consider the anthropic principle [92] . However, invoking the anthropic principle as a way out of the problem entails to abandon the quest for a conventional scientific explanation [91]. . From the observational point of view, the answer is near to w = -1, but possible departures from w = -1 are presently being examined through expansions of the equation of state along the scale factor and other various approaches. A expansion 13 Figure 1. The cosmological constant case (bold line) is compared with evolving models close to w = -1, i.e. a model with w 0 = -1.0 and w a = -1.5 (short dash line) and a model with w 0 = -1.0 and w a = +1.5 (dash-dotted line) often used is [93, 94] :\n\nw(a) = w 0 + w a (1 -a) ( 34\n\n)\n\nwhere the scale factor a = (1 + z) -1 , so w ′ = w a /2 when evaluated at z = 1. Differences between the cosmological constant and departures from it in the equation of state are measurable nowadays (see Fig 1). To reach a higher level of discrimination between dark energy models depends on the progress made in the accuracy of the measurements. The aim of future projects is to determine distances with 1 % accuracy, i.e. with errors in magnitudes of 0.02 mag [95] . If the equation of state of dark energy is very close to -1, one can anticipate decades of improving the supernova measurements. The development that we might witness could be akin to that of the postnewtonian tests of GR where higher levels of precision are being targeted since the middle of the last century into the present. 14 5. The d L test from SNe Ia\n\nA long path has brought us to the present situation where we can already aim at accuracies of 1% in distance. Back in 1968, a first attempt to draw the Hubble diagram from SNe resulted in a spread of σ ∼ 0.6 mag [96] . Such large scatter was due to the inclusion of other types of supernovae, which were later recognized as being of a different type (SNI b/c) and characterized by a large scatter in luminosities [97] . SNe Ia are not standard candles either, but they follow a well known correlation between maximum brightness and rate of decline of the light curve. In 1977, Pskovskii [98] first described the correlation between the brightness at maximum and the rate of decline of the light curve: the brighter SNe Ia have a slower decline of their light curves whereas fainter ones are the faster decliners. A systematic follow-up of SNe Ia confirmed the brightness-decline relation [99, 100] . The intrinsic variation of SNe Ia is written as a linear relationship:\n\nM B = M B0 + α (∆m 15 (B) -β) + 5 log(H 0 /65) ( 35\n\n)\n\n∆m 15 , the parameter of the SNe Ia light curve family, is the number of magnitudes of decline in 15 days after maximum. The value of α, as well as the dispersion of that relation, have been evaluated in samples obtained from 1993 to the present [100] . M B0 is the absolute magnitude in B for a template SNIa of β rate of decline. An intrinsic dispersion of 0.11 mag in that law is found when information in more than one color is available.\n\nIn fact, the 0.11 mag scatter in the brightness-rate of decline is found when one includes supernovae affected by extinction. The scatter is smaller in environments where dust in the host galaxies of the SNe Ia is a lesser effect [101] . Additionally, the family of SNe Ia form a sequence of highly resembling spectra with subtle changes in some spectral features correlated with the light curves shapes [102, 103, 104, 105, 106, 107, 108] . The multiple correlations allow to control errors within the family of SNe Ia [108] . The parameter ∆m 15 requires to have observed the maximum in the light curve of the supernova. This is not always possible, and therefore the various SN collaborations have formulated the brightness-decline correlation in different ways. The High-Z SN Team uses the full shape of the light curve with respect to a template, method refered as MLCS2k2 [109, 110] . The Supernova Cosmology Project collaboration, on the other hand, has introduced the stretch-factor, s, as a parameter to account for the brightnessdecline relationship [111, 2, 112] . The stretch parameter fits the SNIa light curve from premaximum to up to 60 days after maximum. Thus, even if the maximum brightness phase is not observed, the whole light curve provides the decline value. More recently, other ways to measure the decline have been proposed [113, 114, 115] . Measuring the evolution of the rate of expansion of the Universe does not require knowledge of its present value, H 0 . Large projects have been devoted to the determination of H 0 by attempting the absolute calibration of M B from nearby SNe 15 Ia that occur in galaxies with Cepheid-measured distances [116, 117] . Controversy persists, though the most recent values quoted by the two collaborations using this method are closer (in the range 65-75 km s -1 Mpc -1 ) [116, 117] . Though largely ignored as an argument, the mere fact that a SNIa is a explosion of white dwarf bounds very efficiently the lower and upper possible values of H 0 [118] . On the empirical side, the emission at late phases of SNe Ia gives a precise value of H 0 , since such emission is very sensitive to the density profile, the mass of the supernova and the amount of 56 Ni synthesized in the explosion. Those quantities are easily measurable from the spectra [119] . The value favored by this method is H 0 = 68 ± 6 (syst) km s -1 Mpc -1 . This disgresion done, we emphasize the nuisance role of the absolute magnitude of SNe Ia when it comes to dark energy cosmology. The apparent magnitude of supernovae along z give us the cosmology, m B as [2, 51]:\n\nm B = M + 5 log D L (z; Ω M , Ω Λ ) -α(s -1) ( 36\n\n)\n\nwhere s is the stretch value for the supernova, D L ≡ H 0 d L is the \"Hubble-constant-free\" luminosity distance, d L is given by (1) and M ≡ M B -5 log H 0 + 25 is the \"Hubbleconstant-free\" B-band peak absolute magnitude of a s = 1 SN Ia with true absolute peak magnitude M B . Similar use of the nuisance parameter in their light curve fitting is made in [29, 52] .\n\nIn the first determinations of dark energy cosmology, there were four parameters in the fit: the matter density parameter Ω M , the cosmological constant density parameter Ω Λ , as well as the two nuisance parameters, M and α (either α itself as used by the SCP or a equivalent parameter by other collaborations). The four-dimensional (Ω M , Ω Λ , M, α) space, explored as a grid, yields χ 2 and P ∝ e -χ 2 /2 of the fit of the luminosity distance equation to the peak B-band magnitudes and redshifts of the supernovae. After normalization by integrating over the \"nuisance\" parameters, the confidence regions in the Ω M -Ω Λ plane can be obtained. This use of the magnitude-redshift relation m(z) as a function of Ω M and Ω Λ with a sample of high-z SNe Ia of different z, is a very powerful pointer to the allowed values in the Ω M -Ω Λ plane [120, 121] . By 1998, such use yielded important results, impling that Ω Λ > 0 at the 3σ confidence level. For a flat universe (Ω T = 1), the results from the Supernova Cosmology Project meant Ω M =0.28 +0.09 -0.08 (statistical) +0.05 -0.04 (systematics), and the High-Z Supernova Team obtained for a flat universe Ω M =0.24±0.1. The outcoming picture of our universe is that about 20-30% of its density content is in matter and 70-80% in cosmological constant. According to the allowed Ω M and Ω Λ values, the Universe will expand forever, accelerating its rate of expansion.\n\nIf the size of the SNe Ia sample and its redshift range do allow it, one can aim at determining the average value of the index of the equation w or, going a step further, aim at determining the confidence regions on the w 0 -w a plane. For this later purpose, 16 using the information on the global geometry as given by WMAP provides a helpful prior. In Knop et al. (2003) [51], the supernova results were combined with measurements of Ω M from galaxy redshift distorsion data and from the measurement of the distance to the surface of last scattering from WMAP. The confidence regions provided by the measurement of the distance to the surface of last scattering and the SNe Ia confidence regions cross in the Ω M -w plane, thus providing very good complementarity. This work considers an averaged w (given the range in z of the SNe Ia observed). From the fitting assuming w constant, the value is w =-1.06 +0.14 -0.21 (statistical) +0.08 -0.08 (identified systematics). The analysis of the SNLS [30] also aims at an average value of w, allowed by the z range of this survey. Their 71 high-z SNeIa give w =-1.023 ± 0.090 (statistical) ± 0.054 (identified systematics). Recently, the results from ESSENCE, using 60 SNe Ia centered at z= 0.5 point to a similar result with w = -1.05 +0.13 -0-12 (statistical) ± 0.13 (systematics) [122] . In this last work, the supernovae from various collaborations are joined with two separate light curve fitting approaches giving a consistent result for the average value of w. High-z SNe Ia beyond z > 1 enable to have a first look at possible evolution of w(z). The first campaign by Riess et al. (2004) gave the results in the w 0 -w ′ plane, using the Taylor expansion for w(z) as w(z) = w 0 + w ′ z. The best estimates for w 0 and w ′ are w 0 = -1.36 +0.12 -0.28 and w ′ =1.48 +0.81 -0.98 . Those results [29, 52] exclude a fast evolution in the equation of state, and therefore they rule out a number of modified gravity proposals. Moreover, the results are confirmed in [122] where the available SNe Ia at large z, which can constrain w 0 and w a , are incorporated to test with a joined sample encompassing a wide range of z. Significant limits have been placed on the evolution of w(z). Now the potential of the method relies on an improved control of systematic effects. In this area, there has been substantial progress along the last years.\n\nHuge advances have taken place in controlling the uncertainties due to extinction by dust, the universality of the supernova properties, the control of lensing by dark matter and the observational process leading to the the construction of the Hubble diagram of SNe.\n\nThe study with the HST of the host galaxies of the SNe Ia sample of the Supernova Cosmology Project [101] allowed to subclassify the galaxy hosts of the P99 sample [2] . It was found that, at high z, early-type galaxies show a narrower dispersion in SNe Ia properties than late-type galaxies, as they do at low z ([123]). This result is very encouraging as it supersedes previous dispersion values obtained in samples of supernovae in all galaxy types. Supernovae in dust-rich environments, 17 such as spiral galaxies, are more affected by extinction. The supernova magnitudes are corrected regularly from Galactic extinction [124], but the correction by dust residing in the host galaxy or along the line of sight is often not included in the standard fit, as it would require to have extensive color information that might not be available for high-z SNe Ia. When it is included, it is still subject to the uncertainty in the extinction law [110] . Let us, for instance, consider the correction of dimming by dust as entering in the term A B , the magnitudes of extinction in the B band. We take, for instance, a supernova of a given stretch s, and we place its m eff (the magnitude of the equivalent supernova of s = 1) in the Hubble diagram. We might use, as in [2, 51] :\n\nm eff = m B + α(s -1) -A B ( 37\n\n)\n\nThe extinction in the canonical band B of the spectrum (A B ) is determined by the extinction coefficient R B in that band. Any uncertainty in this coefficient as due to variation of dust properties, leads to uncertain estimates of A B . R B is of the order of 4.1 ± 0.5 as properties of dust might change going to high z. If we have an error of 0.1 in the observable E(B -V ) due to dust extinction, it gets amplified as:\n\nA B ≡ R B × E(B -V ) ( 38\n\n)\n\nThe present way of dealing with SNe Ia, including information in several photometric bands of the spectrum where extinction coefficients (R V , R I ) are lower and one gets smaller values in the corresponding A V and A I , allows to control the dust problem.\n\nIn fact, the coefficients R B , R V and R I are also determined by methods determining distances to SNe Ia. In a sample of 133 nearby SNe Ia, it is found in [110] , that the dust in their host galaxies is well described by a mean extinction law with R V ≃ 2.7. In all this, the level at which SNe Ia are treated, including how many bands are used, becomes critical. A use of SNe Ia without taking into account color information leads to a dispersion of 0.17mag. When taking into account color information, the limit goes down to 0.11mag. Limiting the effect of extinction, as in SNeIa in ellipticals should bring down the dispersion. This finding is motivating the ongoing SN searches in clusters of elliptical galaxies [125] . In environments where dust is controlled, such as SNe in clusters rich in elliptical galaxies, the systematic effect introduced by dust extinction can be better controlled. There, one can also control the evolution of dust-properties.\n\nIn Table I , the level of accuracy in the first uses of SNe Ia back in 1998 is shown, with the contributions of each systematic and statistical error. The improvement is given for a future mission able to control the systematics in the best possible way. It is foreseen that several sources of systematic errors will become negligible within the next decade.\n\nThis question resides on whether we know that SNe Ia properties evolve with z. Much has been done along this path. Nowadays, while we know that dust is the unavoidable systematics, several studies confirm the universality of the light curve properties of SNe Ia. The study of a significant sample 18 of SNe Ia at various z by the Supernova Cosmology Project reveals that the rise times to maximum of high-z SNe Ia are similar to the low-z SNe Ia ([126, 112]). The same is found in the SNLS sample ([127]). Statistical evaluation is so far consistent with no difference between the low-z and high-z samples. Moreover, the possibility of existence of an extra parameter in the maximum brightnessrate of decline relation has been carefully examined. Among possible influences, metallicity was one of the obvious ones to consider. Examining supernovae in galaxies with a gradient of metallicity, it is found [128] that there is no evidence for metallicity dependence as an extra parameter in the light curve correlation of SNe Ia. This result comes from an analysis of the SNe Ia light curves of a sample of 62 supernovae in the local Universe. The SNe Ia belong to different populations along a metallicity gradient. This check has been done as well at high-z [129] using 74 SNe Ia (0.17 < z < 0.86) from the SCP sample. No significant correlation between peak SNIa luminosity and metallicity is found.\n\nOften we address the question of the spread of the SNe Ia samples in galaxies of different types: on how many fast SNe Ia versus intermediate or slow decliners are found in the various morphological types. Spirals at low or moderate redshifts should encompass all ranges of variation in the SNe Ia properties since they contain populations with a wide spread in age. The cosmological SN collaborations find that in those samples the one-parameter correlation does give a good description of the variation of the SNe Ia light curves: there is no residual correlation after the light-curve shape correction [112] . One can go a step further and test the SNIa objects themselves. This has been attempted in a intermediate z sample and it is a feasible project at all z [108] . The physical diagrams of SNe Ia at intermediate z and at low z seem to be similar [108] . Those diagrams reveal the composition, velocity gradient and radiation properties of SNeIa [108] . Looking at the inside of the SNe Ia is the best way to test whether they are the same at all z. Such step forward is giving definitive results which reaffirm the validity of the SNe Ia method. The progress done in this domain is summarized in Table I .\n\nThere is another element yet to be introduced here: it deals with the K correction. The K-correction is necessary to calculate the apparent magnitude in a y filter band of a source observed in a x filter. As we are comparing the observed flux of supernovae in a given spectral band along the expansion history, we need to take into account the effect that the photons of the supernova have shifted to redder wavelengths and spread over a different spectral wavelength range and the observed flux is different from that at emission.\n\nm Y (z) = m X (z = 0) + K xy (z) ( 39\n\n)\n\nwhere\n\nK xy = 2.5 log (1 + z) F (λ)S x (λ)dλ F [λ/(1 + z)] S y (λ)dλ + Z y -Z x ( 40\n\n) 19 where F (λ) is the source spectral energy distribution (a SN in this case), and S x (λ) is the transmission of the filter x. The Z y -Z x term accounts for the different zero points of the filters [130] . At present, this term amounts to an uncertainty of 0.02mag in the overall budget of SNeIa as distance indicators, but it can be decreased to 0.01 mag. The comparisons of supernovae with the fine time sequences of spectra along all phases available, are turning the uncertainty in the K-correction into a decreasingly small figure. The large amount of spectral data on SNe Ia, and the good correlation with photometric properties, can indeed make this uncertainty very small.\n\nHubble diagram for high redshift sources. There was an early concern on the systematics and bias that this effect could cause in the determination of the cosmological parameters using SNeIa [131] . It has been shown through the study of lensing in the highest z SNe Ia sample [132] that the magnification distribution of SNeIa matches very well the expectations for an unbiased sample, i.e., their mean magnification factor is consistent with unity. The effect can be very well controlled by studying the galaxy fields [132] . Moreover, SNe Ia should show a negligigle systematics caused by gravitational lensing when having a few SNeIa per redshift bin [122] . This means that in future missions that effect should not have impact in limiting the accuracy of w(z).\n\nThe better we deal with the light curve and spectral information of SNe Ia, the lower becomes the dispersion in the Hubble diagram. If SNe Ia have a systematic error in their use (independently from evolution), this is likely well below the accuracy to which we would like to determine distances (1%). We have not yet hit that brick wall. Using multi-epoch spectral information and accurate light curves in various filters should enable to test the power of these candles and their limiting irreduceable error [108].\n\n5.3.6. A joined sample Supernovae collected along years show the effect of observations treated with various degrees of accuracy. The dispersion around the best fit reflects that fact. For instance, some samples would have information on only two colors and at very few epochs, making it difficult to control reddening, or supernovae had no spectrum confirming that they were SNe Ia. The new samples will naturally have a higher degree of accuracy in magnitude and extinction for each supernova than previous samples. The ones with poorer information have been placed in [29, 52] into the \"silver category (as opposed to \"gold\" category). However, even within the Gold sample the requirements have changed: the latest definition of Gold sample demands higher quality standards than in [29] . In some of these SNe Ia lists, distance moduli of supernovae are assembled but they have been obtained through procedures which deal differently with dust extinction. The Hubble diagram resulting from those compilations would naturally show a large dispersion. In [133], it is argued that a few supernovae of an early High Z SN Search sample are pulling w(z) towards evolution. The exploration 20 done with a method independent from that in [29] showed that the central value in the 1σ confidence varied according to the light curve and reddening fitter of those SNe Ia [134] . Fortunately, supernovae enable to compare results obtained with different light curve fitting approaches. The published photometry by the various collaborations can be used to bring together the SNe Ia by using a consistent procedure into the Hubble diagram. This was done for the SNe Ia considering available up to 2005 in [134] . This is done in [122, 135, 136] for samples including the latest data from the Higher Z collaboration.\n\nIn most of the above discussion on dark energy, the FRW metric is taken as the right metric for our Universe. Modern cosmology supports that the Universe is highly isotropic on average on high scales. The main evidence is coming from the statistical isotropy of the Cosmic Microwave Background radiation. There are also cosmological observations supporting a matter distribution homogeneous on large scales, of the order of 100 h -1 Mpc. Given the observed structures of matter in voids and filaments, it is worth considering whether departures from the homogeneous and isotropic Universe could alter the present discussion, in particular, the possibility that there were no need for dark energy, but instead, that the observations are the result of an effect of inhomogeneity [137] .\n\nAs pointed out in [138] , large samples of supernovae will soon allow to test the luminosity distance relation in various directions and redshift bins. One can consider the luminosity distance d L as a function of direction n and redshift z [138] .\n\nThe direction-averaged luminosity distance is:\n\nd (0) L (z) = 1 4π dΩ n d L (z, n) = (1 + z) z 0 dz ′ H(z ′ ) ( 41\n\n)\n\nThis is the usual value retrieved by the SN collaborations. But its directional dependence d L (z, n) can be a test of the validity of the isotropy assumption. The authors in [138] propose to expand the luminosity distance in terms of spherical harmonics, to obtain the observable multipoles C l (z). This idea was also investigated in [109] , where the motion of our Local Group with respect to the CMB was measured. Results in [138] are in agreement with the results in [109] . Moreover, we can think of using the dispersion in the magnitude of SNe Ia in each direction n to put constrains on anisotropic models. As discussed in [139] , the dispersion of the magnitude in the Hubble diagram is at odds with some simple inhomogeneous models that predict a higher spread in m(z) and also a non-linear behavior. The anisotropic and inhomogeneous models have to confront present and future values on the SNeIa dispersion at each redshift interval z bin and in every direction n:\n\nσ(d L (z bin , n)) d L (z bin , n) = ln(10) 5 σ(m(z bin , n)) ( 42\n\n)\n\n21 The scaling solutions of the regionally averaged cosmologies can simulate the presence of a quintessence field or other negative pressure component [137, 142, 141] . But, any backreaction from inhomogeneity and anisotropy would show in the dispersion in the Hubble diagram. Those challenges have to be met by the non-FRW proposals. Up to now, the exploration of toy models calculating the effect of inhomogenities in the scale factor, find irregularities in the expansion rate beyond what is observed in the SN Ia Hubble diagram. The dispersion in the Hubble diagram of a standard candle is increased [139] .\n\nThe situation concerning the predictions of inhomogeneous models [137, 140, 141] is summarized in [142] . It is concluded that the inhomogeneities that would be able to amount to a d L (z) effect as observed from supernovae must be subhorizon and of strong type and they require a treatment beyond first-order perturbation theory. Research in this field will continue towards quantitative predictions to improve the connection with observational cosmology [143].\n\nThe aim of discriminating between various possibilities for dark energy places stringent requirements on the use of standard candles [144] . Within the two decades from the discovery (1998) and the gathering of space mission data, it is feasible to reduce largely the errors, as mentioned before, and this would reflect into an error of only 1%-2% in d L (z). Errors much bigger than this 1% of error in d L result in the impossibility to distinguish between models. Even at the level of 1% in d L (z) one finds some degeneracy [144, 145] . Space projects aimed at determining the nature of dark energy are counting on a systematic error in the method around 0.02 mag (i.e. 1% in the distance) [95] . A different question is whether from ground, within the planned experiments at work in the present decade, one can converge towards that limit. These questions have been investigated in detail by using Monte Carlo simulations including the systematic errors in m(z) (or d L (z)) to be found in the evaluation of the present value of the equation of state of dark energy w 0 and its first derivative in the scale factor w a [95] (see expression 33). SNAP is a mission that has been specifically designed to gather about 2000 SNe Ia up to z ∼ 1.7. As a single mission gathering supernova data with the same instrumentation, it can eliminate possible photometric offsets and include high-precision photometry out to 1.7 microns [146] . With such large number of SNe Ia, statistical errors in every redshift bin of ∆z = 0.1 become negligible. The error is dominated by the systematics. The total error in each redshift bin is given by:\n\nσ bin = σ 2 st N bin + σ 2 sys ( 43\n\n)\n\n22 with expected systematic error; σ sys = 0.02 z/z max (44)\n\nTable I gives a progress evaluation in the control of high-redshift supernova distance uncertainties.\n\nTABLE I ERROR CONTRIBUTIONS TO HIGH SUPERNOVA DISTANCES in 1998 Systematic Uncertainties (1σ) Magnitude Statistical Uncertainties (1σ) Magnitude Photometric System Zero Point 0.05 Zero Points, S/N 0.17 Evolution < 0.17 K-corrections 0.03 Evolution of Extinction Law 0.02 Extinction 0.10 Gravitational Lensing 0.02 Light curve sampling 0.15 PROSPECTS TOWARDS 1% ERROR IN d L (z) in a space mission Systematic Uncertainties (1σ) Magnitude Statistical Uncertainties (1σ) Magnitude Photometric System Zero Point a 0.01 Zero Points, S/N 0.00* Evolution 0.00* K-corrections 0.00* Evolution of Extinction Law < 0.02 Extinction 0.00* Gravitational Lensing < 0.01 Light curve sampling 0.00* a 0.00* means values of the order of 10 -3 . The statistical errors are for the redshift bin. b In the Table for prospects in a space mission, the statistical error from light curve sampling of SNe Ia is reduced due to the large number of SNe Ia data per redshift bin.\n\nGround-based projects will not be able to achieve such accuracy. They aim at a reduction of systematic errors to 0.04-0.05 mag. Much of what will be feasible from now until the next space era of determination of the equation of state, depends on the use of all the information already gathered in the best way. Spectra gathered so far can be exploited further than what it has been done up to now, in order to reduce systematic errors. In the coming years, a lot can be done to improve the present dispersion of the Hubble diagram of SNe Ia and aim at a scatter of 1% in d L . The above limit is only reachable if all the systematic effects are dealt with to the best.\n\nThe SNe Ia test of w(z) benefits from using information on Ω M obtained by some other method. This can come from weak lensing or from large scale structure [147, 148, 149] .\n\nBaryon acoustic oscillations can also provide a measurement of distance along z, in this case, angular distances d A (z) and of E(z) = H(z)/H 0 [27, 150, 151] . Generally, for any geometry, one has:\n\nA = √ Ω M E(z) 1 3 1 z |Ω K | S |Ω K | z 0 dz ′ E(z ′ ) 2 3 ( 45\n\n)\n\n23 A measurement of d A has been obtained [27] .\n\nA = √ Ω M E(z 1 ) 1/3 1 z 1 z 1 0 dz E(z) 2/3 = 0.469 ± 0.017 ( 46\n\n)\n\nwhere z 1 = 0.35 is the redshift at which the acoustic scale has been measured in the redshift sample. Combining BAO and SNeIa distances one can aim at tracing dark energy all through a high z range. The value of w(z = z rec ) is provided by WMAP and Planck. In between, other methods such as studying the growth of large scale structure (LSS) obtained from Lyα clouds and clusters would depict the growth factor of structure along z. Considering the information at high z derived from LSS, SNe Ia and CMB should lead to learn about early dark energy [152, 153, 90, 148, 154, 155] .\n\nType Ia supernovae can test GR against other alternative theories of gravitation. Theories of higher dimensional gravity have different Friedmann equations from those originating from GR. Some of those modified gravity proposals have already been tested and ruled out through the Hubble diagram of SNeIa up to very high z.\n\nIf dark energy is due to a scalar field either minimally coupled to gravity (i.e. quintessence) or non-minimally coupled to gravity (arising from scalar-tensor gravity or the effective low-energy superstring proposals), the Hubble diagram up to z > 1 obtained by the supernova collaborations gives the opportunity for exhaustive tests. In what concerns scalar-tensor gravity, supernovae are a precision experiment in a way analogous to the observations of the PSR B1913+16 or to solar system tests.\n\nMoreover, supernovae have already provided hints on the form of the potential of the hypothetical scalar field (for the case of minimally coupled field) that could be associated to the acceleration of the Universe.\n\nNonetheless, GR plus a form of cosmological constant is not ruled out observationally. Finding out how is w close to z = 0 is one of the most intriguing questions. Hopefully, this will be clarified by the samples to come. The goal of obtaining a consistent picture on w(z) up tp z ∼ 1.5 from the present samples is approached through the various SN procedures used by the existing collaborations.\n\nUltimately, supernovae together with complementary methods just recently discovered shall select the correct ideas on dark energy and gravity. If any modified gravity theory has a cosmological effect, those methods should start to see it.\n\nAcknowledgements I would like to thank Alex G. Kim and Marek Kowalski for reading the manuscript and pointing to useful suggestions. This work has been supported by grant AYA2006-05369.\n\n24 References [1] Riess A G et al. (The High Z SN Search Collaboration) 1998 Astron. J. 116 1009 [2] Perlmutter S et al.(The Supernova Cosmology Project) 1999 ApJ 517 565 [3] Einstein A 1917 Sitzungsber. Preuss. Akad. Wiss. Berlin 142-152 [4] Brans C H and Dicke R H 1961 Phys. Rev 124 925 [5] Will C 1993 Theory and experiment in gravitational physics, Cambridge (London) [6] Green M B, Schwarz H and Witten E. 1988 Superstring Theory (Cambridge University Press, Cambridge) [7] Arkani-Hamed N, Dimopoulos N and Dvali G 1998 Phys. Lett. B 429 263 (Preprint hep-ph/9807344) [8] Dvali G Gabadadze G and Porrati M 2000 Phys Lett B 485 208 [9] Deffayet C Dvali G and Gavadadze G 2002 Phys. Rev. D 65 044023 [10] Adelberger E G, Heckel B and Nelson A E 2003 Ann. Rev. Nucl. Part. Phys. 53 77 [11] Adelberger E G, Heckel B and Hoyle C D 2005 Physics World 18 41 [12] Adelberger E G et al. 2007 Phys. Rev. Lett. 98 131104 (Preprint hep-th/0611223) [13] Kneller J P and Steigman G 2003 Phys Rev D 67 063501 [14] Koyama K and Maartens R 2006 JCAP 601 016 [15] Maartens R 2004 Living Rev. Rel. 7 7 [16] Spergel D N et al. (WMAP Collaboration) 2003 ApJ Supp 148 175 (Preprint astro-ph/0302209) [17] Sahni V and Starobinsky A 2006 Int. J. M. Phys D 15 2105 (Preprint astro-ph/0610026) [18] Calcagni G, de Carlos B and De Felice A 2006 Nucl. Phys. B 752 404 (Preprint hep-th/0604201) [19] de Felice A Hindmarsh M and Trodden M 2006 JCAP 0608 005 (Preprint astro-ph/0604154) [20] Gorbunov D, Koyama K and Sibiryakov S 2006 Phys. Rev. D 73 044016 (Preprint hep-th/0512097) [21] Padilla A 2007 (Preprint hep-th/0610093) [22] Trodden M 2006 in From Quantum to Cosmos: Fundamental Physics Research in Space, Washington (Preprint astro-ph/0607510) [23] Lukas A, Ovrut B.A., Stelle K.S and Waldram D 1999 Phys. Rev. D 59 086001 (Preprint hep-th/9803235) [24] Binetruy P, Deffayet C and Langlois D 2000 Nucl. Phys. B 565 269 (Preprint hep-th/9905012) [25] Fairbairn M and Goobar A 2005 Phys. Lett. B 642 432 (Preprint astro-ph/0511029) [26] Maartens R and Majerotto E 2006 Phys. Rev. D 74 023004 (Preprint astro-ph/0603353) [27] Eisenstein D J et al. 2005 ApJ 633 560 [28] Wang Y and Mukherjee P 2006 Astrophys. J 650 1 (Preprint astro-ph/0604051) [29] Riess A G et al. (the Higher-Z Team) 2004 ApJ 607 665 [30] Astier P et al. (The SNLS Collaboration) 2006 A & A 447 31 (Preprint astro-ph/0510447) [31] Dvali G and Turner M 2003 Fermilab-Pub-03-040-A (Preprint astro-ph/0301510) [32] Randall L and Sundrum R 1999 Phys. Rev. Lett 83 3370 [33] Randall L and Sundrum R 1999 Phys. Rev. Lett 126 207 [34] Binetruy P, Deffayet C, Ellwanger U and Langlois D 2000 Phys. Lett. B 477 285 [35] Langlois D 2002 Int J. Mod Phy A 17 2701 (Preprint gr-qc/0205004) [36] Brax P and van de Bruck C 2003 Class. Quan. Grav. 20 201 (Preprint hep-th/0303095) [37] Easson D A 2000 Int. J. Mod. Phys. A 16 4803 [38] Durrer R 2005, hep-th/0507066 [39] Weller J and Albrecht A 2001 Phys. Rev. D 86 1939 [40] Ratra B and Peebles P.J.E. 1988 Phys. Rev D 37 3406 [41] Wetterich C 1988 Nucl. Phys. B 302 668 [42] Steinhardt P J, Wang L and Zlatev I 1999 Phys. Rev D 59 123504 [43] Peebles P J E and Ratra B 2003 Rev. Mod. Phys 75 559 [44] Copeland E J, Sami M and Tsujikawa S 2006 Int. J. Mod. Phys. D 15 1753 (Preprint hep/th/063057) [45] Uzan J.-P 2003 Rev. Mod. Phys 75 403 [46] Bertolami O and Martins PJ 2000 Phys. Rev. D 61 064007 25 [47] Li C Holz D and Cooray A 2006, astro-ph/0611093 [48] Huterer T and Turner M S 1999 Phys. Rev. D 60 1301 [49] Huterer T and Turner M S 2001 Phys. Rev. D 64 3527 [50] Sahlén M Liddle A R and Parkinson D 2007 Phys. Rev D 75 023502 (Preprint astro-ph/0610812) [51] Knop R A, et al. (The Supernova Cosmology Project) 2003 Ap J 598 102 [52] Riess A G et al. (the Higher-Z Team) 2007 ApJ 659 98 [53] Brans C H 2005 in Contributions to Santa Clara 2004 1st. International Workshop on Gravitation and Cosmology (Preprint gr-qc/0506063) [54] Esposito-Farese G. 2004, in Phi in the Sky: The Quest for Cosmological Scalar Fields, AIP Conference Proceeding 736 35 [55] Amendola L 1999 Phys. Rev. D 62 043511 (Preprint astro-ph/9908023) [56] Amendola L and Quercellini C 2003 Phys. Rev. D 68 023514 (Preprint astro-ph/0303228) [57] Boisseau B. Esposito-Farese G Polarski D and Starobinsky A 2000 Phys. Rev. Lett 85 2236 (Preprint gr-qc/0001066) [58] Esposito-Farese G and Polarski D. 2001, Phys. Rev. D 63, 063504 [59] Perivolaropoulos L 2005 JCAP 0510 001 [60] Teyssandier P and Tourrenc P 1983 J.Math. Phys 24 2793 [61] Gasperini M and Veneziano G 2003 Phys. Rept 373 1 [62] Barrow J D and Cotsakis S 1988 Phys. Lett. B 214 515 [63] Barrow J D and Maeda K 1990 Nucl. Phys B 341 290 [64] Holden D J and Wands D 2000 Phys. Rev. D 61 043506 [65] Wands D 1994 Class Quant Grav 11 269 [66] Starobinsky A A 1980 Phy Lett B 91 99 [67] Chiba T 2003 Phys. Lett. B 575 ( Preprint astro-ph/0307338) [68] Capozziello S, Carloni S. and Trosi A. 2003 Int. J. Mod. Phys. D 12 1969 (Preprint astro-ph/0303041) [69] Carroll S M, de Felice A and Trodden M 2005 Phys. Rev. D 71 023525 [70] Mena O, Santiago J and Weller J 2006 Phys. Rev. Lett 96 [71] Sotirou T P 2006 Class. Quantum Grav. 23 5117 [72] Amendola L, Polarski D and Tsujikawa S 2007 Phys. Rev. Lett 98 131302 (Preprint astro-ph/0603703) (astro-ph/0603703) [73] Antoniadis I 1992 in Proceedings of String theory and Quantum gravity, Trieste 1992 286 (Preprint hep-th/9211055) [74] Antoniadis I, Rizos J and Tamvakis K 1994 Nucl. Phys. B 415 497, (Preprint hep-th/9305025) [75] Neupane I P 2006 Class. Quant. Grav. 23 7493 [76] Gross D J and Sloan J H 1987 Nucl. Phys B 291, 41 [77] Caldwell RR 2002 Phys. Lett. B 545 23 [78] Nojiri S, Odintsov S D and Sasaki M 2005 Phys. Rev. D 71 123509 [79] Lue A and Starkman G D 2004 Phys. Rev. D 70 101501 [80] Chimento L P Lazkoz R Maartens R and Quirós T 2006 JCAP 0609 004 [81] Bouhmadi-Lopez M, Gonzalez-Diaz P F and Martin-Moruno P 2006 (Preprint gr-qc/0612135) [82] Koivisto T and Mota D F 2007 Phys. Lett. B 644 104 (Preprint astro-ph/0606078) [83] Ellis J, Kalara S, Olive K. A. and Wetterich C 1989 Phys. Lett. B 228 264 [84] Wetterich C 1994 AA 301 321 (Preprint hep-th/9408025) [85] Brax P and Martin J 2006 JCAP 0611 008 [86] Damour T Piazza F and Veneziano G 2002 Phys. Rev D 66 046007 [87] Brax P and Martin J 2006 (Preprint hep-th/0612208) [88] Hoyle CD et al. 2004 Phys. Rev. D 70, 042004 [89] Murphy T W http://physics.ucsd.edu/ ∼tmurphy/apollo/doc/multiplex.pdf [90] Doran M Robberts G and Wetterich C 2007 Phys. Rev D 75 023003 (Preprint astro-ph/0609814) [91] Witten E. 2000 Lecture at DM2000, Marina del Rey (Preprint hep-ph/0002297) 26 [92] Susskind L 2003 (Preprint hep-th/0302219) [93] Chevalier M and Polarski D 2001 Int J. Mod. Phys. D 10 213 (Preprint gr-qc/0009008) [94] Linder E V 2003 Phys. Rev. Lett 90 091301 [95] Kim A G, Linder E V, Miquel R and Mostek N 2004 MNRAS 347 909 [96] Kowal C T 1968 AJ 73 1021 [97] Filippenko A V and Riess A G 1998 Phys. Rep. 307 31 [98] Pskovskii Y P. 1977 Soviet Astron. 21, 675 [99] Phillips M M 1993 ApJ 413 L105 [100] Phillips M M et al. 1999 AJ 118 1766 [101] Sullivan et al. (The Supernova Cosmology Project) 2003 MNRAS 340 1057 [102] Nugent P, Kim A and Perlmutter S 2002 PASP 114 803 [103] Leibundgut B 2002 ARAA 39 67 [104] Matheson T. et al. 2005 (ESSENCE) AJ 129 2352 [105] Hook I M et al (The Supernova Cosmology Project) 2005 AJ 130 2788 [106] Lidman C et al. (The Supernova Cosmology Project) 2005 A A 430 843 [107] Blondin S et al. (ESSENCE) 2006 AJ 131 1648 [108] http://www.am.ub.es/ bcnde/DARKENO.html [109] Riess A G, Press W H and Kirshner R P 1995 ApJ 438 L17 [110] Jha S Riess A G and Kirshner R P 2007 ApJ Astrophys. J 659 122 [111] Perlmutter S et al. (The Supernova Cosmology Project) 1998 Nature 391 51 [112] Goldhaber G et al. (The Supernova Cosmology Project) 2001 ApJ 558 359 [113] Tonry J.L. et al. (the High Z SN Search Collaboration) 2003 ApJ 594 1 [114] Barris B J et al. (The High Z SN Search Collaboration) 2004 ApJ 602 571 [115] Clocchiatti A et al. (The High Z SN Search Collaboration) 2006 ApJ 642 1 [116] Freedman W L et al. 2001 ApJ 553 47 [117] Sandage A et al. 2006 Astrophys. J 653 843 (Preprint astro-ph/0603647) [118] Ruiz-Lapuente P 1997 Astr. Soc. Pac. Conf. Ser. 126 207 (Preprint astro-ph/9710019) [119] Ruiz-Lapuente P 1996 Ap. J. Lett 465 83 (Preprint astro-ph/9604044) [120] Goobar A and Perlmutter S 1995 ApJ 450,14 [121] Garnavich P M et al. 1998 ApJ 509 74 [122] Wood-Vasey W M et al. (ESSENCE) 2007 (Preprint astro-ph/0701041) [123] Branch D, Romanishin W and Baron E 1995 ApJ 465 73 (Preprint astro-ph/9510071) [124] Schlegel D. J., Finkbeiner M and Davis M. 1998 ApJ.S 500 525 [125] Perlmutter S 2005 HST Proposal-GO 10496 [126] Aldering G, Knop R and Nugent P 2000 AJ 119 210 [127] Conley A. et al. (The SNLS Collaboration) 2006 AJ 132 1707 [128] Ivanov V D, Hamuy M, and Pinto P A 2000 ApJ 542 588 [129] Quimby R et al. (The Supernova Cosmology Project) 2002 BAAS 201 2305 [130] Kim A G, Goobar A and Perlmutter S. 1996 PASP 108, 190 [131] Amanullah R, Mörtsell E and Goobar A 2003 A & A 397 819 [132] Jonsson J, Dahlen T, Goobar A, Gunnarson C, Mortsell E and Lee K 2006 ApJ 639 991 [133] Nesseris S and Perivolaropoulos L 2007 (Preprint astro-ph/0612653) J. Cosmol. Astropart. Phys. 0702 025 [134] Ruiz-Lapuente P et al. 2006 http://www.am.ub.es/ bcnde/DARKENO.html Ruiz-Lapuente P 2007 Invited review at Bernard's Cosmic Stories, Valencia June 2006 [135] Rubin D et al. (The Supernova Cosmology Project) 2006 BAAS 209 9003R [136] Kowalski et al. (The Supernova Cosmology Project) 2007, in preparation [137] Buchert T, Larena J and Alimi J-M 2006 Class Quant. Grav 23 6379 [138] Bonvin C, Durrer R and Kunz M 2006 Phys. Rev. Lett 96 191302 [139] Discussions on inhomogeneous cosmologies at the conference Key Approches to Dark Energy Barcelona 2006. This author thanks all the participants for addressing this topic.\n\n27 [140] Rasanen S 2006 JCAP 0611 003 [141] Kolb E W, Matarrese S, Notari A and Riotto A 2005 (Preprint astro-ph/0506534) [142] Célérier M-N 2000 Astron. Astrophys 353 63 [143] Buchert T 2007 gr-qc/0612166 [144] Maor I and Brustein R 2003 Phys. Rev D 67 103508 [145] España-Bonet C and Ruiz-Lapuente P 2005 (Preprint hep-ph/0503210) [146] http://snap.lbl.org [147] Heavens A F, Kitching T D and Taylor A N 2006 MNRAS 373 105 [148] Tegmark M et al. 2006 Phys. Rev. D 74 123507 (Preprint astro-ph/0608632) [149] Percival W J et al. 2007 Astrophys. J 657 645 (Preprint astro-ph/0608635) [150] Nichol R 2006 in Cosmology, Galaxy Formation and Astroparticle Physics on the Pathway to SKA (eds Klöckner H-R et al.) [151] Yamamoto K Bassett B A Nichol R C and Suto Y 2006 Phys Rev D 74 063525 (Preprint astro-ph/0605278) [152] Frieman J A , Huterer D, Linder E V and Turner M S 2003 Phys. Rev. D 67 083505 (Preprint astro-ph/0208100) [153] Wang Y and Freese K 2006 Phys. Lett. B 632 449 [154] Dunkley J, Bucher M, Ferreira P G, Moodley K and Skordis C 2005 Phys. Rev. Lett. 95 261303 [155] Kunz M, Trotta R and Parkinson D 2006 Phys. Rev. D 74 023503"},"sections":{"kind":"list like","value":[{"section_type":"OTHER","section_title":"Untitled Section","text":"2 Contents 1. Introduction 2. Gravity and extra dimensions 3. The dark energy field and scalar-tensor gravity 4. The cosmological constant? 5. The d L test from high-z SNe Ia 6. Testing the adequacy of the FRW metric 7. Next decade experiments 8. Complementarity 9. Conclusion 3 1. Introduction\n\nThe discovery of the acceleration of the expansion of the universe [1, 2] has revived questions concerning the domain of validity of Einstein's General Relativity. At present, the data gathered on the expansion rate do not disclose whether the acceleration is due to a component formally equivalent to the cosmological constant introduced by Einstein [3], or whether we are finding the effective behavior of a theory of a wider scope whose low energy limit slightly departs from GR. The bare Einstein's equations (without cosmological constant) for a Universe with a FRW metric and dust-like matter imply a continuous deceleration of the expansion rate. However, a Universe containing a fluid with an equation of state p = wρ with index w < -1/3 overcomes the deceleration when the density of this fluid dominates over that of the dust-like matter. In this context, the cosmological constant, if it is positive and added to the equations, balances the deceleration by acting as a fluid with an equation of state p = -ρ.\n\nThe cosmological constant was originally in the metric part of Einstein's equations, and alternative metric theories of gravity provide a wealth of terms in that side. When describing a homogeneous and isotropic universe within a FRW model, it is frequent to come as close as possible to the standard Friedmann-Einstein equation for the expansion rate by bringing terms to the right-hand side within the so-called \"effective index\" w eff of the cosmic fluid whose equation of state is p = w eff ρ.\n\nIn the field of gravitation, ideas that appeared in the classical domain several decades ago, are now being brought back within the different context of quantum gravity theories. Scalar-tensor theories of gravitation are being reconsidered. Scalar fields coupled to gravity were proposed as a way to incorporate Mach inertial induction. They gave rise to the Brans-Dicke theory of gravitation [4] , which is well restricted by solar system experiments [5] . Superstring theory and M-theory introduce a number of quantum scalar fields, the dilaton, the radion, the moduli [6] . They can play a role in the late acceleration of the Universe. On the other hand, brane worlds inspired by M-theory provide modifications to gravity and new equations of cosmic expansion [7] . In the brane world context, gravity lives in the bulk of 3+1+d dimensions while the rest of the interactions are confined to the 3+1 brane. Some braneworld proposals do not explain the observed cosmic expansion. However, it was realized [8, 9] that it is possible to obtain late-time acceleration with one extra dimension. This possibility is interesting as it accounts for the observed cosmic expansion without the need of a cosmological constant term. Acceleration results from the induced gravity term in the brane. The number and size of additional dimensions has been explored in a wide range of proposals. The existence of additional dimensions and their size are constrained by tests of the gravity law [10, 11, 12] . At the time when gravity theories of the scalar-tensor type and others were examined, the solar system tests were the most restrictive ones. Nowadays, the theories have to confront the cosmological observations. The theories have to allow Big Bang 4 nucleosynthesis [13] . They have to fullfil the constraints from the growth of density perturbations and give correct predictions for the evolution of the growth factor. Such growth function of density perturbations requires careful evaluation for several modified gravity scenarios [14, 15] . Ultimately, gravity theories have to be consistent with the CMB data and with the expansion rate of the Universe provided by supernovae. Supernovae enable to measure the rate of the expansion of the Universe along redshift z. They do so by tracing the luminosity distance along z for a standard candle. If the Universe is flat, as inferred from WMAP [16] , the d L relation is simply:\n\nd L (z) = cH -1 0 (1 + z) z 0 dz ′ H(z ′ ) ( 1\n\n)\n\nwhile in the general case,\n\nd L (z) = cH -1 0 (1 + z)| Ω K | -1/2 sinn | Ω K | 1/2 z 0 H 0 dz ′ H(z ′ ) ( 2\n\n)\n\nwhere sinn(x) = sin(x), x, or sinh(x) for closed, flat, and open models respectively, where the curvature parameter Ω K , is defined as Ω K = Ω T -1. H(z) is the Hubble parameter. For a FRW universe containing matter and a general form of matter-energy X with index of the equation of state w X , H(z) is given by:\n\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω M (1 + z) 3 + Ω X (1 + z) 3(1+w X ) ( 3\n\n)\n\nwhere H 0 is the present value of the Hubble parameter and\n\nΩ K ≡ -k H 2 0 a 2 0 ( 4\n\n)\n\nwhere k accounts for the geometry and a 0 is the present value of the scale factor.\n\nΩ M = 8πG 3H 2 0 ρ M ( 5\n\n) Ω X is the density parameter for dark energy.\n\nWhen w(z) varies along z, H(z) can be expressed as:\n\nH(z) 2 = H 2 0 Ω K (1 + z) 2 + Ω M (1 + z) 3 + Ω X exp 3 z 0 1 + w(z ′ ) 1 + z ′ dz ′ ( 6\n\n)\n\nOne sees from (1-2), that d L contains w(z) through a double integral. This leads to limitations in recovering w(z) if d L data along z are scarce and with large errorbars. Data are often given as m(z) which relates to d L (z) as:\n\nm(z) = 5 logd L (z) + [M + 25 -5 log(H 0 )] ( 7\n\n)\n\n5 The quantity within brackets is M ≡ M -5 log H 0 + 25, the \"Hubble-constant-free\" peak absolute magnitude of a supernova. The data are given sometimes in the form of distance moduli:\n\nµ = 5 log d L + 25 ( 8\n\n)\n\nThe measurement of w(z) through d L (z) is the standard approach, though possibilities of measuring directly H(z) have been investigated (for a review see [17] ). This Hubble law in (3) is derived from the Einstein-Hilbert action. We will review other possibilities related to expansion histories arising from modified gravity theories and effective theories which contain scalar fields accelerating the cosmic expansion. In the present overview we concentrate on the empirical evaluation of those theories. Some of the models to be explored here might contain a ghost problem or have a subclass amongst them that present quantum instabilities. We refer for this topic to recent updates in [18, 19, 20, 21, 22] ."},{"section_type":"OTHER","section_title":"Gravity and extra dimensions","text":"The idea of allowing gravity to live in extra dimensions arose to explain the hierarchy problem and constitutes one of the major challenges to the classical picture of a 4dimensional description of gravity [7] . The Planck scale of 10 19 GeV, relevant for gravitation, is much larger than the electroweak scale, of about 1 TeV of the standard model of particle physics. According to the proposal by Arkani-Hamed, Dimopoulos and Dvali [7], the Planck scale is not a fundamental scale, but its enormity is simply a consequence of the large size of new extra dimensions. While gravitons can propagate in the new extra dimensions, the standard model fields are confined to our submanifold of 3 + 1 dimensions, the brane, which is embedded in a higher dimensional bulk. In this scenario, the Planck scale in the bulk can be much smaller than in the 4dimensional brane as\n\nM 2 P,4 = R n M n+2 P,n+4 ( 9\n\n)\n\nThe existence of extra dimensions in which gravity propagates produces a transition in the gravitation law at the compactification radius, proposed to be of the submillimetre scale.\n\nIn the cosmological context, this idea has some interesting consequences since it predicts a departure from GR. As soon as the cosmological model was implemented for n = 1 extra dimensions [23, 24, 9, 8] , it was noted that the brane cosmology leads to Friedmann-like equations very different from the standard ones [24] . In the 5dimensional model by Dvali, Gabadadze and Porrati (DGP) whose cosmology is explored by Deffayet and collaborators in [9] , the competition between the five-dimensional term of the Einstein-Hilbert action and the four-dimensional term induced in the brane leads to a modified Friedmann equation which simulates an acceleration term without the need to include a cosmological constant. The transition from the gravity law r -2 to 6 the modified one is characterized by a scale r c , the so-called cross-over scale defined as [9]:\n\nr c ≡ M 2 P 2M 3 ( 5\n\n) ( 10\n\n)\n\nAt large distances, the five dimensional term takes over and gravity gets weaker [9] . If the parameter r c is chosen to be of the order of H 0 -1 ∼ 10 29 mm, this is a choice for M (5) ∼ 10-100 MeV. At short distances r << r c , Newton's law is modified by a logarithmic repulsion. At large distances r >> r c , the modification involves a factor r rc . The Hubble expansion law shows significant departures from the standard Friedmann expansion law:\n\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω rc + Ω rc + α Ω α (1 + z) 3(1+wα) 2 ( 11\n\n)\n\nwhere the sum over α stands for a sum over each component with equation of state w α , Ω α being the density parameter:\n\nΩ α ≡ ρ 0 α 3M 2 Pl H 2 0 a 3(1+wα) 0 ( 12\n\n)\n\nand\n\nΩ rc ≡ 1 4r 2 c H 2 0 ( 13\n\n)\n\nIn the matter-dominated epoch, the expansion equation is:\n\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω rc + Ω rc + Ω M (1 + z) 3 2 ( 14\n\n)\n\nIn the original 5D model in [8] , M 5 was estimated to be ∼ 1 TeV, which corresponds to a cross-over scale r c of 10 15 cm. Such r c led to incompatibilities with the solar system data. When M 5 is taken to be M 5 < ∼ 1 GeV, i.e. r c > ∼ 10 25 cm, the conflict disappears. High-z supernovae give useful constraints to the parameter Ω rc in (10) [25, 26] . In [26] , two samples of high-z supernovae are used in conjunction with measurements from baryon oscillations [27] and the CMB shift parameter [28] to estimate the favored values for Ω rc and Ω M . Ω rc = 0.125 for the supernova data in the Riess sample from 2004 [29] and Ω rc = 0.13 for the SNLS data [30] . It is found in [26] that, for a flat Universe, this 5D modified gravity model gives a worse fit than GR plus cosmological constant. Further supernova samples would reexamine the empirical compatibility of the DGP model and other modified gravity models involving more dimensions. Those models can be explored through a phenomenological description. The Hubble law arising from gravity with extra dimensions can be expressed by adding H α terms to the standard Friedmann equation [31] :\n\nH 2 - H α r 2-α c = 8πGρ M 3 ( 15\n\n)\n\n7 where r c is the cross-over radius, i.e. the transition radius from a regime to another in the gravity law, and α is the parameter to be empirically determined. From redshift z= 0 to z= 2, the above Friedmann equation gives an effective index of the equation of state [31]: w eff ≈ -1 + 0.3α (16) Such effective equation of state evolves too fast to be compatible with present SNe Ia data, unless α is small. Present data favor α close to 0 [25]. Though the most simple implementations lack empirical confirmation, room is left for several models. Cosmological SNe Ia are useful tests in restricting the cross-over radius that relates to departures from GR.\n\nA different kind of model in which the brane is embedded in a 5D bulk has been proposed by Randall and Sundrum [32, 33] . Contrary to the previous approach, here the bulk has a AdS 5 geometry as opposed to the flat geometry of the DGP model. In the original Randall-Sundrum (R-S) model, the brane has tension σ, i.e. cosmological constant, as well as the bulk Λ 5 . The bulk cosmological constant is negative. The resulting Hubble expansion in the brane is the result of the compensating effect of both terms. To make zero the effective cosmological constant in the brane, Λ 4 , a fine tuning between the brane tension and the bulk cosmological constant is required. The original proposal sets the effective cosmological constant, Λ 4 , to zero. Generally, a tiny deviation from exact compensation [34] would lead to an effective cosmological constant at late times. Modifications of the original R-S proposal include in the brane, in addition to tension (cosmological constant), a matter-energy density ρ. The total brane density ρ B is:\n\nρ B = σ + ρ ( 17\n\n)\n\nThis gives the modified Friedmann equation [35] :\n\nH 2 = κ 4 36 σ 2 - 1 l 2 + κ 4 18 σρ + κ 4 36 ρ 2 + C a 4 ( 18\n\n)\n\nwhere κ is the inverse Planck mass M -1 P and l is related to Λ 5 . Generalizations of the Randall-Sundrum scenario in cosmology consider the AdS 5 bulk geometry containing the fifth-dimensional cosmological constant Λ 5 , but explore arbitrary densities in the brane and in the bulk.\n\nExplorations of braneworld ideas that allow a late acceleration of the Universe can be found in [36, 37, 38] .\n\n3. The dark energy scalar field and scalar-tensor gravity"},{"section_type":"OTHER","section_title":"Early developments","text":"Before the development of braneworld scenarios, dark energy as associated to a scalar field was considered in analogous way to research done within the context of inflation 8 [40] . This theoretical possibility was entertained a decade before there was evidence of a non-zero cosmological constant from supernovae and that the CMB results indicated a flat Universe. The favored Ω T = 1 value by inflation and the evidence that Ω M < 1 led to think about a possible Λ term. In the recent expansion history a scalar field could play a role in analogy with inflation, where a scalar field dubbed the inflaton gives rise to the exponential phase of early acceleration. The scalar field responsible for the late acceleration, later named quintessence, is proposed as a new massless scalar field which induces a cosmological constant behavior at late times. In this original setting, the field is minimally coupled to gravity, i.e., thus the inclusion of the field remains within the framework of GR. The interaction of the field with the matter fields is set to be negligible. The scalar field responsible for dark energy can be identified by recovering its potential, or equivalently its effective equation of state. The presence of such scalar field φ, would induce an action:\n\nS = d 4 x √ -g 1 2 g µν ∂ µ φ∂ ν φ -V (φ) ( 19\n\n)\n\nThe field contributes to the stress-energy momentum tensor with an effective mass density and pressure:\n\nρ φ = 1 2 φ2 + V (φ) p φ = 1 2 φ2 -V (φ) ( 20\n\n)\n\nIn this case, the effective value of the equation of state w eff depends on the form of the potential V (φ) and it can evolve with time [39] .\n\nw eff = p φ ρ φ ( 21\n\n)\n\nA search for general properties of this potential, in analogy to the search for the inflation potential, leads to suggestions for potentials with tracking behavior [40, 41, 42, 43] . Several potentials have been proposed, from the original V = κ/φ α [40] to other recent proposals [44] . The dark energy scalar field that does not couple with matter has the feature of avoiding effects on variations of constants and other effects subject to experimental evidence [45, 46] . There is already a reconstruction of what could be the potential of the minimally coupled field V (φ) obtained from the SNLS SNe Ia [47] following reconstruction ideas given in [48, 49] . Another reconstruction of V (φ) using SNe Ia from various collaborations and considering Padé approximant expansions of the potential is found in [50] . By now, many proposals have been discarded, since the equation of state between z= 0 and z= 1.5 evolves very slowly as shown by high-z SNe Ia data [2, 51, 29, 52] . 9 3.2. Scalar-tensor gravity One can still consider the case of a scalar field that is coupled to other matter fields or to gravity. Some of these proposals come from the ballpark of superstring theory. The most well-known scalar-tensor gravity proposal, i.e. Brans-Dicke theory, incorporated the coupling of a field to gravity to account for the Mach induction on Newton's constant. In Brans-Dicke gravity [53, 4] , Newton's constant is replaced by a dynamical field G leading to the generalized Friedmann equation:\n\nH 2 = 8π 3 ω 2πφ 2 ρ + 1 2 φ2 + V (φ) - 3 2ω Hφ φ ( 22\n\n)\n\nThe Brans-Dicke action contains a φ 2 R term:\n\nS = √ -g d 4 x ± 1 8ω φ 2 R ∓ 1 2 g µν φ ,µ φ ,ν -V (φ) + L matter ( 23\n\n)\n\nThe theory has been well tested [5]. Observationally, for other scalar-tensor theories, we aim at retrieving the self-interaction potential of the scalar V (φ). and the coupling function to matter and gravity A(φ) in the matter term of the action [54]:\n\nS matter (ψ, A 2 (φ)g µν ) ( 24\n\n)\n\nIn the case of quintessence (minimally coupled field) A(φ) = 1. The present supernova data have explored the potential for a minimally coupled field [47] . In the non-minimally coupled case, also referred to as coupled quintessence or coupled dark energy [55, 56] , the CMB data can put constraints on the strength of scalar gravity as compared to ordinary tensorial gravity [55] . However, there are limitations to the recovery of the form of the coupling and the coupled dark energy potential from CMB observations [55] . Supernovae together with observational constraints on the growth of density perturbations would allow to retrieve A(φ) [57, 58] . We expand A(φ) in its derivatives α 0 , β 0 and higher orders:\n\nln A(φ) ≡ α 0 (φ -φ 0 ) + 1 2 β 0 (φ -φ 0 ) 2 + O(φ -φ 0 ) 3 ( 25\n\n)\n\nBy exploring the empirical evidence it is found [54] that solar system tests and binary pulsar tests allow to impose precise bounds on the first and second derivatives of the matter-scalar coupling function, while SNe Ia could \"a priori\" allow to reconstruct the full shape of the function A(φ) or the higher order derivatives. It is argued in [57, 58, 59] that the knowledge of the luminosity distance and the density fluctuations δρ/ρ as functions of redshift z is sufficient to reconstruct the potential V (φ) and the coupling function A(φ).\n\nThese results can be generalized to proposed actions that can be re-expressed as a theory of interacting matter fields in general relativity [60] . Actions containing e -Φ R are motivated from the lowest order effective action including the dilaton Φ [61] . They can be brought by redefinition of the field to an Einstein frame in which the field is minimally coupled to the metric.\n\n10 A number of interesting cases have been examined and have been expressed in terms of a scalar field which is minimally coupled to the metric, but nonminimally coupled to the other fields [62, 63, 64] . The self-interaction potential shows an explicit energy transfer between the scalar field and the matter fields. Related to the previous exploration is that of f(R) gravity or extended curvature gravity [65], also known as \"c-essence\" when including an inverse power of R [67] . Those modified gravity theories with a Lagrangian containing an arbitrary function of R are equivalent to a particular class of scalar-tensor theories of gravity. For instance, the action\n\nS = 1 2κ 2 d 4 x √ -g R - µ 4 R + S matter (g µν ) ( 26\n\n)\n\nis found to be equivalent to [67]:\n\nS = 1 2κ 2 d 4 x √ -g 1 + µ 4 φ 2 R - 2µ 4 φ + S matter (g µν ) ( 27\n\n)\n\nThe equivalence is generalized to an arbitrary function of R [60, 65]:\n\nS = 1 2κ 2 d 4 x √ -g F (R) + S matter (g µν ) ( 28\n\n)\n\nwith an equivalent action [60, 65]:\n\nS = 1 2κ 2 d 4 x √ -g (F (φ) + F ′ (φ)(R -φ)) + S matter (g µν ) ( 29\n\n)\n\nwhere F ′ (φ) = dF/dφ. The introduction of a canonical scalar field ϕ such that F ′ (φ) = exp( 2/3 κϕ), allows to rewrite the action with F(R) as:\n\nS = d 4 x √ -g E 1 2κ 2 R E - 1 2 (∇ϕ) 2 -V (ϕ) + S matter (g E µν /F ′ (φ(ϕ)), V (ϕ) = (φ(ϕ)F ′ (φ(ϕ)) -F (φ(ϕ)))/2κ 2 F ′ (φ(ϕ)) 2 ( 30\n\n)\n\nThe gravity described by g E µν is the Einstein-scalar system and g µν (= g E µν /F ′ (φ)). g µν is a metric of scalar-tensor gravity which is, in this case, subject to constraints coming from solar system experiments. Again, a classification of when one recovers an action equivalent to the Einstein-Hilbert case and when one recovers a Brans-Dicketype action in f(R) gravity is given in [60] . When considering a general function of R, it is not always possible to obtain a description equivalent to Einstein GR plus additional fields minimally coupled to the metric but coupled to other fields. When this is possible, we have in the matter term the coupling between the matter fields ψ and the scalar field φ. Examples of f(R) casted in terms of fields can be found in [62, 64, 66, 67] . For the Starobinsky model [66] with F (R) = R + R 2 /M 2 with M ∼ 10 12 GeV, for instance, the effective potential can be rewritten, in terms of the scalar field ϕ, as\n\nV (ϕ) = M 2 e -2 √ 2/3 κϕ 8κ 2 (e √ 2/3 κϕ -1) 2 ( 31\n\n)\n\n11 Actions with generalized inverse powers of R have been investigate as well [68, 69, 70, 71]. The supernovae gathered up to now are compatible with some f(R) gravity models and incompatible with others [68, 70, 72]. More general scalar-tensor theories than the ones here defined, such as those where the potential depends on several scalar fields V (φ n ) and matter-scalar coupling A(φ n ), give rise to a wider phenomenology. In the superstring scenario, nonminimal coupling of the scalar fields to the spacetime curvature leads to effective actions with similarities to scalar-tensor proposals. Some scalars present in the effective action arise from compactification to four dimensions.\n\nThose are structure moduli which do not couple directly with the spacetime curvature tensor. One modulus field σ associated with the overall size of the internal compactificaton couples to Einstein gravity via Riemann curvature invariants, such as the Gauss-Bonnet invariant [74] . If one considers the Gauss-Bonnet term, at least an extra function, the coupling to the Gauss-Bonnet term, W (σ) needs to be determined.\n\nIn the following, we review attempts to include such enlarged action and their test with cosmological d L data."},{"section_type":"OTHER","section_title":"Effective low energy action with R GB and other terms","text":"The Gauss-Bonnet term can be found in the effective low energy string Lagrangian and in brane theories [73, 74, 75, 35, 76] . The case of the simplest action containing the Gauss-Bonnet term can be written as:\n\nS = √ -g R 2κ 2 - 1 2 (∂ µ φ) 2 - √ -g W (σ) R 2 GB + S matter [matter ; g µν ] ( 32\n\n)\n\nThis Lagrangian includes the Gauss-Bonnet term coupled to a field φ ≡ σ, for instance the modulus field in [74] where W (σ) is the coupling of the modulus field to the Gauss-Bonnet term, R GB .\n\nR 2 GB ≡ R µνρσ R µνρσ -4R µν R µν + R 2 ( 33\n\n)\n\nThe possible coupling of a scalar field to the Gauss-Bonnet topological invariant can be constrained if one takes into account cosmological supernovae and solar system tests.\n\nFrom supernova data gathered in [29] and with a parameterized analysis of the evolution, it is found that the present value of the equation of state w 0 is below -1. Though for the measurement of the index of the equation of state at z = 0 one awaits a large enough nearby supernova sample, the results presented in [29, 52] move to examine the possibility of w eff < -1. In the original exploration of the idea [77], a Universe which evolves towards w eff < -1 would suffer a super-aceleration phase, that has come to be called the Big Rip. From an action which includes the Gauss-Bonnet term, one can obtain w eff < -1 for a canonical scalar field (positive kinetic energy) [78, 35] . It avoids the Big Rip for w eff < -1. To add a word on this, this is not the main motivation for 12 Gauss-Bonnet gravity, since variations of the braneworld scenario can also get w eff < -1 and avoid the Big Rip [79, 80, 81] . The interest on the GB term is justified by its presence in a variety of gravity proposals from string and M-Theory. Recently some tests using supernovae have already been done [82] . The extra GB term opens naturally many possibilities for the evolution of the scale factor. The supernova test succeeds for a range of effective low energy actions including a Gauss-Bonnet term [82]."},{"section_type":"OTHER","section_title":"Fifth force and equivalence principle constraints","text":"The fields that interact with the dark energy scalar field are constrained by a large number of tests [45] . The possibility that the dark energy scalar couples to nucleons and photons has been investigated in relation to implications for a fifth force [83, 84] . Within current scenarios of particle physics such as MSSM and MSUGRA the role of this dark energy field shows some observational signatures of the fifth force type as well [85] . For instance, the interaction of the dark energy scalar field φ with the Higgs doublet H u and H d leads to an action where the field couples differently to each of the Higgs ψ u and ψ d and one expects a violation of the weak equivalence principle. In the string scenario, where radion, dilaton and moduli can be responsible for the early and late acceleration, one has as well some possible strong couplings of the fields with matter [86, 75, 87]. Thus, the cosmological context has to be considered together with classical gravitational tests on Earth. Thus far, the torsion balance experiments set strict limits on the variation of Newton's law using different substances and locations. There is a wide range of tests being done to examine the predictions from scalar-tensor proposals and modified gravity ideas [12, 10, 11, 88, 89] . The empirical bounds on some types of couplings are tight and worth considering together with the cosmological probes."},{"section_type":"OTHER","section_title":"Cosmological constant?","text":"Finding that the negative pressure component gives w = -1, would likely be the biggest challenge to gravitation theories. At present, SNe Ia have not provided such evidence, though averaged values of w are close to -1 at redshifts up to z∼ 1. To probe that w stays -1 up to the surface of last scattering, involves refining several other methods (growth of density perturbations, integrated Sachs-Wolfe effect) [90] . From the view point of string theorists [91] , there is no natural explanation for the vanishing or extreme smallness of the vacuum energy. A way to address the small value of the cosmological constant in the context of a stringy landscape is to consider the anthropic principle [92] . However, invoking the anthropic principle as a way out of the problem entails to abandon the quest for a conventional scientific explanation [91]. . From the observational point of view, the answer is near to w = -1, but possible departures from w = -1 are presently being examined through expansions of the equation of state along the scale factor and other various approaches. A expansion 13 Figure 1. The cosmological constant case (bold line) is compared with evolving models close to w = -1, i.e. a model with w 0 = -1.0 and w a = -1.5 (short dash line) and a model with w 0 = -1.0 and w a = +1.5 (dash-dotted line) often used is [93, 94] :\n\nw(a) = w 0 + w a (1 -a) ( 34\n\n)\n\nwhere the scale factor a = (1 + z) -1 , so w ′ = w a /2 when evaluated at z = 1. Differences between the cosmological constant and departures from it in the equation of state are measurable nowadays (see Fig 1). To reach a higher level of discrimination between dark energy models depends on the progress made in the accuracy of the measurements. The aim of future projects is to determine distances with 1 % accuracy, i.e. with errors in magnitudes of 0.02 mag [95] . If the equation of state of dark energy is very close to -1, one can anticipate decades of improving the supernova measurements. The development that we might witness could be akin to that of the postnewtonian tests of GR where higher levels of precision are being targeted since the middle of the last century into the present. 14 5. The d L test from SNe Ia"},{"section_type":"OTHER","section_title":"Overview","text":"A long path has brought us to the present situation where we can already aim at accuracies of 1% in distance. Back in 1968, a first attempt to draw the Hubble diagram from SNe resulted in a spread of σ ∼ 0.6 mag [96] . Such large scatter was due to the inclusion of other types of supernovae, which were later recognized as being of a different type (SNI b/c) and characterized by a large scatter in luminosities [97] . SNe Ia are not standard candles either, but they follow a well known correlation between maximum brightness and rate of decline of the light curve. In 1977, Pskovskii [98] first described the correlation between the brightness at maximum and the rate of decline of the light curve: the brighter SNe Ia have a slower decline of their light curves whereas fainter ones are the faster decliners. A systematic follow-up of SNe Ia confirmed the brightness-decline relation [99, 100] . The intrinsic variation of SNe Ia is written as a linear relationship:\n\nM B = M B0 + α (∆m 15 (B) -β) + 5 log(H 0 /65) ( 35\n\n)\n\n∆m 15 , the parameter of the SNe Ia light curve family, is the number of magnitudes of decline in 15 days after maximum. The value of α, as well as the dispersion of that relation, have been evaluated in samples obtained from 1993 to the present [100] . M B0 is the absolute magnitude in B for a template SNIa of β rate of decline. An intrinsic dispersion of 0.11 mag in that law is found when information in more than one color is available.\n\nIn fact, the 0.11 mag scatter in the brightness-rate of decline is found when one includes supernovae affected by extinction. The scatter is smaller in environments where dust in the host galaxies of the SNe Ia is a lesser effect [101] . Additionally, the family of SNe Ia form a sequence of highly resembling spectra with subtle changes in some spectral features correlated with the light curves shapes [102, 103, 104, 105, 106, 107, 108] . The multiple correlations allow to control errors within the family of SNe Ia [108] . The parameter ∆m 15 requires to have observed the maximum in the light curve of the supernova. This is not always possible, and therefore the various SN collaborations have formulated the brightness-decline correlation in different ways. The High-Z SN Team uses the full shape of the light curve with respect to a template, method refered as MLCS2k2 [109, 110] . The Supernova Cosmology Project collaboration, on the other hand, has introduced the stretch-factor, s, as a parameter to account for the brightnessdecline relationship [111, 2, 112] . The stretch parameter fits the SNIa light curve from premaximum to up to 60 days after maximum. Thus, even if the maximum brightness phase is not observed, the whole light curve provides the decline value. More recently, other ways to measure the decline have been proposed [113, 114, 115] . Measuring the evolution of the rate of expansion of the Universe does not require knowledge of its present value, H 0 . Large projects have been devoted to the determination of H 0 by attempting the absolute calibration of M B from nearby SNe 15 Ia that occur in galaxies with Cepheid-measured distances [116, 117] . Controversy persists, though the most recent values quoted by the two collaborations using this method are closer (in the range 65-75 km s -1 Mpc -1 ) [116, 117] . Though largely ignored as an argument, the mere fact that a SNIa is a explosion of white dwarf bounds very efficiently the lower and upper possible values of H 0 [118] . On the empirical side, the emission at late phases of SNe Ia gives a precise value of H 0 , since such emission is very sensitive to the density profile, the mass of the supernova and the amount of 56 Ni synthesized in the explosion. Those quantities are easily measurable from the spectra [119] . The value favored by this method is H 0 = 68 ± 6 (syst) km s -1 Mpc -1 . This disgresion done, we emphasize the nuisance role of the absolute magnitude of SNe Ia when it comes to dark energy cosmology. The apparent magnitude of supernovae along z give us the cosmology, m B as [2, 51]:\n\nm B = M + 5 log D L (z; Ω M , Ω Λ ) -α(s -1) ( 36\n\n)\n\nwhere s is the stretch value for the supernova, D L ≡ H 0 d L is the \"Hubble-constant-free\" luminosity distance, d L is given by (1) and M ≡ M B -5 log H 0 + 25 is the \"Hubbleconstant-free\" B-band peak absolute magnitude of a s = 1 SN Ia with true absolute peak magnitude M B . Similar use of the nuisance parameter in their light curve fitting is made in [29, 52] .\n\nIn the first determinations of dark energy cosmology, there were four parameters in the fit: the matter density parameter Ω M , the cosmological constant density parameter Ω Λ , as well as the two nuisance parameters, M and α (either α itself as used by the SCP or a equivalent parameter by other collaborations). The four-dimensional (Ω M , Ω Λ , M, α) space, explored as a grid, yields χ 2 and P ∝ e -χ 2 /2 of the fit of the luminosity distance equation to the peak B-band magnitudes and redshifts of the supernovae. After normalization by integrating over the \"nuisance\" parameters, the confidence regions in the Ω M -Ω Λ plane can be obtained. This use of the magnitude-redshift relation m(z) as a function of Ω M and Ω Λ with a sample of high-z SNe Ia of different z, is a very powerful pointer to the allowed values in the Ω M -Ω Λ plane [120, 121] . By 1998, such use yielded important results, impling that Ω Λ > 0 at the 3σ confidence level. For a flat universe (Ω T = 1), the results from the Supernova Cosmology Project meant Ω M =0.28 +0.09 -0.08 (statistical) +0.05 -0.04 (systematics), and the High-Z Supernova Team obtained for a flat universe Ω M =0.24±0.1. The outcoming picture of our universe is that about 20-30% of its density content is in matter and 70-80% in cosmological constant. According to the allowed Ω M and Ω Λ values, the Universe will expand forever, accelerating its rate of expansion."},{"section_type":"RESULTS","section_title":"Results on w","text":"If the size of the SNe Ia sample and its redshift range do allow it, one can aim at determining the average value of the index of the equation w or, going a step further, aim at determining the confidence regions on the w 0 -w a plane. For this later purpose, 16 using the information on the global geometry as given by WMAP provides a helpful prior. In Knop et al. (2003) [51], the supernova results were combined with measurements of Ω M from galaxy redshift distorsion data and from the measurement of the distance to the surface of last scattering from WMAP. The confidence regions provided by the measurement of the distance to the surface of last scattering and the SNe Ia confidence regions cross in the Ω M -w plane, thus providing very good complementarity. This work considers an averaged w (given the range in z of the SNe Ia observed). From the fitting assuming w constant, the value is w =-1.06 +0.14 -0.21 (statistical) +0.08 -0.08 (identified systematics). The analysis of the SNLS [30] also aims at an average value of w, allowed by the z range of this survey. Their 71 high-z SNeIa give w =-1.023 ± 0.090 (statistical) ± 0.054 (identified systematics). Recently, the results from ESSENCE, using 60 SNe Ia centered at z= 0.5 point to a similar result with w = -1.05 +0.13 -0-12 (statistical) ± 0.13 (systematics) [122] . In this last work, the supernovae from various collaborations are joined with two separate light curve fitting approaches giving a consistent result for the average value of w. High-z SNe Ia beyond z > 1 enable to have a first look at possible evolution of w(z). The first campaign by Riess et al. (2004) gave the results in the w 0 -w ′ plane, using the Taylor expansion for w(z) as w(z) = w 0 + w ′ z. The best estimates for w 0 and w ′ are w 0 = -1.36 +0.12 -0.28 and w ′ =1.48 +0.81 -0.98 . Those results [29, 52] exclude a fast evolution in the equation of state, and therefore they rule out a number of modified gravity proposals. Moreover, the results are confirmed in [122] where the available SNe Ia at large z, which can constrain w 0 and w a , are incorporated to test with a joined sample encompassing a wide range of z. Significant limits have been placed on the evolution of w(z). Now the potential of the method relies on an improved control of systematic effects. In this area, there has been substantial progress along the last years."},{"section_type":"RESULTS","section_title":"Results on systematic uncertainties","text":"Huge advances have taken place in controlling the uncertainties due to extinction by dust, the universality of the supernova properties, the control of lensing by dark matter and the observational process leading to the the construction of the Hubble diagram of SNe.\n\nThe study with the HST of the host galaxies of the SNe Ia sample of the Supernova Cosmology Project [101] allowed to subclassify the galaxy hosts of the P99 sample [2] . It was found that, at high z, early-type galaxies show a narrower dispersion in SNe Ia properties than late-type galaxies, as they do at low z ([123]). This result is very encouraging as it supersedes previous dispersion values obtained in samples of supernovae in all galaxy types. Supernovae in dust-rich environments, 17 such as spiral galaxies, are more affected by extinction. The supernova magnitudes are corrected regularly from Galactic extinction [124], but the correction by dust residing in the host galaxy or along the line of sight is often not included in the standard fit, as it would require to have extensive color information that might not be available for high-z SNe Ia. When it is included, it is still subject to the uncertainty in the extinction law [110] . Let us, for instance, consider the correction of dimming by dust as entering in the term A B , the magnitudes of extinction in the B band. We take, for instance, a supernova of a given stretch s, and we place its m eff (the magnitude of the equivalent supernova of s = 1) in the Hubble diagram. We might use, as in [2, 51] :\n\nm eff = m B + α(s -1) -A B ( 37\n\n)\n\nThe extinction in the canonical band B of the spectrum (A B ) is determined by the extinction coefficient R B in that band. Any uncertainty in this coefficient as due to variation of dust properties, leads to uncertain estimates of A B . R B is of the order of 4.1 ± 0.5 as properties of dust might change going to high z. If we have an error of 0.1 in the observable E(B -V ) due to dust extinction, it gets amplified as:\n\nA B ≡ R B × E(B -V ) ( 38\n\n)\n\nThe present way of dealing with SNe Ia, including information in several photometric bands of the spectrum where extinction coefficients (R V , R I ) are lower and one gets smaller values in the corresponding A V and A I , allows to control the dust problem.\n\nIn fact, the coefficients R B , R V and R I are also determined by methods determining distances to SNe Ia. In a sample of 133 nearby SNe Ia, it is found in [110] , that the dust in their host galaxies is well described by a mean extinction law with R V ≃ 2.7. In all this, the level at which SNe Ia are treated, including how many bands are used, becomes critical. A use of SNe Ia without taking into account color information leads to a dispersion of 0.17mag. When taking into account color information, the limit goes down to 0.11mag. Limiting the effect of extinction, as in SNeIa in ellipticals should bring down the dispersion. This finding is motivating the ongoing SN searches in clusters of elliptical galaxies [125] . In environments where dust is controlled, such as SNe in clusters rich in elliptical galaxies, the systematic effect introduced by dust extinction can be better controlled. There, one can also control the evolution of dust-properties.\n\nIn Table I , the level of accuracy in the first uses of SNe Ia back in 1998 is shown, with the contributions of each systematic and statistical error. The improvement is given for a future mission able to control the systematics in the best possible way. It is foreseen that several sources of systematic errors will become negligible within the next decade."},{"section_type":"OTHER","section_title":"Evolution or Population drift","text":"This question resides on whether we know that SNe Ia properties evolve with z. Much has been done along this path. Nowadays, while we know that dust is the unavoidable systematics, several studies confirm the universality of the light curve properties of SNe Ia. The study of a significant sample 18 of SNe Ia at various z by the Supernova Cosmology Project reveals that the rise times to maximum of high-z SNe Ia are similar to the low-z SNe Ia ([126, 112]). The same is found in the SNLS sample ([127]). Statistical evaluation is so far consistent with no difference between the low-z and high-z samples. Moreover, the possibility of existence of an extra parameter in the maximum brightnessrate of decline relation has been carefully examined. Among possible influences, metallicity was one of the obvious ones to consider. Examining supernovae in galaxies with a gradient of metallicity, it is found [128] that there is no evidence for metallicity dependence as an extra parameter in the light curve correlation of SNe Ia. This result comes from an analysis of the SNe Ia light curves of a sample of 62 supernovae in the local Universe. The SNe Ia belong to different populations along a metallicity gradient. This check has been done as well at high-z [129] using 74 SNe Ia (0.17 < z < 0.86) from the SCP sample. No significant correlation between peak SNIa luminosity and metallicity is found.\n\nOften we address the question of the spread of the SNe Ia samples in galaxies of different types: on how many fast SNe Ia versus intermediate or slow decliners are found in the various morphological types. Spirals at low or moderate redshifts should encompass all ranges of variation in the SNe Ia properties since they contain populations with a wide spread in age. The cosmological SN collaborations find that in those samples the one-parameter correlation does give a good description of the variation of the SNe Ia light curves: there is no residual correlation after the light-curve shape correction [112] . One can go a step further and test the SNIa objects themselves. This has been attempted in a intermediate z sample and it is a feasible project at all z [108] . The physical diagrams of SNe Ia at intermediate z and at low z seem to be similar [108] . Those diagrams reveal the composition, velocity gradient and radiation properties of SNeIa [108] . Looking at the inside of the SNe Ia is the best way to test whether they are the same at all z. Such step forward is giving definitive results which reaffirm the validity of the SNe Ia method. The progress done in this domain is summarized in Table I ."},{"section_type":"OTHER","section_title":"K-corrections","text":"There is another element yet to be introduced here: it deals with the K correction. The K-correction is necessary to calculate the apparent magnitude in a y filter band of a source observed in a x filter. As we are comparing the observed flux of supernovae in a given spectral band along the expansion history, we need to take into account the effect that the photons of the supernova have shifted to redder wavelengths and spread over a different spectral wavelength range and the observed flux is different from that at emission.\n\nm Y (z) = m X (z = 0) + K xy (z) ( 39\n\n)\n\nwhere\n\nK xy = 2.5 log (1 + z) F (λ)S x (λ)dλ F [λ/(1 + z)] S y (λ)dλ + Z y -Z x ( 40\n\n) 19 where F (λ) is the source spectral energy distribution (a SN in this case), and S x (λ) is the transmission of the filter x. The Z y -Z x term accounts for the different zero points of the filters [130] . At present, this term amounts to an uncertainty of 0.02mag in the overall budget of SNeIa as distance indicators, but it can be decreased to 0.01 mag. The comparisons of supernovae with the fine time sequences of spectra along all phases available, are turning the uncertainty in the K-correction into a decreasingly small figure. The large amount of spectral data on SNe Ia, and the good correlation with photometric properties, can indeed make this uncertainty very small."},{"section_type":"OTHER","section_title":"Gravitational lensing of SNe Ia Gravitational lensing causes dispersion in the","text":"Hubble diagram for high redshift sources. There was an early concern on the systematics and bias that this effect could cause in the determination of the cosmological parameters using SNeIa [131] . It has been shown through the study of lensing in the highest z SNe Ia sample [132] that the magnification distribution of SNeIa matches very well the expectations for an unbiased sample, i.e., their mean magnification factor is consistent with unity. The effect can be very well controlled by studying the galaxy fields [132] . Moreover, SNe Ia should show a negligigle systematics caused by gravitational lensing when having a few SNeIa per redshift bin [122] . This means that in future missions that effect should not have impact in limiting the accuracy of w(z)."},{"section_type":"OTHER","section_title":"The limit of SNeIa as calibrated candles","text":"The better we deal with the light curve and spectral information of SNe Ia, the lower becomes the dispersion in the Hubble diagram. If SNe Ia have a systematic error in their use (independently from evolution), this is likely well below the accuracy to which we would like to determine distances (1%). We have not yet hit that brick wall. Using multi-epoch spectral information and accurate light curves in various filters should enable to test the power of these candles and their limiting irreduceable error [108].\n\n5.3.6. A joined sample Supernovae collected along years show the effect of observations treated with various degrees of accuracy. The dispersion around the best fit reflects that fact. For instance, some samples would have information on only two colors and at very few epochs, making it difficult to control reddening, or supernovae had no spectrum confirming that they were SNe Ia. The new samples will naturally have a higher degree of accuracy in magnitude and extinction for each supernova than previous samples. The ones with poorer information have been placed in [29, 52] into the \"silver category (as opposed to \"gold\" category). However, even within the Gold sample the requirements have changed: the latest definition of Gold sample demands higher quality standards than in [29] . In some of these SNe Ia lists, distance moduli of supernovae are assembled but they have been obtained through procedures which deal differently with dust extinction. The Hubble diagram resulting from those compilations would naturally show a large dispersion. In [133], it is argued that a few supernovae of an early High Z SN Search sample are pulling w(z) towards evolution. The exploration 20 done with a method independent from that in [29] showed that the central value in the 1σ confidence varied according to the light curve and reddening fitter of those SNe Ia [134] . Fortunately, supernovae enable to compare results obtained with different light curve fitting approaches. The published photometry by the various collaborations can be used to bring together the SNe Ia by using a consistent procedure into the Hubble diagram. This was done for the SNe Ia considering available up to 2005 in [134] . This is done in [122, 135, 136] for samples including the latest data from the Higher Z collaboration."},{"section_type":"OTHER","section_title":"Testing the adequacy of a FRW metric","text":"In most of the above discussion on dark energy, the FRW metric is taken as the right metric for our Universe. Modern cosmology supports that the Universe is highly isotropic on average on high scales. The main evidence is coming from the statistical isotropy of the Cosmic Microwave Background radiation. There are also cosmological observations supporting a matter distribution homogeneous on large scales, of the order of 100 h -1 Mpc. Given the observed structures of matter in voids and filaments, it is worth considering whether departures from the homogeneous and isotropic Universe could alter the present discussion, in particular, the possibility that there were no need for dark energy, but instead, that the observations are the result of an effect of inhomogeneity [137] .\n\nAs pointed out in [138] , large samples of supernovae will soon allow to test the luminosity distance relation in various directions and redshift bins. One can consider the luminosity distance d L as a function of direction n and redshift z [138] .\n\nThe direction-averaged luminosity distance is:\n\nd (0) L (z) = 1 4π dΩ n d L (z, n) = (1 + z) z 0 dz ′ H(z ′ ) ( 41\n\n)\n\nThis is the usual value retrieved by the SN collaborations. But its directional dependence d L (z, n) can be a test of the validity of the isotropy assumption. The authors in [138] propose to expand the luminosity distance in terms of spherical harmonics, to obtain the observable multipoles C l (z). This idea was also investigated in [109] , where the motion of our Local Group with respect to the CMB was measured. Results in [138] are in agreement with the results in [109] . Moreover, we can think of using the dispersion in the magnitude of SNe Ia in each direction n to put constrains on anisotropic models. As discussed in [139] , the dispersion of the magnitude in the Hubble diagram is at odds with some simple inhomogeneous models that predict a higher spread in m(z) and also a non-linear behavior. The anisotropic and inhomogeneous models have to confront present and future values on the SNeIa dispersion at each redshift interval z bin and in every direction n:\n\nσ(d L (z bin , n)) d L (z bin , n) = ln(10) 5 σ(m(z bin , n)) ( 42\n\n)\n\n21 The scaling solutions of the regionally averaged cosmologies can simulate the presence of a quintessence field or other negative pressure component [137, 142, 141] . But, any backreaction from inhomogeneity and anisotropy would show in the dispersion in the Hubble diagram. Those challenges have to be met by the non-FRW proposals. Up to now, the exploration of toy models calculating the effect of inhomogenities in the scale factor, find irregularities in the expansion rate beyond what is observed in the SN Ia Hubble diagram. The dispersion in the Hubble diagram of a standard candle is increased [139] .\n\nThe situation concerning the predictions of inhomogeneous models [137, 140, 141] is summarized in [142] . It is concluded that the inhomogeneities that would be able to amount to a d L (z) effect as observed from supernovae must be subhorizon and of strong type and they require a treatment beyond first-order perturbation theory. Research in this field will continue towards quantitative predictions to improve the connection with observational cosmology [143]."},{"section_type":"METHOD","section_title":"Next decade experiments","text":"The aim of discriminating between various possibilities for dark energy places stringent requirements on the use of standard candles [144] . Within the two decades from the discovery (1998) and the gathering of space mission data, it is feasible to reduce largely the errors, as mentioned before, and this would reflect into an error of only 1%-2% in d L (z). Errors much bigger than this 1% of error in d L result in the impossibility to distinguish between models. Even at the level of 1% in d L (z) one finds some degeneracy [144, 145] . Space projects aimed at determining the nature of dark energy are counting on a systematic error in the method around 0.02 mag (i.e. 1% in the distance) [95] . A different question is whether from ground, within the planned experiments at work in the present decade, one can converge towards that limit. These questions have been investigated in detail by using Monte Carlo simulations including the systematic errors in m(z) (or d L (z)) to be found in the evaluation of the present value of the equation of state of dark energy w 0 and its first derivative in the scale factor w a [95] (see expression 33). SNAP is a mission that has been specifically designed to gather about 2000 SNe Ia up to z ∼ 1.7. As a single mission gathering supernova data with the same instrumentation, it can eliminate possible photometric offsets and include high-precision photometry out to 1.7 microns [146] . With such large number of SNe Ia, statistical errors in every redshift bin of ∆z = 0.1 become negligible. The error is dominated by the systematics. The total error in each redshift bin is given by:\n\nσ bin = σ 2 st N bin + σ 2 sys ( 43\n\n)\n\n22 with expected systematic error; σ sys = 0.02 z/z max (44)\n\nTable I gives a progress evaluation in the control of high-redshift supernova distance uncertainties.\n\nTABLE I ERROR CONTRIBUTIONS TO HIGH SUPERNOVA DISTANCES in 1998 Systematic Uncertainties (1σ) Magnitude Statistical Uncertainties (1σ) Magnitude Photometric System Zero Point 0.05 Zero Points, S/N 0.17 Evolution < 0.17 K-corrections 0.03 Evolution of Extinction Law 0.02 Extinction 0.10 Gravitational Lensing 0.02 Light curve sampling 0.15 PROSPECTS TOWARDS 1% ERROR IN d L (z) in a space mission Systematic Uncertainties (1σ) Magnitude Statistical Uncertainties (1σ) Magnitude Photometric System Zero Point a 0.01 Zero Points, S/N 0.00* Evolution 0.00* K-corrections 0.00* Evolution of Extinction Law < 0.02 Extinction 0.00* Gravitational Lensing < 0.01 Light curve sampling 0.00* a 0.00* means values of the order of 10 -3 . The statistical errors are for the redshift bin. b In the Table for prospects in a space mission, the statistical error from light curve sampling of SNe Ia is reduced due to the large number of SNe Ia data per redshift bin.\n\nGround-based projects will not be able to achieve such accuracy. They aim at a reduction of systematic errors to 0.04-0.05 mag. Much of what will be feasible from now until the next space era of determination of the equation of state, depends on the use of all the information already gathered in the best way. Spectra gathered so far can be exploited further than what it has been done up to now, in order to reduce systematic errors. In the coming years, a lot can be done to improve the present dispersion of the Hubble diagram of SNe Ia and aim at a scatter of 1% in d L . The above limit is only reachable if all the systematic effects are dealt with to the best."},{"section_type":"OTHER","section_title":"Complementarity","text":"The SNe Ia test of w(z) benefits from using information on Ω M obtained by some other method. This can come from weak lensing or from large scale structure [147, 148, 149] .\n\nBaryon acoustic oscillations can also provide a measurement of distance along z, in this case, angular distances d A (z) and of E(z) = H(z)/H 0 [27, 150, 151] . Generally, for any geometry, one has:\n\nA = √ Ω M E(z) 1 3 1 z |Ω K | S |Ω K | z 0 dz ′ E(z ′ ) 2 3 ( 45\n\n)\n\n23 A measurement of d A has been obtained [27] .\n\nA = √ Ω M E(z 1 ) 1/3 1 z 1 z 1 0 dz E(z) 2/3 = 0.469 ± 0.017 ( 46\n\n)\n\nwhere z 1 = 0.35 is the redshift at which the acoustic scale has been measured in the redshift sample. Combining BAO and SNeIa distances one can aim at tracing dark energy all through a high z range. The value of w(z = z rec ) is provided by WMAP and Planck. In between, other methods such as studying the growth of large scale structure (LSS) obtained from Lyα clouds and clusters would depict the growth factor of structure along z. Considering the information at high z derived from LSS, SNe Ia and CMB should lead to learn about early dark energy [152, 153, 90, 148, 154, 155] ."},{"section_type":"CONCLUSION","section_title":"Conclusions","text":"Type Ia supernovae can test GR against other alternative theories of gravitation. Theories of higher dimensional gravity have different Friedmann equations from those originating from GR. Some of those modified gravity proposals have already been tested and ruled out through the Hubble diagram of SNeIa up to very high z.\n\nIf dark energy is due to a scalar field either minimally coupled to gravity (i.e. quintessence) or non-minimally coupled to gravity (arising from scalar-tensor gravity or the effective low-energy superstring proposals), the Hubble diagram up to z > 1 obtained by the supernova collaborations gives the opportunity for exhaustive tests. In what concerns scalar-tensor gravity, supernovae are a precision experiment in a way analogous to the observations of the PSR B1913+16 or to solar system tests.\n\nMoreover, supernovae have already provided hints on the form of the potential of the hypothetical scalar field (for the case of minimally coupled field) that could be associated to the acceleration of the Universe.\n\nNonetheless, GR plus a form of cosmological constant is not ruled out observationally. Finding out how is w close to z = 0 is one of the most intriguing questions. Hopefully, this will be clarified by the samples to come. The goal of obtaining a consistent picture on w(z) up tp z ∼ 1.5 from the present samples is approached through the various SN procedures used by the existing collaborations.\n\nUltimately, supernovae together with complementary methods just recently discovered shall select the correct ideas on dark energy and gravity. If any modified gravity theory has a cosmological effect, those methods should start to see it.\n\nAcknowledgements I would like to thank Alex G. Kim and Marek Kowalski for reading the manuscript and pointing to useful suggestions. This work has been supported by grant AYA2006-05369.\n\n24 References [1] Riess A G et al. (The High Z SN Search Collaboration) 1998 Astron. J. 116 1009 [2] Perlmutter S et al.(The Supernova Cosmology Project) 1999 ApJ 517 565 [3] Einstein A 1917 Sitzungsber. Preuss. Akad. Wiss. Berlin 142-152 [4] Brans C H and Dicke R H 1961 Phys. Rev 124 925 [5] Will C 1993 Theory and experiment in gravitational physics, Cambridge (London) [6] Green M B, Schwarz H and Witten E. 1988 Superstring Theory (Cambridge University Press, Cambridge) [7] Arkani-Hamed N, Dimopoulos N and Dvali G 1998 Phys. Lett. B 429 263 (Preprint hep-ph/9807344) [8] Dvali G Gabadadze G and Porrati M 2000 Phys Lett B 485 208 [9] Deffayet C Dvali G and Gavadadze G 2002 Phys. Rev. D 65 044023 [10] Adelberger E G, Heckel B and Nelson A E 2003 Ann. Rev. Nucl. Part. Phys. 53 77 [11] Adelberger E G, Heckel B and Hoyle C D 2005 Physics World 18 41 [12] Adelberger E G et al. 2007 Phys. Rev. Lett. 98 131104 (Preprint hep-th/0611223) [13] Kneller J P and Steigman G 2003 Phys Rev D 67 063501 [14] Koyama K and Maartens R 2006 JCAP 601 016 [15] Maartens R 2004 Living Rev. Rel. 7 7 [16] Spergel D N et al. (WMAP Collaboration) 2003 ApJ Supp 148 175 (Preprint astro-ph/0302209) [17] Sahni V and Starobinsky A 2006 Int. J. M. Phys D 15 2105 (Preprint astro-ph/0610026) [18] Calcagni G, de Carlos B and De Felice A 2006 Nucl. Phys. B 752 404 (Preprint hep-th/0604201) [19] de Felice A Hindmarsh M and Trodden M 2006 JCAP 0608 005 (Preprint astro-ph/0604154) [20] Gorbunov D, Koyama K and Sibiryakov S 2006 Phys. Rev. D 73 044016 (Preprint hep-th/0512097) [21] Padilla A 2007 (Preprint hep-th/0610093) [22] Trodden M 2006 in From Quantum to Cosmos: Fundamental Physics Research in Space, Washington (Preprint astro-ph/0607510) [23] Lukas A, Ovrut B.A., Stelle K.S and Waldram D 1999 Phys. Rev. D 59 086001 (Preprint hep-th/9803235) [24] Binetruy P, Deffayet C and Langlois D 2000 Nucl. Phys. B 565 269 (Preprint hep-th/9905012) [25] Fairbairn M and Goobar A 2005 Phys. Lett. B 642 432 (Preprint astro-ph/0511029) [26] Maartens R and Majerotto E 2006 Phys. Rev. D 74 023004 (Preprint astro-ph/0603353) [27] Eisenstein D J et al. 2005 ApJ 633 560 [28] Wang Y and Mukherjee P 2006 Astrophys. J 650 1 (Preprint astro-ph/0604051) [29] Riess A G et al. (the Higher-Z Team) 2004 ApJ 607 665 [30] Astier P et al. (The SNLS Collaboration) 2006 A & A 447 31 (Preprint astro-ph/0510447) [31] Dvali G and Turner M 2003 Fermilab-Pub-03-040-A (Preprint astro-ph/0301510) [32] Randall L and Sundrum R 1999 Phys. Rev. Lett 83 3370 [33] Randall L and Sundrum R 1999 Phys. Rev. Lett 126 207 [34] Binetruy P, Deffayet C, Ellwanger U and Langlois D 2000 Phys. Lett. B 477 285 [35] Langlois D 2002 Int J. Mod Phy A 17 2701 (Preprint gr-qc/0205004) [36] Brax P and van de Bruck C 2003 Class. Quan. Grav. 20 201 (Preprint hep-th/0303095) [37] Easson D A 2000 Int. J. Mod. Phys. A 16 4803 [38] Durrer R 2005, hep-th/0507066 [39] Weller J and Albrecht A 2001 Phys. Rev. D 86 1939 [40] Ratra B and Peebles P.J.E. 1988 Phys. Rev D 37 3406 [41] Wetterich C 1988 Nucl. Phys. B 302 668 [42] Steinhardt P J, Wang L and Zlatev I 1999 Phys. Rev D 59 123504 [43] Peebles P J E and Ratra B 2003 Rev. Mod. Phys 75 559 [44] Copeland E J, Sami M and Tsujikawa S 2006 Int. J. Mod. Phys. D 15 1753 (Preprint hep/th/063057) [45] Uzan J.-P 2003 Rev. Mod. Phys 75 403 [46] Bertolami O and Martins PJ 2000 Phys. Rev. D 61 064007 25 [47] Li C Holz D and Cooray A 2006, astro-ph/0611093 [48] Huterer T and Turner M S 1999 Phys. Rev. D 60 1301 [49] Huterer T and Turner M S 2001 Phys. Rev. D 64 3527 [50] Sahlén M Liddle A R and Parkinson D 2007 Phys. Rev D 75 023502 (Preprint astro-ph/0610812) [51] Knop R A, et al. (The Supernova Cosmology Project) 2003 Ap J 598 102 [52] Riess A G et al. (the Higher-Z Team) 2007 ApJ 659 98 [53] Brans C H 2005 in Contributions to Santa Clara 2004 1st. International Workshop on Gravitation and Cosmology (Preprint gr-qc/0506063) [54] Esposito-Farese G. 2004, in Phi in the Sky: The Quest for Cosmological Scalar Fields, AIP Conference Proceeding 736 35 [55] Amendola L 1999 Phys. Rev. D 62 043511 (Preprint astro-ph/9908023) [56] Amendola L and Quercellini C 2003 Phys. Rev. D 68 023514 (Preprint astro-ph/0303228) [57] Boisseau B. Esposito-Farese G Polarski D and Starobinsky A 2000 Phys. Rev. Lett 85 2236 (Preprint gr-qc/0001066) [58] Esposito-Farese G and Polarski D. 2001, Phys. Rev. D 63, 063504 [59] Perivolaropoulos L 2005 JCAP 0510 001 [60] Teyssandier P and Tourrenc P 1983 J.Math. Phys 24 2793 [61] Gasperini M and Veneziano G 2003 Phys. Rept 373 1 [62] Barrow J D and Cotsakis S 1988 Phys. Lett. B 214 515 [63] Barrow J D and Maeda K 1990 Nucl. Phys B 341 290 [64] Holden D J and Wands D 2000 Phys. Rev. D 61 043506 [65] Wands D 1994 Class Quant Grav 11 269 [66] Starobinsky A A 1980 Phy Lett B 91 99 [67] Chiba T 2003 Phys. Lett. B 575 ( Preprint astro-ph/0307338) [68] Capozziello S, Carloni S. and Trosi A. 2003 Int. J. Mod. Phys. D 12 1969 (Preprint astro-ph/0303041) [69] Carroll S M, de Felice A and Trodden M 2005 Phys. Rev. D 71 023525 [70] Mena O, Santiago J and Weller J 2006 Phys. Rev. Lett 96 [71] Sotirou T P 2006 Class. Quantum Grav. 23 5117 [72] Amendola L, Polarski D and Tsujikawa S 2007 Phys. Rev. Lett 98 131302 (Preprint astro-ph/0603703) (astro-ph/0603703) [73] Antoniadis I 1992 in Proceedings of String theory and Quantum gravity, Trieste 1992 286 (Preprint hep-th/9211055) [74] Antoniadis I, Rizos J and Tamvakis K 1994 Nucl. Phys. B 415 497, (Preprint hep-th/9305025) [75] Neupane I P 2006 Class. Quant. Grav. 23 7493 [76] Gross D J and Sloan J H 1987 Nucl. Phys B 291, 41 [77] Caldwell RR 2002 Phys. Lett. B 545 23 [78] Nojiri S, Odintsov S D and Sasaki M 2005 Phys. Rev. D 71 123509 [79] Lue A and Starkman G D 2004 Phys. Rev. D 70 101501 [80] Chimento L P Lazkoz R Maartens R and Quirós T 2006 JCAP 0609 004 [81] Bouhmadi-Lopez M, Gonzalez-Diaz P F and Martin-Moruno P 2006 (Preprint gr-qc/0612135) [82] Koivisto T and Mota D F 2007 Phys. Lett. B 644 104 (Preprint astro-ph/0606078) [83] Ellis J, Kalara S, Olive K. A. and Wetterich C 1989 Phys. Lett. B 228 264 [84] Wetterich C 1994 AA 301 321 (Preprint hep-th/9408025) [85] Brax P and Martin J 2006 JCAP 0611 008 [86] Damour T Piazza F and Veneziano G 2002 Phys. Rev D 66 046007 [87] Brax P and Martin J 2006 (Preprint hep-th/0612208) [88] Hoyle CD et al. 2004 Phys. Rev. D 70, 042004 [89] Murphy T W http://physics.ucsd.edu/ ∼tmurphy/apollo/doc/multiplex.pdf [90] Doran M Robberts G and Wetterich C 2007 Phys. Rev D 75 023003 (Preprint astro-ph/0609814) [91] Witten E. 2000 Lecture at DM2000, Marina del Rey (Preprint hep-ph/0002297) 26 [92] Susskind L 2003 (Preprint hep-th/0302219) [93] Chevalier M and Polarski D 2001 Int J. Mod. Phys. D 10 213 (Preprint gr-qc/0009008) [94] Linder E V 2003 Phys. Rev. Lett 90 091301 [95] Kim A G, Linder E V, Miquel R and Mostek N 2004 MNRAS 347 909 [96] Kowal C T 1968 AJ 73 1021 [97] Filippenko A V and Riess A G 1998 Phys. Rep. 307 31 [98] Pskovskii Y P. 1977 Soviet Astron. 21, 675 [99] Phillips M M 1993 ApJ 413 L105 [100] Phillips M M et al. 1999 AJ 118 1766 [101] Sullivan et al. (The Supernova Cosmology Project) 2003 MNRAS 340 1057 [102] Nugent P, Kim A and Perlmutter S 2002 PASP 114 803 [103] Leibundgut B 2002 ARAA 39 67 [104] Matheson T. et al. 2005 (ESSENCE) AJ 129 2352 [105] Hook I M et al (The Supernova Cosmology Project) 2005 AJ 130 2788 [106] Lidman C et al. (The Supernova Cosmology Project) 2005 A A 430 843 [107] Blondin S et al. (ESSENCE) 2006 AJ 131 1648 [108] http://www.am.ub.es/ bcnde/DARKENO.html [109] Riess A G, Press W H and Kirshner R P 1995 ApJ 438 L17 [110] Jha S Riess A G and Kirshner R P 2007 ApJ Astrophys. J 659 122 [111] Perlmutter S et al. (The Supernova Cosmology Project) 1998 Nature 391 51 [112] Goldhaber G et al. (The Supernova Cosmology Project) 2001 ApJ 558 359 [113] Tonry J.L. et al. (the High Z SN Search Collaboration) 2003 ApJ 594 1 [114] Barris B J et al. (The High Z SN Search Collaboration) 2004 ApJ 602 571 [115] Clocchiatti A et al. (The High Z SN Search Collaboration) 2006 ApJ 642 1 [116] Freedman W L et al. 2001 ApJ 553 47 [117] Sandage A et al. 2006 Astrophys. J 653 843 (Preprint astro-ph/0603647) [118] Ruiz-Lapuente P 1997 Astr. Soc. Pac. Conf. Ser. 126 207 (Preprint astro-ph/9710019) [119] Ruiz-Lapuente P 1996 Ap. J. Lett 465 83 (Preprint astro-ph/9604044) [120] Goobar A and Perlmutter S 1995 ApJ 450,14 [121] Garnavich P M et al. 1998 ApJ 509 74 [122] Wood-Vasey W M et al. (ESSENCE) 2007 (Preprint astro-ph/0701041) [123] Branch D, Romanishin W and Baron E 1995 ApJ 465 73 (Preprint astro-ph/9510071) [124] Schlegel D. J., Finkbeiner M and Davis M. 1998 ApJ.S 500 525 [125] Perlmutter S 2005 HST Proposal-GO 10496 [126] Aldering G, Knop R and Nugent P 2000 AJ 119 210 [127] Conley A. et al. (The SNLS Collaboration) 2006 AJ 132 1707 [128] Ivanov V D, Hamuy M, and Pinto P A 2000 ApJ 542 588 [129] Quimby R et al. (The Supernova Cosmology Project) 2002 BAAS 201 2305 [130] Kim A G, Goobar A and Perlmutter S. 1996 PASP 108, 190 [131] Amanullah R, Mörtsell E and Goobar A 2003 A & A 397 819 [132] Jonsson J, Dahlen T, Goobar A, Gunnarson C, Mortsell E and Lee K 2006 ApJ 639 991 [133] Nesseris S and Perivolaropoulos L 2007 (Preprint astro-ph/0612653) J. Cosmol. Astropart. Phys. 0702 025 [134] Ruiz-Lapuente P et al. 2006 http://www.am.ub.es/ bcnde/DARKENO.html Ruiz-Lapuente P 2007 Invited review at Bernard's Cosmic Stories, Valencia June 2006 [135] Rubin D et al. (The Supernova Cosmology Project) 2006 BAAS 209 9003R [136] Kowalski et al. (The Supernova Cosmology Project) 2007, in preparation [137] Buchert T, Larena J and Alimi J-M 2006 Class Quant. Grav 23 6379 [138] Bonvin C, Durrer R and Kunz M 2006 Phys. Rev. Lett 96 191302 [139] Discussions on inhomogeneous cosmologies at the conference Key Approches to Dark Energy Barcelona 2006. This author thanks all the participants for addressing this topic.\n\n27 [140] Rasanen S 2006 JCAP 0611 003 [141] Kolb E W, Matarrese S, Notari A and Riotto A 2005 (Preprint astro-ph/0506534) [142] Célérier M-N 2000 Astron. Astrophys 353 63 [143] Buchert T 2007 gr-qc/0612166 [144] Maor I and Brustein R 2003 Phys. Rev D 67 103508 [145] España-Bonet C and Ruiz-Lapuente P 2005 (Preprint hep-ph/0503210) [146] http://snap.lbl.org [147] Heavens A F, Kitching T D and Taylor A N 2006 MNRAS 373 105 [148] Tegmark M et al. 2006 Phys. Rev. D 74 123507 (Preprint astro-ph/0608632) [149] Percival W J et al. 2007 Astrophys. J 657 645 (Preprint astro-ph/0608635) [150] Nichol R 2006 in Cosmology, Galaxy Formation and Astroparticle Physics on the Pathway to SKA (eds Klöckner H-R et al.) [151] Yamamoto K Bassett B A Nichol R C and Suto Y 2006 Phys Rev D 74 063525 (Preprint astro-ph/0605278) [152] Frieman J A , Huterer D, Linder E V and Turner M S 2003 Phys. Rev. D 67 083505 (Preprint astro-ph/0208100) [153] Wang Y and Freese K 2006 Phys. Lett. B 632 449 [154] Dunkley J, Bucher M, Ferreira P G, Moodley K and Skordis C 2005 Phys. Rev. Lett. 95 261303 [155] Kunz M, Trotta R and Parkinson D 2006 Phys. Rev. D 74 023503"}],"string":"[\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"2 Contents 1. Introduction 2. Gravity and extra dimensions 3. The dark energy field and scalar-tensor gravity 4. The cosmological constant? 5. The d L test from high-z SNe Ia 6. Testing the adequacy of the FRW metric 7. Next decade experiments 8. Complementarity 9. Conclusion 3 1. Introduction\\n\\nThe discovery of the acceleration of the expansion of the universe [1, 2] has revived questions concerning the domain of validity of Einstein's General Relativity. At present, the data gathered on the expansion rate do not disclose whether the acceleration is due to a component formally equivalent to the cosmological constant introduced by Einstein [3], or whether we are finding the effective behavior of a theory of a wider scope whose low energy limit slightly departs from GR. The bare Einstein's equations (without cosmological constant) for a Universe with a FRW metric and dust-like matter imply a continuous deceleration of the expansion rate. However, a Universe containing a fluid with an equation of state p = wρ with index w < -1/3 overcomes the deceleration when the density of this fluid dominates over that of the dust-like matter. In this context, the cosmological constant, if it is positive and added to the equations, balances the deceleration by acting as a fluid with an equation of state p = -ρ.\\n\\nThe cosmological constant was originally in the metric part of Einstein's equations, and alternative metric theories of gravity provide a wealth of terms in that side. When describing a homogeneous and isotropic universe within a FRW model, it is frequent to come as close as possible to the standard Friedmann-Einstein equation for the expansion rate by bringing terms to the right-hand side within the so-called \\\"effective index\\\" w eff of the cosmic fluid whose equation of state is p = w eff ρ.\\n\\nIn the field of gravitation, ideas that appeared in the classical domain several decades ago, are now being brought back within the different context of quantum gravity theories. Scalar-tensor theories of gravitation are being reconsidered. Scalar fields coupled to gravity were proposed as a way to incorporate Mach inertial induction. They gave rise to the Brans-Dicke theory of gravitation [4] , which is well restricted by solar system experiments [5] . Superstring theory and M-theory introduce a number of quantum scalar fields, the dilaton, the radion, the moduli [6] . They can play a role in the late acceleration of the Universe. On the other hand, brane worlds inspired by M-theory provide modifications to gravity and new equations of cosmic expansion [7] . In the brane world context, gravity lives in the bulk of 3+1+d dimensions while the rest of the interactions are confined to the 3+1 brane. Some braneworld proposals do not explain the observed cosmic expansion. However, it was realized [8, 9] that it is possible to obtain late-time acceleration with one extra dimension. This possibility is interesting as it accounts for the observed cosmic expansion without the need of a cosmological constant term. Acceleration results from the induced gravity term in the brane. The number and size of additional dimensions has been explored in a wide range of proposals. The existence of additional dimensions and their size are constrained by tests of the gravity law [10, 11, 12] . At the time when gravity theories of the scalar-tensor type and others were examined, the solar system tests were the most restrictive ones. Nowadays, the theories have to confront the cosmological observations. The theories have to allow Big Bang 4 nucleosynthesis [13] . They have to fullfil the constraints from the growth of density perturbations and give correct predictions for the evolution of the growth factor. Such growth function of density perturbations requires careful evaluation for several modified gravity scenarios [14, 15] . Ultimately, gravity theories have to be consistent with the CMB data and with the expansion rate of the Universe provided by supernovae. Supernovae enable to measure the rate of the expansion of the Universe along redshift z. They do so by tracing the luminosity distance along z for a standard candle. If the Universe is flat, as inferred from WMAP [16] , the d L relation is simply:\\n\\nd L (z) = cH -1 0 (1 + z) z 0 dz ′ H(z ′ ) ( 1\\n\\n)\\n\\nwhile in the general case,\\n\\nd L (z) = cH -1 0 (1 + z)| Ω K | -1/2 sinn | Ω K | 1/2 z 0 H 0 dz ′ H(z ′ ) ( 2\\n\\n)\\n\\nwhere sinn(x) = sin(x), x, or sinh(x) for closed, flat, and open models respectively, where the curvature parameter Ω K , is defined as Ω K = Ω T -1. H(z) is the Hubble parameter. For a FRW universe containing matter and a general form of matter-energy X with index of the equation of state w X , H(z) is given by:\\n\\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω M (1 + z) 3 + Ω X (1 + z) 3(1+w X ) ( 3\\n\\n)\\n\\nwhere H 0 is the present value of the Hubble parameter and\\n\\nΩ K ≡ -k H 2 0 a 2 0 ( 4\\n\\n)\\n\\nwhere k accounts for the geometry and a 0 is the present value of the scale factor.\\n\\nΩ M = 8πG 3H 2 0 ρ M ( 5\\n\\n) Ω X is the density parameter for dark energy.\\n\\nWhen w(z) varies along z, H(z) can be expressed as:\\n\\nH(z) 2 = H 2 0 Ω K (1 + z) 2 + Ω M (1 + z) 3 + Ω X exp 3 z 0 1 + w(z ′ ) 1 + z ′ dz ′ ( 6\\n\\n)\\n\\nOne sees from (1-2), that d L contains w(z) through a double integral. This leads to limitations in recovering w(z) if d L data along z are scarce and with large errorbars. Data are often given as m(z) which relates to d L (z) as:\\n\\nm(z) = 5 logd L (z) + [M + 25 -5 log(H 0 )] ( 7\\n\\n)\\n\\n5 The quantity within brackets is M ≡ M -5 log H 0 + 25, the \\\"Hubble-constant-free\\\" peak absolute magnitude of a supernova. The data are given sometimes in the form of distance moduli:\\n\\nµ = 5 log d L + 25 ( 8\\n\\n)\\n\\nThe measurement of w(z) through d L (z) is the standard approach, though possibilities of measuring directly H(z) have been investigated (for a review see [17] ). This Hubble law in (3) is derived from the Einstein-Hilbert action. We will review other possibilities related to expansion histories arising from modified gravity theories and effective theories which contain scalar fields accelerating the cosmic expansion. In the present overview we concentrate on the empirical evaluation of those theories. Some of the models to be explored here might contain a ghost problem or have a subclass amongst them that present quantum instabilities. We refer for this topic to recent updates in [18, 19, 20, 21, 22] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Gravity and extra dimensions\",\n \"text\": \"The idea of allowing gravity to live in extra dimensions arose to explain the hierarchy problem and constitutes one of the major challenges to the classical picture of a 4dimensional description of gravity [7] . The Planck scale of 10 19 GeV, relevant for gravitation, is much larger than the electroweak scale, of about 1 TeV of the standard model of particle physics. According to the proposal by Arkani-Hamed, Dimopoulos and Dvali [7], the Planck scale is not a fundamental scale, but its enormity is simply a consequence of the large size of new extra dimensions. While gravitons can propagate in the new extra dimensions, the standard model fields are confined to our submanifold of 3 + 1 dimensions, the brane, which is embedded in a higher dimensional bulk. In this scenario, the Planck scale in the bulk can be much smaller than in the 4dimensional brane as\\n\\nM 2 P,4 = R n M n+2 P,n+4 ( 9\\n\\n)\\n\\nThe existence of extra dimensions in which gravity propagates produces a transition in the gravitation law at the compactification radius, proposed to be of the submillimetre scale.\\n\\nIn the cosmological context, this idea has some interesting consequences since it predicts a departure from GR. As soon as the cosmological model was implemented for n = 1 extra dimensions [23, 24, 9, 8] , it was noted that the brane cosmology leads to Friedmann-like equations very different from the standard ones [24] . In the 5dimensional model by Dvali, Gabadadze and Porrati (DGP) whose cosmology is explored by Deffayet and collaborators in [9] , the competition between the five-dimensional term of the Einstein-Hilbert action and the four-dimensional term induced in the brane leads to a modified Friedmann equation which simulates an acceleration term without the need to include a cosmological constant. The transition from the gravity law r -2 to 6 the modified one is characterized by a scale r c , the so-called cross-over scale defined as [9]:\\n\\nr c ≡ M 2 P 2M 3 ( 5\\n\\n) ( 10\\n\\n)\\n\\nAt large distances, the five dimensional term takes over and gravity gets weaker [9] . If the parameter r c is chosen to be of the order of H 0 -1 ∼ 10 29 mm, this is a choice for M (5) ∼ 10-100 MeV. At short distances r << r c , Newton's law is modified by a logarithmic repulsion. At large distances r >> r c , the modification involves a factor r rc . The Hubble expansion law shows significant departures from the standard Friedmann expansion law:\\n\\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω rc + Ω rc + α Ω α (1 + z) 3(1+wα) 2 ( 11\\n\\n)\\n\\nwhere the sum over α stands for a sum over each component with equation of state w α , Ω α being the density parameter:\\n\\nΩ α ≡ ρ 0 α 3M 2 Pl H 2 0 a 3(1+wα) 0 ( 12\\n\\n)\\n\\nand\\n\\nΩ rc ≡ 1 4r 2 c H 2 0 ( 13\\n\\n)\\n\\nIn the matter-dominated epoch, the expansion equation is:\\n\\nH 2 (z) = H 2 0 Ω K (1 + z) 2 + Ω rc + Ω rc + Ω M (1 + z) 3 2 ( 14\\n\\n)\\n\\nIn the original 5D model in [8] , M 5 was estimated to be ∼ 1 TeV, which corresponds to a cross-over scale r c of 10 15 cm. Such r c led to incompatibilities with the solar system data. When M 5 is taken to be M 5 < ∼ 1 GeV, i.e. r c > ∼ 10 25 cm, the conflict disappears. High-z supernovae give useful constraints to the parameter Ω rc in (10) [25, 26] . In [26] , two samples of high-z supernovae are used in conjunction with measurements from baryon oscillations [27] and the CMB shift parameter [28] to estimate the favored values for Ω rc and Ω M . Ω rc = 0.125 for the supernova data in the Riess sample from 2004 [29] and Ω rc = 0.13 for the SNLS data [30] . It is found in [26] that, for a flat Universe, this 5D modified gravity model gives a worse fit than GR plus cosmological constant. Further supernova samples would reexamine the empirical compatibility of the DGP model and other modified gravity models involving more dimensions. Those models can be explored through a phenomenological description. The Hubble law arising from gravity with extra dimensions can be expressed by adding H α terms to the standard Friedmann equation [31] :\\n\\nH 2 - H α r 2-α c = 8πGρ M 3 ( 15\\n\\n)\\n\\n7 where r c is the cross-over radius, i.e. the transition radius from a regime to another in the gravity law, and α is the parameter to be empirically determined. From redshift z= 0 to z= 2, the above Friedmann equation gives an effective index of the equation of state [31]: w eff ≈ -1 + 0.3α (16) Such effective equation of state evolves too fast to be compatible with present SNe Ia data, unless α is small. Present data favor α close to 0 [25]. Though the most simple implementations lack empirical confirmation, room is left for several models. Cosmological SNe Ia are useful tests in restricting the cross-over radius that relates to departures from GR.\\n\\nA different kind of model in which the brane is embedded in a 5D bulk has been proposed by Randall and Sundrum [32, 33] . Contrary to the previous approach, here the bulk has a AdS 5 geometry as opposed to the flat geometry of the DGP model. In the original Randall-Sundrum (R-S) model, the brane has tension σ, i.e. cosmological constant, as well as the bulk Λ 5 . The bulk cosmological constant is negative. The resulting Hubble expansion in the brane is the result of the compensating effect of both terms. To make zero the effective cosmological constant in the brane, Λ 4 , a fine tuning between the brane tension and the bulk cosmological constant is required. The original proposal sets the effective cosmological constant, Λ 4 , to zero. Generally, a tiny deviation from exact compensation [34] would lead to an effective cosmological constant at late times. Modifications of the original R-S proposal include in the brane, in addition to tension (cosmological constant), a matter-energy density ρ. The total brane density ρ B is:\\n\\nρ B = σ + ρ ( 17\\n\\n)\\n\\nThis gives the modified Friedmann equation [35] :\\n\\nH 2 = κ 4 36 σ 2 - 1 l 2 + κ 4 18 σρ + κ 4 36 ρ 2 + C a 4 ( 18\\n\\n)\\n\\nwhere κ is the inverse Planck mass M -1 P and l is related to Λ 5 . Generalizations of the Randall-Sundrum scenario in cosmology consider the AdS 5 bulk geometry containing the fifth-dimensional cosmological constant Λ 5 , but explore arbitrary densities in the brane and in the bulk.\\n\\nExplorations of braneworld ideas that allow a late acceleration of the Universe can be found in [36, 37, 38] .\\n\\n3. The dark energy scalar field and scalar-tensor gravity\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Early developments\",\n \"text\": \"Before the development of braneworld scenarios, dark energy as associated to a scalar field was considered in analogous way to research done within the context of inflation 8 [40] . This theoretical possibility was entertained a decade before there was evidence of a non-zero cosmological constant from supernovae and that the CMB results indicated a flat Universe. The favored Ω T = 1 value by inflation and the evidence that Ω M < 1 led to think about a possible Λ term. In the recent expansion history a scalar field could play a role in analogy with inflation, where a scalar field dubbed the inflaton gives rise to the exponential phase of early acceleration. The scalar field responsible for the late acceleration, later named quintessence, is proposed as a new massless scalar field which induces a cosmological constant behavior at late times. In this original setting, the field is minimally coupled to gravity, i.e., thus the inclusion of the field remains within the framework of GR. The interaction of the field with the matter fields is set to be negligible. The scalar field responsible for dark energy can be identified by recovering its potential, or equivalently its effective equation of state. The presence of such scalar field φ, would induce an action:\\n\\nS = d 4 x √ -g 1 2 g µν ∂ µ φ∂ ν φ -V (φ) ( 19\\n\\n)\\n\\nThe field contributes to the stress-energy momentum tensor with an effective mass density and pressure:\\n\\nρ φ = 1 2 φ2 + V (φ) p φ = 1 2 φ2 -V (φ) ( 20\\n\\n)\\n\\nIn this case, the effective value of the equation of state w eff depends on the form of the potential V (φ) and it can evolve with time [39] .\\n\\nw eff = p φ ρ φ ( 21\\n\\n)\\n\\nA search for general properties of this potential, in analogy to the search for the inflation potential, leads to suggestions for potentials with tracking behavior [40, 41, 42, 43] . Several potentials have been proposed, from the original V = κ/φ α [40] to other recent proposals [44] . The dark energy scalar field that does not couple with matter has the feature of avoiding effects on variations of constants and other effects subject to experimental evidence [45, 46] . There is already a reconstruction of what could be the potential of the minimally coupled field V (φ) obtained from the SNLS SNe Ia [47] following reconstruction ideas given in [48, 49] . Another reconstruction of V (φ) using SNe Ia from various collaborations and considering Padé approximant expansions of the potential is found in [50] . By now, many proposals have been discarded, since the equation of state between z= 0 and z= 1.5 evolves very slowly as shown by high-z SNe Ia data [2, 51, 29, 52] . 9 3.2. Scalar-tensor gravity One can still consider the case of a scalar field that is coupled to other matter fields or to gravity. Some of these proposals come from the ballpark of superstring theory. The most well-known scalar-tensor gravity proposal, i.e. Brans-Dicke theory, incorporated the coupling of a field to gravity to account for the Mach induction on Newton's constant. In Brans-Dicke gravity [53, 4] , Newton's constant is replaced by a dynamical field G leading to the generalized Friedmann equation:\\n\\nH 2 = 8π 3 ω 2πφ 2 ρ + 1 2 φ2 + V (φ) - 3 2ω Hφ φ ( 22\\n\\n)\\n\\nThe Brans-Dicke action contains a φ 2 R term:\\n\\nS = √ -g d 4 x ± 1 8ω φ 2 R ∓ 1 2 g µν φ ,µ φ ,ν -V (φ) + L matter ( 23\\n\\n)\\n\\nThe theory has been well tested [5]. Observationally, for other scalar-tensor theories, we aim at retrieving the self-interaction potential of the scalar V (φ). and the coupling function to matter and gravity A(φ) in the matter term of the action [54]:\\n\\nS matter (ψ, A 2 (φ)g µν ) ( 24\\n\\n)\\n\\nIn the case of quintessence (minimally coupled field) A(φ) = 1. The present supernova data have explored the potential for a minimally coupled field [47] . In the non-minimally coupled case, also referred to as coupled quintessence or coupled dark energy [55, 56] , the CMB data can put constraints on the strength of scalar gravity as compared to ordinary tensorial gravity [55] . However, there are limitations to the recovery of the form of the coupling and the coupled dark energy potential from CMB observations [55] . Supernovae together with observational constraints on the growth of density perturbations would allow to retrieve A(φ) [57, 58] . We expand A(φ) in its derivatives α 0 , β 0 and higher orders:\\n\\nln A(φ) ≡ α 0 (φ -φ 0 ) + 1 2 β 0 (φ -φ 0 ) 2 + O(φ -φ 0 ) 3 ( 25\\n\\n)\\n\\nBy exploring the empirical evidence it is found [54] that solar system tests and binary pulsar tests allow to impose precise bounds on the first and second derivatives of the matter-scalar coupling function, while SNe Ia could \\\"a priori\\\" allow to reconstruct the full shape of the function A(φ) or the higher order derivatives. It is argued in [57, 58, 59] that the knowledge of the luminosity distance and the density fluctuations δρ/ρ as functions of redshift z is sufficient to reconstruct the potential V (φ) and the coupling function A(φ).\\n\\nThese results can be generalized to proposed actions that can be re-expressed as a theory of interacting matter fields in general relativity [60] . Actions containing e -Φ R are motivated from the lowest order effective action including the dilaton Φ [61] . They can be brought by redefinition of the field to an Einstein frame in which the field is minimally coupled to the metric.\\n\\n10 A number of interesting cases have been examined and have been expressed in terms of a scalar field which is minimally coupled to the metric, but nonminimally coupled to the other fields [62, 63, 64] . The self-interaction potential shows an explicit energy transfer between the scalar field and the matter fields. Related to the previous exploration is that of f(R) gravity or extended curvature gravity [65], also known as \\\"c-essence\\\" when including an inverse power of R [67] . Those modified gravity theories with a Lagrangian containing an arbitrary function of R are equivalent to a particular class of scalar-tensor theories of gravity. For instance, the action\\n\\nS = 1 2κ 2 d 4 x √ -g R - µ 4 R + S matter (g µν ) ( 26\\n\\n)\\n\\nis found to be equivalent to [67]:\\n\\nS = 1 2κ 2 d 4 x √ -g 1 + µ 4 φ 2 R - 2µ 4 φ + S matter (g µν ) ( 27\\n\\n)\\n\\nThe equivalence is generalized to an arbitrary function of R [60, 65]:\\n\\nS = 1 2κ 2 d 4 x √ -g F (R) + S matter (g µν ) ( 28\\n\\n)\\n\\nwith an equivalent action [60, 65]:\\n\\nS = 1 2κ 2 d 4 x √ -g (F (φ) + F ′ (φ)(R -φ)) + S matter (g µν ) ( 29\\n\\n)\\n\\nwhere F ′ (φ) = dF/dφ. The introduction of a canonical scalar field ϕ such that F ′ (φ) = exp( 2/3 κϕ), allows to rewrite the action with F(R) as:\\n\\nS = d 4 x √ -g E 1 2κ 2 R E - 1 2 (∇ϕ) 2 -V (ϕ) + S matter (g E µν /F ′ (φ(ϕ)), V (ϕ) = (φ(ϕ)F ′ (φ(ϕ)) -F (φ(ϕ)))/2κ 2 F ′ (φ(ϕ)) 2 ( 30\\n\\n)\\n\\nThe gravity described by g E µν is the Einstein-scalar system and g µν (= g E µν /F ′ (φ)). g µν is a metric of scalar-tensor gravity which is, in this case, subject to constraints coming from solar system experiments. Again, a classification of when one recovers an action equivalent to the Einstein-Hilbert case and when one recovers a Brans-Dicketype action in f(R) gravity is given in [60] . When considering a general function of R, it is not always possible to obtain a description equivalent to Einstein GR plus additional fields minimally coupled to the metric but coupled to other fields. When this is possible, we have in the matter term the coupling between the matter fields ψ and the scalar field φ. Examples of f(R) casted in terms of fields can be found in [62, 64, 66, 67] . For the Starobinsky model [66] with F (R) = R + R 2 /M 2 with M ∼ 10 12 GeV, for instance, the effective potential can be rewritten, in terms of the scalar field ϕ, as\\n\\nV (ϕ) = M 2 e -2 √ 2/3 κϕ 8κ 2 (e √ 2/3 κϕ -1) 2 ( 31\\n\\n)\\n\\n11 Actions with generalized inverse powers of R have been investigate as well [68, 69, 70, 71]. The supernovae gathered up to now are compatible with some f(R) gravity models and incompatible with others [68, 70, 72]. More general scalar-tensor theories than the ones here defined, such as those where the potential depends on several scalar fields V (φ n ) and matter-scalar coupling A(φ n ), give rise to a wider phenomenology. In the superstring scenario, nonminimal coupling of the scalar fields to the spacetime curvature leads to effective actions with similarities to scalar-tensor proposals. Some scalars present in the effective action arise from compactification to four dimensions.\\n\\nThose are structure moduli which do not couple directly with the spacetime curvature tensor. One modulus field σ associated with the overall size of the internal compactificaton couples to Einstein gravity via Riemann curvature invariants, such as the Gauss-Bonnet invariant [74] . If one considers the Gauss-Bonnet term, at least an extra function, the coupling to the Gauss-Bonnet term, W (σ) needs to be determined.\\n\\nIn the following, we review attempts to include such enlarged action and their test with cosmological d L data.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Effective low energy action with R GB and other terms\",\n \"text\": \"The Gauss-Bonnet term can be found in the effective low energy string Lagrangian and in brane theories [73, 74, 75, 35, 76] . The case of the simplest action containing the Gauss-Bonnet term can be written as:\\n\\nS = √ -g R 2κ 2 - 1 2 (∂ µ φ) 2 - √ -g W (σ) R 2 GB + S matter [matter ; g µν ] ( 32\\n\\n)\\n\\nThis Lagrangian includes the Gauss-Bonnet term coupled to a field φ ≡ σ, for instance the modulus field in [74] where W (σ) is the coupling of the modulus field to the Gauss-Bonnet term, R GB .\\n\\nR 2 GB ≡ R µνρσ R µνρσ -4R µν R µν + R 2 ( 33\\n\\n)\\n\\nThe possible coupling of a scalar field to the Gauss-Bonnet topological invariant can be constrained if one takes into account cosmological supernovae and solar system tests.\\n\\nFrom supernova data gathered in [29] and with a parameterized analysis of the evolution, it is found that the present value of the equation of state w 0 is below -1. Though for the measurement of the index of the equation of state at z = 0 one awaits a large enough nearby supernova sample, the results presented in [29, 52] move to examine the possibility of w eff < -1. In the original exploration of the idea [77], a Universe which evolves towards w eff < -1 would suffer a super-aceleration phase, that has come to be called the Big Rip. From an action which includes the Gauss-Bonnet term, one can obtain w eff < -1 for a canonical scalar field (positive kinetic energy) [78, 35] . It avoids the Big Rip for w eff < -1. To add a word on this, this is not the main motivation for 12 Gauss-Bonnet gravity, since variations of the braneworld scenario can also get w eff < -1 and avoid the Big Rip [79, 80, 81] . The interest on the GB term is justified by its presence in a variety of gravity proposals from string and M-Theory. Recently some tests using supernovae have already been done [82] . The extra GB term opens naturally many possibilities for the evolution of the scale factor. The supernova test succeeds for a range of effective low energy actions including a Gauss-Bonnet term [82].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Fifth force and equivalence principle constraints\",\n \"text\": \"The fields that interact with the dark energy scalar field are constrained by a large number of tests [45] . The possibility that the dark energy scalar couples to nucleons and photons has been investigated in relation to implications for a fifth force [83, 84] . Within current scenarios of particle physics such as MSSM and MSUGRA the role of this dark energy field shows some observational signatures of the fifth force type as well [85] . For instance, the interaction of the dark energy scalar field φ with the Higgs doublet H u and H d leads to an action where the field couples differently to each of the Higgs ψ u and ψ d and one expects a violation of the weak equivalence principle. In the string scenario, where radion, dilaton and moduli can be responsible for the early and late acceleration, one has as well some possible strong couplings of the fields with matter [86, 75, 87]. Thus, the cosmological context has to be considered together with classical gravitational tests on Earth. Thus far, the torsion balance experiments set strict limits on the variation of Newton's law using different substances and locations. There is a wide range of tests being done to examine the predictions from scalar-tensor proposals and modified gravity ideas [12, 10, 11, 88, 89] . The empirical bounds on some types of couplings are tight and worth considering together with the cosmological probes.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Cosmological constant?\",\n \"text\": \"Finding that the negative pressure component gives w = -1, would likely be the biggest challenge to gravitation theories. At present, SNe Ia have not provided such evidence, though averaged values of w are close to -1 at redshifts up to z∼ 1. To probe that w stays -1 up to the surface of last scattering, involves refining several other methods (growth of density perturbations, integrated Sachs-Wolfe effect) [90] . From the view point of string theorists [91] , there is no natural explanation for the vanishing or extreme smallness of the vacuum energy. A way to address the small value of the cosmological constant in the context of a stringy landscape is to consider the anthropic principle [92] . However, invoking the anthropic principle as a way out of the problem entails to abandon the quest for a conventional scientific explanation [91]. . From the observational point of view, the answer is near to w = -1, but possible departures from w = -1 are presently being examined through expansions of the equation of state along the scale factor and other various approaches. A expansion 13 Figure 1. The cosmological constant case (bold line) is compared with evolving models close to w = -1, i.e. a model with w 0 = -1.0 and w a = -1.5 (short dash line) and a model with w 0 = -1.0 and w a = +1.5 (dash-dotted line) often used is [93, 94] :\\n\\nw(a) = w 0 + w a (1 -a) ( 34\\n\\n)\\n\\nwhere the scale factor a = (1 + z) -1 , so w ′ = w a /2 when evaluated at z = 1. Differences between the cosmological constant and departures from it in the equation of state are measurable nowadays (see Fig 1). To reach a higher level of discrimination between dark energy models depends on the progress made in the accuracy of the measurements. The aim of future projects is to determine distances with 1 % accuracy, i.e. with errors in magnitudes of 0.02 mag [95] . If the equation of state of dark energy is very close to -1, one can anticipate decades of improving the supernova measurements. The development that we might witness could be akin to that of the postnewtonian tests of GR where higher levels of precision are being targeted since the middle of the last century into the present. 14 5. The d L test from SNe Ia\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Overview\",\n \"text\": \"A long path has brought us to the present situation where we can already aim at accuracies of 1% in distance. Back in 1968, a first attempt to draw the Hubble diagram from SNe resulted in a spread of σ ∼ 0.6 mag [96] . Such large scatter was due to the inclusion of other types of supernovae, which were later recognized as being of a different type (SNI b/c) and characterized by a large scatter in luminosities [97] . SNe Ia are not standard candles either, but they follow a well known correlation between maximum brightness and rate of decline of the light curve. In 1977, Pskovskii [98] first described the correlation between the brightness at maximum and the rate of decline of the light curve: the brighter SNe Ia have a slower decline of their light curves whereas fainter ones are the faster decliners. A systematic follow-up of SNe Ia confirmed the brightness-decline relation [99, 100] . The intrinsic variation of SNe Ia is written as a linear relationship:\\n\\nM B = M B0 + α (∆m 15 (B) -β) + 5 log(H 0 /65) ( 35\\n\\n)\\n\\n∆m 15 , the parameter of the SNe Ia light curve family, is the number of magnitudes of decline in 15 days after maximum. The value of α, as well as the dispersion of that relation, have been evaluated in samples obtained from 1993 to the present [100] . M B0 is the absolute magnitude in B for a template SNIa of β rate of decline. An intrinsic dispersion of 0.11 mag in that law is found when information in more than one color is available.\\n\\nIn fact, the 0.11 mag scatter in the brightness-rate of decline is found when one includes supernovae affected by extinction. The scatter is smaller in environments where dust in the host galaxies of the SNe Ia is a lesser effect [101] . Additionally, the family of SNe Ia form a sequence of highly resembling spectra with subtle changes in some spectral features correlated with the light curves shapes [102, 103, 104, 105, 106, 107, 108] . The multiple correlations allow to control errors within the family of SNe Ia [108] . The parameter ∆m 15 requires to have observed the maximum in the light curve of the supernova. This is not always possible, and therefore the various SN collaborations have formulated the brightness-decline correlation in different ways. The High-Z SN Team uses the full shape of the light curve with respect to a template, method refered as MLCS2k2 [109, 110] . The Supernova Cosmology Project collaboration, on the other hand, has introduced the stretch-factor, s, as a parameter to account for the brightnessdecline relationship [111, 2, 112] . The stretch parameter fits the SNIa light curve from premaximum to up to 60 days after maximum. Thus, even if the maximum brightness phase is not observed, the whole light curve provides the decline value. More recently, other ways to measure the decline have been proposed [113, 114, 115] . Measuring the evolution of the rate of expansion of the Universe does not require knowledge of its present value, H 0 . Large projects have been devoted to the determination of H 0 by attempting the absolute calibration of M B from nearby SNe 15 Ia that occur in galaxies with Cepheid-measured distances [116, 117] . Controversy persists, though the most recent values quoted by the two collaborations using this method are closer (in the range 65-75 km s -1 Mpc -1 ) [116, 117] . Though largely ignored as an argument, the mere fact that a SNIa is a explosion of white dwarf bounds very efficiently the lower and upper possible values of H 0 [118] . On the empirical side, the emission at late phases of SNe Ia gives a precise value of H 0 , since such emission is very sensitive to the density profile, the mass of the supernova and the amount of 56 Ni synthesized in the explosion. Those quantities are easily measurable from the spectra [119] . The value favored by this method is H 0 = 68 ± 6 (syst) km s -1 Mpc -1 . This disgresion done, we emphasize the nuisance role of the absolute magnitude of SNe Ia when it comes to dark energy cosmology. The apparent magnitude of supernovae along z give us the cosmology, m B as [2, 51]:\\n\\nm B = M + 5 log D L (z; Ω M , Ω Λ ) -α(s -1) ( 36\\n\\n)\\n\\nwhere s is the stretch value for the supernova, D L ≡ H 0 d L is the \\\"Hubble-constant-free\\\" luminosity distance, d L is given by (1) and M ≡ M B -5 log H 0 + 25 is the \\\"Hubbleconstant-free\\\" B-band peak absolute magnitude of a s = 1 SN Ia with true absolute peak magnitude M B . Similar use of the nuisance parameter in their light curve fitting is made in [29, 52] .\\n\\nIn the first determinations of dark energy cosmology, there were four parameters in the fit: the matter density parameter Ω M , the cosmological constant density parameter Ω Λ , as well as the two nuisance parameters, M and α (either α itself as used by the SCP or a equivalent parameter by other collaborations). The four-dimensional (Ω M , Ω Λ , M, α) space, explored as a grid, yields χ 2 and P ∝ e -χ 2 /2 of the fit of the luminosity distance equation to the peak B-band magnitudes and redshifts of the supernovae. After normalization by integrating over the \\\"nuisance\\\" parameters, the confidence regions in the Ω M -Ω Λ plane can be obtained. This use of the magnitude-redshift relation m(z) as a function of Ω M and Ω Λ with a sample of high-z SNe Ia of different z, is a very powerful pointer to the allowed values in the Ω M -Ω Λ plane [120, 121] . By 1998, such use yielded important results, impling that Ω Λ > 0 at the 3σ confidence level. For a flat universe (Ω T = 1), the results from the Supernova Cosmology Project meant Ω M =0.28 +0.09 -0.08 (statistical) +0.05 -0.04 (systematics), and the High-Z Supernova Team obtained for a flat universe Ω M =0.24±0.1. The outcoming picture of our universe is that about 20-30% of its density content is in matter and 70-80% in cosmological constant. According to the allowed Ω M and Ω Λ values, the Universe will expand forever, accelerating its rate of expansion.\"\n },\n {\n \"section_type\": \"RESULTS\",\n \"section_title\": \"Results on w\",\n \"text\": \"If the size of the SNe Ia sample and its redshift range do allow it, one can aim at determining the average value of the index of the equation w or, going a step further, aim at determining the confidence regions on the w 0 -w a plane. For this later purpose, 16 using the information on the global geometry as given by WMAP provides a helpful prior. In Knop et al. (2003) [51], the supernova results were combined with measurements of Ω M from galaxy redshift distorsion data and from the measurement of the distance to the surface of last scattering from WMAP. The confidence regions provided by the measurement of the distance to the surface of last scattering and the SNe Ia confidence regions cross in the Ω M -w plane, thus providing very good complementarity. This work considers an averaged w (given the range in z of the SNe Ia observed). From the fitting assuming w constant, the value is w =-1.06 +0.14 -0.21 (statistical) +0.08 -0.08 (identified systematics). The analysis of the SNLS [30] also aims at an average value of w, allowed by the z range of this survey. Their 71 high-z SNeIa give w =-1.023 ± 0.090 (statistical) ± 0.054 (identified systematics). Recently, the results from ESSENCE, using 60 SNe Ia centered at z= 0.5 point to a similar result with w = -1.05 +0.13 -0-12 (statistical) ± 0.13 (systematics) [122] . In this last work, the supernovae from various collaborations are joined with two separate light curve fitting approaches giving a consistent result for the average value of w. High-z SNe Ia beyond z > 1 enable to have a first look at possible evolution of w(z). The first campaign by Riess et al. (2004) gave the results in the w 0 -w ′ plane, using the Taylor expansion for w(z) as w(z) = w 0 + w ′ z. The best estimates for w 0 and w ′ are w 0 = -1.36 +0.12 -0.28 and w ′ =1.48 +0.81 -0.98 . Those results [29, 52] exclude a fast evolution in the equation of state, and therefore they rule out a number of modified gravity proposals. Moreover, the results are confirmed in [122] where the available SNe Ia at large z, which can constrain w 0 and w a , are incorporated to test with a joined sample encompassing a wide range of z. Significant limits have been placed on the evolution of w(z). Now the potential of the method relies on an improved control of systematic effects. In this area, there has been substantial progress along the last years.\"\n },\n {\n \"section_type\": \"RESULTS\",\n \"section_title\": \"Results on systematic uncertainties\",\n \"text\": \"Huge advances have taken place in controlling the uncertainties due to extinction by dust, the universality of the supernova properties, the control of lensing by dark matter and the observational process leading to the the construction of the Hubble diagram of SNe.\\n\\nThe study with the HST of the host galaxies of the SNe Ia sample of the Supernova Cosmology Project [101] allowed to subclassify the galaxy hosts of the P99 sample [2] . It was found that, at high z, early-type galaxies show a narrower dispersion in SNe Ia properties than late-type galaxies, as they do at low z ([123]). This result is very encouraging as it supersedes previous dispersion values obtained in samples of supernovae in all galaxy types. Supernovae in dust-rich environments, 17 such as spiral galaxies, are more affected by extinction. The supernova magnitudes are corrected regularly from Galactic extinction [124], but the correction by dust residing in the host galaxy or along the line of sight is often not included in the standard fit, as it would require to have extensive color information that might not be available for high-z SNe Ia. When it is included, it is still subject to the uncertainty in the extinction law [110] . Let us, for instance, consider the correction of dimming by dust as entering in the term A B , the magnitudes of extinction in the B band. We take, for instance, a supernova of a given stretch s, and we place its m eff (the magnitude of the equivalent supernova of s = 1) in the Hubble diagram. We might use, as in [2, 51] :\\n\\nm eff = m B + α(s -1) -A B ( 37\\n\\n)\\n\\nThe extinction in the canonical band B of the spectrum (A B ) is determined by the extinction coefficient R B in that band. Any uncertainty in this coefficient as due to variation of dust properties, leads to uncertain estimates of A B . R B is of the order of 4.1 ± 0.5 as properties of dust might change going to high z. If we have an error of 0.1 in the observable E(B -V ) due to dust extinction, it gets amplified as:\\n\\nA B ≡ R B × E(B -V ) ( 38\\n\\n)\\n\\nThe present way of dealing with SNe Ia, including information in several photometric bands of the spectrum where extinction coefficients (R V , R I ) are lower and one gets smaller values in the corresponding A V and A I , allows to control the dust problem.\\n\\nIn fact, the coefficients R B , R V and R I are also determined by methods determining distances to SNe Ia. In a sample of 133 nearby SNe Ia, it is found in [110] , that the dust in their host galaxies is well described by a mean extinction law with R V ≃ 2.7. In all this, the level at which SNe Ia are treated, including how many bands are used, becomes critical. A use of SNe Ia without taking into account color information leads to a dispersion of 0.17mag. When taking into account color information, the limit goes down to 0.11mag. Limiting the effect of extinction, as in SNeIa in ellipticals should bring down the dispersion. This finding is motivating the ongoing SN searches in clusters of elliptical galaxies [125] . In environments where dust is controlled, such as SNe in clusters rich in elliptical galaxies, the systematic effect introduced by dust extinction can be better controlled. There, one can also control the evolution of dust-properties.\\n\\nIn Table I , the level of accuracy in the first uses of SNe Ia back in 1998 is shown, with the contributions of each systematic and statistical error. The improvement is given for a future mission able to control the systematics in the best possible way. It is foreseen that several sources of systematic errors will become negligible within the next decade.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Evolution or Population drift\",\n \"text\": \"This question resides on whether we know that SNe Ia properties evolve with z. Much has been done along this path. Nowadays, while we know that dust is the unavoidable systematics, several studies confirm the universality of the light curve properties of SNe Ia. The study of a significant sample 18 of SNe Ia at various z by the Supernova Cosmology Project reveals that the rise times to maximum of high-z SNe Ia are similar to the low-z SNe Ia ([126, 112]). The same is found in the SNLS sample ([127]). Statistical evaluation is so far consistent with no difference between the low-z and high-z samples. Moreover, the possibility of existence of an extra parameter in the maximum brightnessrate of decline relation has been carefully examined. Among possible influences, metallicity was one of the obvious ones to consider. Examining supernovae in galaxies with a gradient of metallicity, it is found [128] that there is no evidence for metallicity dependence as an extra parameter in the light curve correlation of SNe Ia. This result comes from an analysis of the SNe Ia light curves of a sample of 62 supernovae in the local Universe. The SNe Ia belong to different populations along a metallicity gradient. This check has been done as well at high-z [129] using 74 SNe Ia (0.17 < z < 0.86) from the SCP sample. No significant correlation between peak SNIa luminosity and metallicity is found.\\n\\nOften we address the question of the spread of the SNe Ia samples in galaxies of different types: on how many fast SNe Ia versus intermediate or slow decliners are found in the various morphological types. Spirals at low or moderate redshifts should encompass all ranges of variation in the SNe Ia properties since they contain populations with a wide spread in age. The cosmological SN collaborations find that in those samples the one-parameter correlation does give a good description of the variation of the SNe Ia light curves: there is no residual correlation after the light-curve shape correction [112] . One can go a step further and test the SNIa objects themselves. This has been attempted in a intermediate z sample and it is a feasible project at all z [108] . The physical diagrams of SNe Ia at intermediate z and at low z seem to be similar [108] . Those diagrams reveal the composition, velocity gradient and radiation properties of SNeIa [108] . Looking at the inside of the SNe Ia is the best way to test whether they are the same at all z. Such step forward is giving definitive results which reaffirm the validity of the SNe Ia method. The progress done in this domain is summarized in Table I .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"K-corrections\",\n \"text\": \"There is another element yet to be introduced here: it deals with the K correction. The K-correction is necessary to calculate the apparent magnitude in a y filter band of a source observed in a x filter. As we are comparing the observed flux of supernovae in a given spectral band along the expansion history, we need to take into account the effect that the photons of the supernova have shifted to redder wavelengths and spread over a different spectral wavelength range and the observed flux is different from that at emission.\\n\\nm Y (z) = m X (z = 0) + K xy (z) ( 39\\n\\n)\\n\\nwhere\\n\\nK xy = 2.5 log (1 + z) F (λ)S x (λ)dλ F [λ/(1 + z)] S y (λ)dλ + Z y -Z x ( 40\\n\\n) 19 where F (λ) is the source spectral energy distribution (a SN in this case), and S x (λ) is the transmission of the filter x. The Z y -Z x term accounts for the different zero points of the filters [130] . At present, this term amounts to an uncertainty of 0.02mag in the overall budget of SNeIa as distance indicators, but it can be decreased to 0.01 mag. The comparisons of supernovae with the fine time sequences of spectra along all phases available, are turning the uncertainty in the K-correction into a decreasingly small figure. The large amount of spectral data on SNe Ia, and the good correlation with photometric properties, can indeed make this uncertainty very small.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Gravitational lensing of SNe Ia Gravitational lensing causes dispersion in the\",\n \"text\": \"Hubble diagram for high redshift sources. There was an early concern on the systematics and bias that this effect could cause in the determination of the cosmological parameters using SNeIa [131] . It has been shown through the study of lensing in the highest z SNe Ia sample [132] that the magnification distribution of SNeIa matches very well the expectations for an unbiased sample, i.e., their mean magnification factor is consistent with unity. The effect can be very well controlled by studying the galaxy fields [132] . Moreover, SNe Ia should show a negligigle systematics caused by gravitational lensing when having a few SNeIa per redshift bin [122] . This means that in future missions that effect should not have impact in limiting the accuracy of w(z).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The limit of SNeIa as calibrated candles\",\n \"text\": \"The better we deal with the light curve and spectral information of SNe Ia, the lower becomes the dispersion in the Hubble diagram. If SNe Ia have a systematic error in their use (independently from evolution), this is likely well below the accuracy to which we would like to determine distances (1%). We have not yet hit that brick wall. Using multi-epoch spectral information and accurate light curves in various filters should enable to test the power of these candles and their limiting irreduceable error [108].\\n\\n5.3.6. A joined sample Supernovae collected along years show the effect of observations treated with various degrees of accuracy. The dispersion around the best fit reflects that fact. For instance, some samples would have information on only two colors and at very few epochs, making it difficult to control reddening, or supernovae had no spectrum confirming that they were SNe Ia. The new samples will naturally have a higher degree of accuracy in magnitude and extinction for each supernova than previous samples. The ones with poorer information have been placed in [29, 52] into the \\\"silver category (as opposed to \\\"gold\\\" category). However, even within the Gold sample the requirements have changed: the latest definition of Gold sample demands higher quality standards than in [29] . In some of these SNe Ia lists, distance moduli of supernovae are assembled but they have been obtained through procedures which deal differently with dust extinction. The Hubble diagram resulting from those compilations would naturally show a large dispersion. In [133], it is argued that a few supernovae of an early High Z SN Search sample are pulling w(z) towards evolution. The exploration 20 done with a method independent from that in [29] showed that the central value in the 1σ confidence varied according to the light curve and reddening fitter of those SNe Ia [134] . Fortunately, supernovae enable to compare results obtained with different light curve fitting approaches. The published photometry by the various collaborations can be used to bring together the SNe Ia by using a consistent procedure into the Hubble diagram. This was done for the SNe Ia considering available up to 2005 in [134] . This is done in [122, 135, 136] for samples including the latest data from the Higher Z collaboration.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Testing the adequacy of a FRW metric\",\n \"text\": \"In most of the above discussion on dark energy, the FRW metric is taken as the right metric for our Universe. Modern cosmology supports that the Universe is highly isotropic on average on high scales. The main evidence is coming from the statistical isotropy of the Cosmic Microwave Background radiation. There are also cosmological observations supporting a matter distribution homogeneous on large scales, of the order of 100 h -1 Mpc. Given the observed structures of matter in voids and filaments, it is worth considering whether departures from the homogeneous and isotropic Universe could alter the present discussion, in particular, the possibility that there were no need for dark energy, but instead, that the observations are the result of an effect of inhomogeneity [137] .\\n\\nAs pointed out in [138] , large samples of supernovae will soon allow to test the luminosity distance relation in various directions and redshift bins. One can consider the luminosity distance d L as a function of direction n and redshift z [138] .\\n\\nThe direction-averaged luminosity distance is:\\n\\nd (0) L (z) = 1 4π dΩ n d L (z, n) = (1 + z) z 0 dz ′ H(z ′ ) ( 41\\n\\n)\\n\\nThis is the usual value retrieved by the SN collaborations. But its directional dependence d L (z, n) can be a test of the validity of the isotropy assumption. The authors in [138] propose to expand the luminosity distance in terms of spherical harmonics, to obtain the observable multipoles C l (z). This idea was also investigated in [109] , where the motion of our Local Group with respect to the CMB was measured. Results in [138] are in agreement with the results in [109] . Moreover, we can think of using the dispersion in the magnitude of SNe Ia in each direction n to put constrains on anisotropic models. As discussed in [139] , the dispersion of the magnitude in the Hubble diagram is at odds with some simple inhomogeneous models that predict a higher spread in m(z) and also a non-linear behavior. The anisotropic and inhomogeneous models have to confront present and future values on the SNeIa dispersion at each redshift interval z bin and in every direction n:\\n\\nσ(d L (z bin , n)) d L (z bin , n) = ln(10) 5 σ(m(z bin , n)) ( 42\\n\\n)\\n\\n21 The scaling solutions of the regionally averaged cosmologies can simulate the presence of a quintessence field or other negative pressure component [137, 142, 141] . But, any backreaction from inhomogeneity and anisotropy would show in the dispersion in the Hubble diagram. Those challenges have to be met by the non-FRW proposals. Up to now, the exploration of toy models calculating the effect of inhomogenities in the scale factor, find irregularities in the expansion rate beyond what is observed in the SN Ia Hubble diagram. The dispersion in the Hubble diagram of a standard candle is increased [139] .\\n\\nThe situation concerning the predictions of inhomogeneous models [137, 140, 141] is summarized in [142] . It is concluded that the inhomogeneities that would be able to amount to a d L (z) effect as observed from supernovae must be subhorizon and of strong type and they require a treatment beyond first-order perturbation theory. Research in this field will continue towards quantitative predictions to improve the connection with observational cosmology [143].\"\n },\n {\n \"section_type\": \"METHOD\",\n \"section_title\": \"Next decade experiments\",\n \"text\": \"The aim of discriminating between various possibilities for dark energy places stringent requirements on the use of standard candles [144] . Within the two decades from the discovery (1998) and the gathering of space mission data, it is feasible to reduce largely the errors, as mentioned before, and this would reflect into an error of only 1%-2% in d L (z). Errors much bigger than this 1% of error in d L result in the impossibility to distinguish between models. Even at the level of 1% in d L (z) one finds some degeneracy [144, 145] . Space projects aimed at determining the nature of dark energy are counting on a systematic error in the method around 0.02 mag (i.e. 1% in the distance) [95] . A different question is whether from ground, within the planned experiments at work in the present decade, one can converge towards that limit. These questions have been investigated in detail by using Monte Carlo simulations including the systematic errors in m(z) (or d L (z)) to be found in the evaluation of the present value of the equation of state of dark energy w 0 and its first derivative in the scale factor w a [95] (see expression 33). SNAP is a mission that has been specifically designed to gather about 2000 SNe Ia up to z ∼ 1.7. As a single mission gathering supernova data with the same instrumentation, it can eliminate possible photometric offsets and include high-precision photometry out to 1.7 microns [146] . With such large number of SNe Ia, statistical errors in every redshift bin of ∆z = 0.1 become negligible. The error is dominated by the systematics. The total error in each redshift bin is given by:\\n\\nσ bin = σ 2 st N bin + σ 2 sys ( 43\\n\\n)\\n\\n22 with expected systematic error; σ sys = 0.02 z/z max (44)\\n\\nTable I gives a progress evaluation in the control of high-redshift supernova distance uncertainties.\\n\\nTABLE I ERROR CONTRIBUTIONS TO HIGH SUPERNOVA DISTANCES in 1998 Systematic Uncertainties (1σ) Magnitude Statistical Uncertainties (1σ) Magnitude Photometric System Zero Point 0.05 Zero Points, S/N 0.17 Evolution < 0.17 K-corrections 0.03 Evolution of Extinction Law 0.02 Extinction 0.10 Gravitational Lensing 0.02 Light curve sampling 0.15 PROSPECTS TOWARDS 1% ERROR IN d L (z) in a space mission Systematic Uncertainties (1σ) Magnitude Statistical Uncertainties (1σ) Magnitude Photometric System Zero Point a 0.01 Zero Points, S/N 0.00* Evolution 0.00* K-corrections 0.00* Evolution of Extinction Law < 0.02 Extinction 0.00* Gravitational Lensing < 0.01 Light curve sampling 0.00* a 0.00* means values of the order of 10 -3 . The statistical errors are for the redshift bin. b In the Table for prospects in a space mission, the statistical error from light curve sampling of SNe Ia is reduced due to the large number of SNe Ia data per redshift bin.\\n\\nGround-based projects will not be able to achieve such accuracy. They aim at a reduction of systematic errors to 0.04-0.05 mag. Much of what will be feasible from now until the next space era of determination of the equation of state, depends on the use of all the information already gathered in the best way. Spectra gathered so far can be exploited further than what it has been done up to now, in order to reduce systematic errors. In the coming years, a lot can be done to improve the present dispersion of the Hubble diagram of SNe Ia and aim at a scatter of 1% in d L . The above limit is only reachable if all the systematic effects are dealt with to the best.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Complementarity\",\n \"text\": \"The SNe Ia test of w(z) benefits from using information on Ω M obtained by some other method. This can come from weak lensing or from large scale structure [147, 148, 149] .\\n\\nBaryon acoustic oscillations can also provide a measurement of distance along z, in this case, angular distances d A (z) and of E(z) = H(z)/H 0 [27, 150, 151] . Generally, for any geometry, one has:\\n\\nA = √ Ω M E(z) 1 3 1 z |Ω K | S |Ω K | z 0 dz ′ E(z ′ ) 2 3 ( 45\\n\\n)\\n\\n23 A measurement of d A has been obtained [27] .\\n\\nA = √ Ω M E(z 1 ) 1/3 1 z 1 z 1 0 dz E(z) 2/3 = 0.469 ± 0.017 ( 46\\n\\n)\\n\\nwhere z 1 = 0.35 is the redshift at which the acoustic scale has been measured in the redshift sample. Combining BAO and SNeIa distances one can aim at tracing dark energy all through a high z range. The value of w(z = z rec ) is provided by WMAP and Planck. In between, other methods such as studying the growth of large scale structure (LSS) obtained from Lyα clouds and clusters would depict the growth factor of structure along z. Considering the information at high z derived from LSS, SNe Ia and CMB should lead to learn about early dark energy [152, 153, 90, 148, 154, 155] .\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions\",\n \"text\": \"Type Ia supernovae can test GR against other alternative theories of gravitation. Theories of higher dimensional gravity have different Friedmann equations from those originating from GR. Some of those modified gravity proposals have already been tested and ruled out through the Hubble diagram of SNeIa up to very high z.\\n\\nIf dark energy is due to a scalar field either minimally coupled to gravity (i.e. quintessence) or non-minimally coupled to gravity (arising from scalar-tensor gravity or the effective low-energy superstring proposals), the Hubble diagram up to z > 1 obtained by the supernova collaborations gives the opportunity for exhaustive tests. In what concerns scalar-tensor gravity, supernovae are a precision experiment in a way analogous to the observations of the PSR B1913+16 or to solar system tests.\\n\\nMoreover, supernovae have already provided hints on the form of the potential of the hypothetical scalar field (for the case of minimally coupled field) that could be associated to the acceleration of the Universe.\\n\\nNonetheless, GR plus a form of cosmological constant is not ruled out observationally. Finding out how is w close to z = 0 is one of the most intriguing questions. Hopefully, this will be clarified by the samples to come. The goal of obtaining a consistent picture on w(z) up tp z ∼ 1.5 from the present samples is approached through the various SN procedures used by the existing collaborations.\\n\\nUltimately, supernovae together with complementary methods just recently discovered shall select the correct ideas on dark energy and gravity. If any modified gravity theory has a cosmological effect, those methods should start to see it.\\n\\nAcknowledgements I would like to thank Alex G. Kim and Marek Kowalski for reading the manuscript and pointing to useful suggestions. This work has been supported by grant AYA2006-05369.\\n\\n24 References [1] Riess A G et al. (The High Z SN Search Collaboration) 1998 Astron. J. 116 1009 [2] Perlmutter S et al.(The Supernova Cosmology Project) 1999 ApJ 517 565 [3] Einstein A 1917 Sitzungsber. Preuss. Akad. Wiss. Berlin 142-152 [4] Brans C H and Dicke R H 1961 Phys. Rev 124 925 [5] Will C 1993 Theory and experiment in gravitational physics, Cambridge (London) [6] Green M B, Schwarz H and Witten E. 1988 Superstring Theory (Cambridge University Press, Cambridge) [7] Arkani-Hamed N, Dimopoulos N and Dvali G 1998 Phys. Lett. B 429 263 (Preprint hep-ph/9807344) [8] Dvali G Gabadadze G and Porrati M 2000 Phys Lett B 485 208 [9] Deffayet C Dvali G and Gavadadze G 2002 Phys. Rev. D 65 044023 [10] Adelberger E G, Heckel B and Nelson A E 2003 Ann. Rev. Nucl. Part. Phys. 53 77 [11] Adelberger E G, Heckel B and Hoyle C D 2005 Physics World 18 41 [12] Adelberger E G et al. 2007 Phys. Rev. Lett. 98 131104 (Preprint hep-th/0611223) [13] Kneller J P and Steigman G 2003 Phys Rev D 67 063501 [14] Koyama K and Maartens R 2006 JCAP 601 016 [15] Maartens R 2004 Living Rev. Rel. 7 7 [16] Spergel D N et al. (WMAP Collaboration) 2003 ApJ Supp 148 175 (Preprint astro-ph/0302209) [17] Sahni V and Starobinsky A 2006 Int. J. M. Phys D 15 2105 (Preprint astro-ph/0610026) [18] Calcagni G, de Carlos B and De Felice A 2006 Nucl. Phys. B 752 404 (Preprint hep-th/0604201) [19] de Felice A Hindmarsh M and Trodden M 2006 JCAP 0608 005 (Preprint astro-ph/0604154) [20] Gorbunov D, Koyama K and Sibiryakov S 2006 Phys. Rev. D 73 044016 (Preprint hep-th/0512097) [21] Padilla A 2007 (Preprint hep-th/0610093) [22] Trodden M 2006 in From Quantum to Cosmos: Fundamental Physics Research in Space, Washington (Preprint astro-ph/0607510) [23] Lukas A, Ovrut B.A., Stelle K.S and Waldram D 1999 Phys. Rev. D 59 086001 (Preprint hep-th/9803235) [24] Binetruy P, Deffayet C and Langlois D 2000 Nucl. Phys. B 565 269 (Preprint hep-th/9905012) [25] Fairbairn M and Goobar A 2005 Phys. Lett. B 642 432 (Preprint astro-ph/0511029) [26] Maartens R and Majerotto E 2006 Phys. Rev. D 74 023004 (Preprint astro-ph/0603353) [27] Eisenstein D J et al. 2005 ApJ 633 560 [28] Wang Y and Mukherjee P 2006 Astrophys. J 650 1 (Preprint astro-ph/0604051) [29] Riess A G et al. (the Higher-Z Team) 2004 ApJ 607 665 [30] Astier P et al. (The SNLS Collaboration) 2006 A & A 447 31 (Preprint astro-ph/0510447) [31] Dvali G and Turner M 2003 Fermilab-Pub-03-040-A (Preprint astro-ph/0301510) [32] Randall L and Sundrum R 1999 Phys. Rev. Lett 83 3370 [33] Randall L and Sundrum R 1999 Phys. Rev. Lett 126 207 [34] Binetruy P, Deffayet C, Ellwanger U and Langlois D 2000 Phys. Lett. B 477 285 [35] Langlois D 2002 Int J. Mod Phy A 17 2701 (Preprint gr-qc/0205004) [36] Brax P and van de Bruck C 2003 Class. Quan. Grav. 20 201 (Preprint hep-th/0303095) [37] Easson D A 2000 Int. J. Mod. Phys. A 16 4803 [38] Durrer R 2005, hep-th/0507066 [39] Weller J and Albrecht A 2001 Phys. Rev. D 86 1939 [40] Ratra B and Peebles P.J.E. 1988 Phys. Rev D 37 3406 [41] Wetterich C 1988 Nucl. Phys. B 302 668 [42] Steinhardt P J, Wang L and Zlatev I 1999 Phys. Rev D 59 123504 [43] Peebles P J E and Ratra B 2003 Rev. Mod. Phys 75 559 [44] Copeland E J, Sami M and Tsujikawa S 2006 Int. J. Mod. Phys. D 15 1753 (Preprint hep/th/063057) [45] Uzan J.-P 2003 Rev. Mod. Phys 75 403 [46] Bertolami O and Martins PJ 2000 Phys. Rev. D 61 064007 25 [47] Li C Holz D and Cooray A 2006, astro-ph/0611093 [48] Huterer T and Turner M S 1999 Phys. Rev. D 60 1301 [49] Huterer T and Turner M S 2001 Phys. Rev. D 64 3527 [50] Sahlén M Liddle A R and Parkinson D 2007 Phys. Rev D 75 023502 (Preprint astro-ph/0610812) [51] Knop R A, et al. (The Supernova Cosmology Project) 2003 Ap J 598 102 [52] Riess A G et al. (the Higher-Z Team) 2007 ApJ 659 98 [53] Brans C H 2005 in Contributions to Santa Clara 2004 1st. International Workshop on Gravitation and Cosmology (Preprint gr-qc/0506063) [54] Esposito-Farese G. 2004, in Phi in the Sky: The Quest for Cosmological Scalar Fields, AIP Conference Proceeding 736 35 [55] Amendola L 1999 Phys. Rev. D 62 043511 (Preprint astro-ph/9908023) [56] Amendola L and Quercellini C 2003 Phys. Rev. D 68 023514 (Preprint astro-ph/0303228) [57] Boisseau B. Esposito-Farese G Polarski D and Starobinsky A 2000 Phys. Rev. Lett 85 2236 (Preprint gr-qc/0001066) [58] Esposito-Farese G and Polarski D. 2001, Phys. Rev. D 63, 063504 [59] Perivolaropoulos L 2005 JCAP 0510 001 [60] Teyssandier P and Tourrenc P 1983 J.Math. Phys 24 2793 [61] Gasperini M and Veneziano G 2003 Phys. Rept 373 1 [62] Barrow J D and Cotsakis S 1988 Phys. Lett. B 214 515 [63] Barrow J D and Maeda K 1990 Nucl. Phys B 341 290 [64] Holden D J and Wands D 2000 Phys. Rev. D 61 043506 [65] Wands D 1994 Class Quant Grav 11 269 [66] Starobinsky A A 1980 Phy Lett B 91 99 [67] Chiba T 2003 Phys. Lett. B 575 ( Preprint astro-ph/0307338) [68] Capozziello S, Carloni S. and Trosi A. 2003 Int. J. Mod. Phys. D 12 1969 (Preprint astro-ph/0303041) [69] Carroll S M, de Felice A and Trodden M 2005 Phys. Rev. D 71 023525 [70] Mena O, Santiago J and Weller J 2006 Phys. Rev. Lett 96 [71] Sotirou T P 2006 Class. Quantum Grav. 23 5117 [72] Amendola L, Polarski D and Tsujikawa S 2007 Phys. Rev. Lett 98 131302 (Preprint astro-ph/0603703) (astro-ph/0603703) [73] Antoniadis I 1992 in Proceedings of String theory and Quantum gravity, Trieste 1992 286 (Preprint hep-th/9211055) [74] Antoniadis I, Rizos J and Tamvakis K 1994 Nucl. Phys. B 415 497, (Preprint hep-th/9305025) [75] Neupane I P 2006 Class. Quant. Grav. 23 7493 [76] Gross D J and Sloan J H 1987 Nucl. Phys B 291, 41 [77] Caldwell RR 2002 Phys. Lett. B 545 23 [78] Nojiri S, Odintsov S D and Sasaki M 2005 Phys. Rev. D 71 123509 [79] Lue A and Starkman G D 2004 Phys. Rev. D 70 101501 [80] Chimento L P Lazkoz R Maartens R and Quirós T 2006 JCAP 0609 004 [81] Bouhmadi-Lopez M, Gonzalez-Diaz P F and Martin-Moruno P 2006 (Preprint gr-qc/0612135) [82] Koivisto T and Mota D F 2007 Phys. Lett. B 644 104 (Preprint astro-ph/0606078) [83] Ellis J, Kalara S, Olive K. A. and Wetterich C 1989 Phys. Lett. B 228 264 [84] Wetterich C 1994 AA 301 321 (Preprint hep-th/9408025) [85] Brax P and Martin J 2006 JCAP 0611 008 [86] Damour T Piazza F and Veneziano G 2002 Phys. Rev D 66 046007 [87] Brax P and Martin J 2006 (Preprint hep-th/0612208) [88] Hoyle CD et al. 2004 Phys. Rev. D 70, 042004 [89] Murphy T W http://physics.ucsd.edu/ ∼tmurphy/apollo/doc/multiplex.pdf [90] Doran M Robberts G and Wetterich C 2007 Phys. Rev D 75 023003 (Preprint astro-ph/0609814) [91] Witten E. 2000 Lecture at DM2000, Marina del Rey (Preprint hep-ph/0002297) 26 [92] Susskind L 2003 (Preprint hep-th/0302219) [93] Chevalier M and Polarski D 2001 Int J. Mod. Phys. D 10 213 (Preprint gr-qc/0009008) [94] Linder E V 2003 Phys. Rev. Lett 90 091301 [95] Kim A G, Linder E V, Miquel R and Mostek N 2004 MNRAS 347 909 [96] Kowal C T 1968 AJ 73 1021 [97] Filippenko A V and Riess A G 1998 Phys. Rep. 307 31 [98] Pskovskii Y P. 1977 Soviet Astron. 21, 675 [99] Phillips M M 1993 ApJ 413 L105 [100] Phillips M M et al. 1999 AJ 118 1766 [101] Sullivan et al. (The Supernova Cosmology Project) 2003 MNRAS 340 1057 [102] Nugent P, Kim A and Perlmutter S 2002 PASP 114 803 [103] Leibundgut B 2002 ARAA 39 67 [104] Matheson T. et al. 2005 (ESSENCE) AJ 129 2352 [105] Hook I M et al (The Supernova Cosmology Project) 2005 AJ 130 2788 [106] Lidman C et al. (The Supernova Cosmology Project) 2005 A A 430 843 [107] Blondin S et al. (ESSENCE) 2006 AJ 131 1648 [108] http://www.am.ub.es/ bcnde/DARKENO.html [109] Riess A G, Press W H and Kirshner R P 1995 ApJ 438 L17 [110] Jha S Riess A G and Kirshner R P 2007 ApJ Astrophys. J 659 122 [111] Perlmutter S et al. (The Supernova Cosmology Project) 1998 Nature 391 51 [112] Goldhaber G et al. (The Supernova Cosmology Project) 2001 ApJ 558 359 [113] Tonry J.L. et al. (the High Z SN Search Collaboration) 2003 ApJ 594 1 [114] Barris B J et al. (The High Z SN Search Collaboration) 2004 ApJ 602 571 [115] Clocchiatti A et al. (The High Z SN Search Collaboration) 2006 ApJ 642 1 [116] Freedman W L et al. 2001 ApJ 553 47 [117] Sandage A et al. 2006 Astrophys. J 653 843 (Preprint astro-ph/0603647) [118] Ruiz-Lapuente P 1997 Astr. Soc. Pac. Conf. Ser. 126 207 (Preprint astro-ph/9710019) [119] Ruiz-Lapuente P 1996 Ap. J. Lett 465 83 (Preprint astro-ph/9604044) [120] Goobar A and Perlmutter S 1995 ApJ 450,14 [121] Garnavich P M et al. 1998 ApJ 509 74 [122] Wood-Vasey W M et al. (ESSENCE) 2007 (Preprint astro-ph/0701041) [123] Branch D, Romanishin W and Baron E 1995 ApJ 465 73 (Preprint astro-ph/9510071) [124] Schlegel D. J., Finkbeiner M and Davis M. 1998 ApJ.S 500 525 [125] Perlmutter S 2005 HST Proposal-GO 10496 [126] Aldering G, Knop R and Nugent P 2000 AJ 119 210 [127] Conley A. et al. (The SNLS Collaboration) 2006 AJ 132 1707 [128] Ivanov V D, Hamuy M, and Pinto P A 2000 ApJ 542 588 [129] Quimby R et al. (The Supernova Cosmology Project) 2002 BAAS 201 2305 [130] Kim A G, Goobar A and Perlmutter S. 1996 PASP 108, 190 [131] Amanullah R, Mörtsell E and Goobar A 2003 A & A 397 819 [132] Jonsson J, Dahlen T, Goobar A, Gunnarson C, Mortsell E and Lee K 2006 ApJ 639 991 [133] Nesseris S and Perivolaropoulos L 2007 (Preprint astro-ph/0612653) J. Cosmol. Astropart. Phys. 0702 025 [134] Ruiz-Lapuente P et al. 2006 http://www.am.ub.es/ bcnde/DARKENO.html Ruiz-Lapuente P 2007 Invited review at Bernard's Cosmic Stories, Valencia June 2006 [135] Rubin D et al. (The Supernova Cosmology Project) 2006 BAAS 209 9003R [136] Kowalski et al. (The Supernova Cosmology Project) 2007, in preparation [137] Buchert T, Larena J and Alimi J-M 2006 Class Quant. Grav 23 6379 [138] Bonvin C, Durrer R and Kunz M 2006 Phys. Rev. Lett 96 191302 [139] Discussions on inhomogeneous cosmologies at the conference Key Approches to Dark Energy Barcelona 2006. This author thanks all the participants for addressing this topic.\\n\\n27 [140] Rasanen S 2006 JCAP 0611 003 [141] Kolb E W, Matarrese S, Notari A and Riotto A 2005 (Preprint astro-ph/0506534) [142] Célérier M-N 2000 Astron. Astrophys 353 63 [143] Buchert T 2007 gr-qc/0612166 [144] Maor I and Brustein R 2003 Phys. Rev D 67 103508 [145] España-Bonet C and Ruiz-Lapuente P 2005 (Preprint hep-ph/0503210) [146] http://snap.lbl.org [147] Heavens A F, Kitching T D and Taylor A N 2006 MNRAS 373 105 [148] Tegmark M et al. 2006 Phys. Rev. D 74 123507 (Preprint astro-ph/0608632) [149] Percival W J et al. 2007 Astrophys. J 657 645 (Preprint astro-ph/0608635) [150] Nichol R 2006 in Cosmology, Galaxy Formation and Astroparticle Physics on the Pathway to SKA (eds Klöckner H-R et al.) [151] Yamamoto K Bassett B A Nichol R C and Suto Y 2006 Phys Rev D 74 063525 (Preprint astro-ph/0605278) [152] Frieman J A , Huterer D, Linder E V and Turner M S 2003 Phys. Rev. D 67 083505 (Preprint astro-ph/0208100) [153] Wang Y and Freese K 2006 Phys. Lett. B 632 449 [154] Dunkley J, Bucher M, Ferreira P G, Moodley K and Skordis C 2005 Phys. Rev. Lett. 95 261303 [155] Kunz M, Trotta R and Parkinson D 2006 Phys. Rev. D 74 023503\"\n }\n]","isConversational":false}}},{"rowIdx":186,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1060"},"arxiv_id":{"kind":"string","value":"0704.1060"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1016/j.physletb.2007.05.028"},"content_hash":{"kind":"string","value":"5c344ad79790e17129e4b9cd0d618b59413fdc31e40f21d7653b7ad44038512e"},"title":{"kind":"string","value":"Deformation of Dijkgraaf-Vafa Relation via Spontaneously Broken N=2\n Supersymmetry"},"abstract":{"kind":"string","value":"It is known that the fermionic shift symmetry of the N=1, U(N) gauge model\nwith a superpotential of an adjoint chiral superfield is replaced by the second\n(spontaneously broken) supersymmetry in the N=2, U(N) gauge model with a\nprepotential and Fayet-Iliopoulos parameters. Based on a diagrammatic analysis,\nwe demonstrate how the well-known form of the effective superpotential in the\nformer model is modified in the latter. A set of two equations on the one-point\nfunctions stating the Konishi anomaly is modified accordingly."},"authors":{"kind":"list like","value":["Hiroshi Itoyama","Kazunobu Maruyoshi"],"string":"[\n \"Hiroshi Itoyama\",\n \"Kazunobu Maruyoshi\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"For more than two decades, effective superpotential has been a central object in the nonperturbative study of N = 1 supersymmetric theories. This object is protected from perturbative corrections in the conventional sense [1] , and yet receives important nonperturbative corrections (see for example [2, 3] ). In recent years, analyses from superstring theory have revealed an interesting perturbative window into nonperturbative physics with the use of the gluino condensate superfield variable [4, 5, 6, 7] . In [8], field theoretic discussion based on the model with U(N) gauge group and rigid N = 1 supersymmetry (see eq. (2.2) for its action S N =1 ) is given and this is in accord with the string theory based developments.\n\nSuperstring theory, on the other hand, insists upon maximally extended supersymmetry with no adjustable parameter. A scenario that one may draw is that this extended supersymmetry becomes spontaneously broken to N = 1. Along this vein, a field theory model with U(N) gauge group and rigid N = 2 supersymmetry spontaneously broken to N = 1 has been introduced in [9, 10, 11] (see eq. (2.1) for its action S N =2 ), generalizing the abelian counterpart of [12] . (See also [13] for N = 2 supergravity and [14] for related discussions.) Several properties of this model have been derived.\n\nIn this letter, we make a first analysis on the interplay between the effective superpotential and partially as well as spontaneously broken N = 2 supersymmetry, shedding a light upon the comparison of the two models mentioned above. A key aspect of this comparison is that the fermionic shift symmetry of S N =1 gets replaced by the second (spontaneously broken) supersymmetry of S N =2 . In fact, this is one of the original motivations/results of [9] .\n\nThe fermionic shift symmetry of S N =1 supplies the well-known formula [7, 8] constraining the form of the effective superpotential, which is originally proposed from flux compactification of string theory [15, 16] . Based on a diagrammatic analysis [17] (for a review see [18] ), we are able to state how this form undergoes modifications in the model S N =2 . After giving a few accounts of the model in the next section, we present a diagrammatic analysis of W ef f in section III. Our final understanding is summarized in eq. (3.10). This is followed by a computation of the two-loop contribution to W ef f in section IV. In the final section, we derive a set of two equations on the two generating functions R(z) and T (z) of the one-point functions, generalizing the argument based on the chiral ring and the Konishi anomaly in [8] . We observe a modification from that given in [8] here as well.\n\nII. The U (N ) gauged model with spontaneously broken N = 2 supersymmetry Let us briefly recall a few ingredients of the model, which are needed in what follows. The action [9] given in the Wess-Zumino gauge can be written as S N =2 = d 4 xd 4 θ -i 2 Tr Φe adV ∂F (Φ) ∂Φ h.c. + ξV 0\n\n+ d 4 xd 2 θ - i 4 ∂ 2 F (Φ)\n\n∂Φ a ∂Φ b W a W b + eΦ 0 + m ∂F (Φ) ∂Φ 0 + h.c. , (2.1)\n\nwhere V = V a t a and W α are the vector superfield and the gauge superfield strength respectively and Φ = Φ a t a (a = 0, 1, . . . , N 2 -1) is the chiral superfield * . There are three Fayet-Iliopoulos parameters (e, m, ξ) which are all real. For simplicity, we choose the prepotential as a single trace function of degree n + 2: F (Φ) = n+1 k=1 g k TrΦ k+1 /(k + 1)!. While this action is shown to be invariant under the N = 2 supersymmetry transformations [9, 10], the vacuum breaks half of the N = 2 supersymmetries. Extremizing the scalar potential, we obtain the condition ∂ 2 F ∂Φ 0 ∂Φ 0 = -(e ± iξ)/m, which is a polynomial of order n and this determines the expectation value of the scalar field.\n\nThe action S N =2 in (2.1) is to be compared with that of the N = 1, U(N) gauge model with a single trace tree level superpotential W (Φ):\n\nS N =1 = d 4 xd 4 θ Tr Φe adV Φ + d 4 xd 2 θ Tr (iτ WW + W (Φ)) + h.c. , (2.2)\n\nwhere τ is a complex gauge coupling τ = θ/2π + 4πi/g 2 .\n\nIn [9] , it is checked that the second supersymmetry reduces to the fermionic shift symmetry in the limit m → ∞. The action S N =2 in fact reduces to S N =1 in the limit m, e, ξ → ∞ with mg k (k ≥ 2) fixed [19] . We show that our result reduces to that of [7, 17] in this limit.\n\nIn this letter, we consider the matter-induced part of the effective superpotential by integrating out the massive degrees of freedom Φ:\n\ne i R d 4 x(d 2 θW ef f +h.c.+d 4 θ(nonchiral terms)) = DΦD Φe iS N =2 .\n\n(3.1) * a = 0 corresponds to the overall U (1) part.\n\nLet us take W α (or V ) as the background field † . We consider the case of unbroken U(N) gauge group. For simplicity, we choose Φ = 0 by setting g 1 = -(e ± iξ)/m.\n\nWe are interested in the holomorphic superpotential which does not contain the antiholomorphic couplings ḡk . We can take ḡk = 0 for k ≥ 3 without loss of generality. Collecting the Φ dependent terms, we obtain\n\nSΦ = d 4 xd 4 θ -i 2 Tr Φe adV ∂F (Φ) ∂Φ -(ḡ 1 Φ + ḡ2 2 Φ2 )e adV Φ + d 4 xd 2 θ mḡ 2 2 Tr Φ2 = d 4 xd 4 θTr Φḡ 2 - 2m ∇ 2 + i 4 Φ Φ + i 2 ḡ1 Φ - ∂F ∂Φ Φ . (3.2)\n\nIn the last expression, we have introduced a covariantly anti-chiral superfield Φ = Φe adV , which satisfies ∇ α Φ = 0 (∇ α = e -adV D α e adV ). Eq. (3.2) is quadratic in Φ and can be integrated straightforwardly. As a result, we obtain the following terms,\n\n1 16ḡ 2 ḡ1 Φ - ∂F ∂Φ - 2m ∇ 2 + i 4 Φ -1 ḡ1 Φ - ∂F ∂Φ = (Img 1 ) 2 8m ḡ2 Φ∇ 2 Φ + . . . , (3.3)\n\nwhere . . . denotes the higher order interaction terms, which we will not consider here. Indeed, these interaction vertices are higher order in m -1 compared to the vertices which we consider below. These contribute to our main result (3.10) as higher order corrections in m -1 and do not spoil our conclusion that the effective superpotential is modified from the case of S N =1 (2.2).\n\nReplacing d 2 θ integration by -∇2 /4 and collecting the terms which are not in SΦ, we obtain an action after the Φ integration ‡ :\n\nd 4 xd 2 θTr - (Img 1 ) 2 32mḡ 2 Φ ∇2 ∇ 2 Φ + m n+1 k=2 g k k! Φ k - i 4 n+1 k=3 k-1 s=0 g k k! (WΦ s WΦ k-1-s ) . (3.4)\n\nThe first two terms are already present in the integrations with regard to the action S N =1 (2.2). The last term is new and originates from the gauge kinetic term in eq. (2.1). As we will see below, this last term does contribute to the effective superpotential and becomes responsible for the violation of the well-known relation [7, 8] between the effective superpotential of the gauge theory and the planar free energy of the matrix model having the tree level (bare) superpotential as its potential.\n\n† The simplest background is that consisting of a vanishing gauge field A µ and a constant gaugino λ α , which satisfies {λ α , λ β } = 0 [18] . This configuration implies that traces of more than two W vanish. ‡ In eq. (3.4), it is understood that the generating functional has a renormalized perturbation expansion in which a nonvanishing tadpole is always canceled by a nonvanishing value of the source coupled to Φ. This implies that the tadpole can in practice be ignored.\n\nAfter rescaling Φ → aΦ with a 2 = mḡ 2 /(Img 1 ) 2 , the quadratic part of the action (3.4) reduces to\n\n1 2 Φ -+ m ′ + 1 2 adW α D α Φ - ig ′ 3 2 (2WWΦ 2 + WΦWΦ),\n\nwhere we have used the relation ∇2 ∇ 2 Φ = 16( Φ -adW α D α Φ/2) and introduced m ′ = a 2 mg 2 and g ′ 3 = a 2 g 3 /12. The propagator in the momentum space is\n\n∆(p, π) = ∞ 0 dse -s(p 2 +m ′ + 1 2 adW α πα-ig ′ 3 M ).\n\nThe Grassmann momentum π α is Fourier transformation of superspace coordinate θ α and the matrix M is\n\nM abcd = (WW) da δ bc + (WW) bc δ da + W da W bc , (3.5)\n\nwhere we have exhibited the gauge index dependence explicitly. This matrix is not present in the propagator of [17] . Using eq. (3.5), we are able to insert W without involving the momentum π α .\n\nThe interaction terms in eq. (3.4) are divided into the following two types: type I. m g k a k k! Tr Φ k , k = 3, . . . , n + 1. type II.\n\ni 4 k-1 s=0 g k a k-1 k! Tr(WΦ s WΦ k-1-s ), k = 4, . . . , n + 1.\n\nType I vertices are already present in [17] . Type II vertices are not present in [17] . They insert two W in specific ways.\n\nBefore going on to consider loop diagrams, let us first demonstrate that we have only to consider planar diagrams in our case as well [17, 18] . For a given diagram, we denote by V the number of vertices, by P the number of propagators and by h the number of holes (or index loops). There are V sets of chiral superspace integrations from V vertices. One of them becomes the chiral superspace integration over the effective superpotential, and the number of remaining π α momentum integrations is P -V + 1. These Grassmann integrations must be saturated by 1 2 adW α π α terms in the propagators. Furthermore, we can freely insert W both from the M terms in the propagators and from the type II vertices. If we denote the number of these additional insertions by 2α, the total number of W insertions is 2(P -V + 1 + α). On the other hand, one index loop can accommodate at most two W. Thus we have h ≥ P -V + 1 + α. This implies that only the planar diagrams contribute to the effective superpotential as the Euler number of the diagram is χ = V -P + h.\n\nA planar diagram with h index loops has (h -1) loop momenta. Let us consider the (h -1)-loop planar diagrams (contributing to the (h -1)-loop vacuum amplitude) in which all vertices are type I. Let us, for a moment, ignore the M term of (3.5). The calculation is then the same as that of [17] which we briefly describe. Each diagram is a product of the bosonic part obtained by integrating over the momentum p and the fermionic one coming from the π α integrations. As we have seen in the last paragraph, we have exactly 2(h -1) W insertions in the fermion part. There are two possibilities for these W insertions. The one is to keep one of the index loops empty, filling the remaining index loops with two W. This yields NS h-1 term, where\n\nS = -1 64π 2 Tr U (N ) W α W α .\n\nThe other is to fill each of two index loops chosen with single W, which yields S h-2 w α w α terms where w α = 1 8π TrW α . After calculating the both parts, we perform the Schwinger parameter integrals. Clearly this procedure is universal to every (h -1)-loop planar diagram up to the multiplications by the symmetric factor and by the coupling constants. Therefore every such diagram is a product of these factors with the following expression\n\nP i=1 ds i e -( P s i )m ′ 1 4 h-1 {NhS h-1 + h C 2 2S h-2 w α w α } ≡ P i=1 ds i e -( P s i )m ′ A (h-1) 0 , (3.6)\n\nwhere we have introduced A (h-1) 0\n\n. The factor h of the first term comes from the choice of the empty index loop, and h C 2 of the second term is the combination of inserting two W into different index loops. The most important fact is that the dependence on Schwinger parameters of the bosonic part is cancelled by that of the fermionic part. This explains that the calculation of the effective superpotential of the gauge theory reduces to that of the matrix model [17] .\n\nThere are two types of corrections to A (h-1) 0 . The one is due to the presence of the M terms in the propagators, which we denote by A (h-1) 1 . The other is due to the type II vertices, which is obtained by replacing one of the type I vertices in A (h-1) 0 by the corresponding type II vertex and by summing over all possibilities. We denote this by A (h-1) 2\n\n. We consider them in order.\n\nLet us see the effects of the M term, namely, eq. (3.5). It plays a role of inserting two W further. Thus we will obtain terms which are proportional to S h . Note that we cannot insert more than two W because, in such case, at least one of the index loops has more than two insertions of W. For the parts contributing to NS h-1 , which have an empty index loop, we can further insert W α W α from the first two terms in (3.5). In the case in which they are inserted in the a-th index loop, we obtain S 4 h-1 ig ′ 3 ia s ia TrWW, where i a labels the propagators which form the a-th index loop. The absence of factor N is explained by the absence of an empty index loop. The factor h is not present as we have so far 6 restricted ourselves to the a-th index loop. Summing over all index loops, we obtain the first contribution to A (h-1) 1\n\n: a S 4 h-1 ig ′ 3 ia s ia TrWW = 2ig ′ 3 i s i S 4 h-1\n\nTrWW,\n\nwhere we have used that when all index loops are summed, they pass through each double line propagator exactly twice.\n\nLet us note that the parts contributing to the second term of eq. (3.6) can receive further insertions of W as well. They have two index loops with a single W insertion, for which we can exploit the last term of M. An insertion of this term requires that two index loops share a propagator. Let us define the index A = 1, . . . , h C 2 as labeling the combinations of such two index loops and the index à labeling the cases which have a common propagator in the two index loops chosen. Let us further introduce the index i à labeling the common propagator in case Ã. With these notations, we obtain the second contribution to A (h-1) 1 :\n\n2S h-2 4 h-1 ig ′ 3 i à s i à 1 64π 2 W α ab W αcd W β ba W βdc = ig ′ 3 i s i S 4 h-1 TrW α W α .\n\nPutting all these together, we obtain the contributions from the vertices of type I, P i=1\n\nds i e -( P s i )m ′ (A (h-1) 0 + A (h-1) 1 (s i )) = h m ′P S 4 h-1 N - 16π 2 iP g 3 S hmg 2 + h C 2 2m ′P S 4 h-2 w α w α . (3.7)\n\nIt is important that the above new term has Schwinger parameter dependence aside from the exponential factor. In [17] , it was pointed out that the cancellation of this dependence represents the reduction of the system to the matrix model. The appearance of this new term with Schwinger parameter dependence may spoil this reduction. Note also that this new term does not have an overall factor N, indicating the violation of the well-known relation due to Dijkgraaf-Vafa [7].\n\nWe now turn to the vertices of type II which contain two W insertions. The ℓ-th order vertex in Φ is\n\nTr(2WWΦ ℓ + WΦWΦ ℓ-1 + . . . + WΦ ℓ-1 WΦ). (3.8)\n\nwhere we have omitted the overall factors. The first term inserts two W into an index loop while the remainder insert them into two different index loops. Having done 2(h -1) π α 7 Figure 1: two-loop planar diagrams integrations, we obtain 2(h -1) W insertions. We can therefore use vertex (3.8) only once in a diagram. When this is done, insertion of the M term from the propagator is disallowed.\n\nLet us consider A (h-1) 2 and suppose that one of the type I vertices, Tr Φ ℓ , is replaced by the above vertex (3.8). The first term connects ℓ index loops and we can insert W 2 into ℓ different ways. Thus we obtain S 4 h-1 2ℓTrWW as a contribution to A (h-1) 2\n\n. For the other terms of eq. (3.8), there are in total ℓ(ℓ -1) ways of inserting two W into different index loops. These give\n\n2S h-2 4 h-1 ℓ(ℓ -1) 1 64π 2 W α ab W αcd W β ba W βdc = S 4 h-1 ℓ(ℓ -1)TrWW.\n\nSumming the above two contributions, we obtain S 4 h-1 ℓ(ℓ + 1)TrWW. Thus, in any (h -1)-loop diagram, changing a vertex from type I to type II is equivalent to considering only NS h-1 terms in eq. (3.6) and changing the coupling constant by mg ℓ → 16π 2 ig ℓ+1 S Nh , for ℓ ≥ 3. (3.9) Therefore, we obtain a formula for the contribution from the (h -1)-loop diagrams with P propagators to W ef f in (2.1),\n\nW (h-1) ef f = N ∂F (h-1) ∂S + ∂ 2 F (h-1) ∂S 2 w α w α - 16π 2 iP mg 3 hmg 2 ∂F (h-1) ∂S S m + W (h-1) 2 , (3.10) where W (h-1) 2\n\nis defined by replacing, in the first term, one coupling constant according to eq. (3.9) and summing over all possibilities. We have denoted by F (h-1) the (h -1)-loop contribution to the planar free energy of the matrix model.\n\nAs a sample computation, let us take the two-loop contribution to the effective superpotential. There are two two-loop planar diagrams depicted in Fig. 1 . Collecting all possible 8 insertions of W, we obtain\n\nW ( 2\n\n) ef f = - (mg 3 ) 2 32(mg 2 ) 3 NS 2 - (mg 3 ) 2 16(mg 2 ) 3 Sw α w α + π 2 i(mg 3 ) 3 2(mg 2 ) 4 S 3 m - π 2 i(mg 3 )(mg 4 ) 3(mg 2 ) 3 S 3 m . (4.1)\n\nThe first two terms are the ones which are present in the computation based on [7, 8] with S N =1 . The third one comes from the M term in the propagator and the last one from the type II vertices. Note that, in the limit m → ∞ with mg k (k ≥ 2) fixed, we reproduce the result of [17] . In an arbitrary loop amplitude, the situation is the same: new terms are of order m -1 in this limit.\n\nThe overall U(1) part does not decouple from the SU(N) part. This can be easily seen by translating S into the glueball superfield Ŝ = -1 64π 2 Tr SU (N ) W α W α and extracting the factor in front of w α w α . By the existence of the last two terms in eq. (3.10), it is nonvanishing. For example, in the two-loop example, this part in (4.1) reads\n\nπi(mg 3 )[2(mg 2 )(mg 4 ) -3(mg 3 ) 2 ] 2(mg 2 ) 4 Ŝ2 m w α w α = 0,\n\nAn alternative approach to the effective superpotential is to exploit and extend the properties of the N = 1 chiral ring and the generalized Konishi anomaly equations based on reference [20, 8] . The anomalous Ward identity of our model for the general transformation δΦ = f (Φ, W) is\n\n- 1 64π 2 W α , W α , ∂f ∂Φ ij ij Φ = Trf W ′ (Φ) Φ - i 4 Tr(f F ′′′ (Φ)W α W α ) Φ , (5.1) where W ′′ (Φ) = mF ′′′ (Φ).\n\nIn terms of the two generating functions of chiral one-point functions\n\nR(z) = - 1 64π 2 TrW α W α 1 z -Φ Φ , T (z) = Tr 1 z -Φ Φ ,\n\nthe anomalous Ward identities (5.1) are\n\nR(z) 2 = W ′ (z)R(z) + 1 4 f (z), 2R(z)T (z) = W ′ (z)T (z) + 1 4 c(z) + 16π 2 iF ′′′ (z)R(z) + 1 4 c(z), 9\n\nwhere f (z) and c(z) are polynomials of degree n -1 in z and c(z) is a polynomial of degree n -2:\n\nf (z) = - 1 16π 2 Tr (W ′ (Φ) -W ′ (z))W α W α z -Φ Φ , c(z) = 4 W ′ (Φ) -W ′ (z) z -Φ Φ , c(z) = -i (F ′′′ (Φ) -F ′′′ (z))W α W α z -Φ Φ .\n\nThe last term of eq. (5.1) does not contribute to the equation for R(z) because of the chiral ring relation TrW α W α W β W β = 0. The equation for R(z) is the same as that of [8], which is the loop equation of the matrix model. On the other hand, the equation for T (z) alters from that of [8] .\n\nThe final step of this approach is to express the effective superpotential in terms of R(z) and T (z). Taking a variational derivative of (3.1) with respect to the coupling g k , we obtain\n\n∂W ef f ∂g k = m k! dzz k T (z) + 16π 2 i (k -1)! dzz k-1 R(z).\n\nHence we can determine the effective superpotential up to g k independent terms.\n\nWe thank Kazuhito Fujiwara, Yosuke Imamura, Hiroaki Kanno, Hironobu Kihara, Yasunari Kurita, Kazutoshi Ohta and Makoto Sakaguchi for useful discussions. We are grateful to Hiraku Yonemura for his collaboration at an early stage. This work is supported in part by the Grant-in-Aid for Scientific Research (18540285) from the Ministry of Education, Science and Culture, Japan. Support from the 21 century COE program \"Constitution of wide-angle mathematical basis focused on knots\" is gratefully appreciated. The preliminary version of this work was presented in YITP workshop \"Fundamental Problems and Applications of Quantum Field Theory\", YITP-W-06-16 in Yukawa Institute for Theoretical Physics, Kyoto University (December 14-16 2006) . We wish to acknowledge the participants for stimulating discussions."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. Introduction","text":"For more than two decades, effective superpotential has been a central object in the nonperturbative study of N = 1 supersymmetric theories. This object is protected from perturbative corrections in the conventional sense [1] , and yet receives important nonperturbative corrections (see for example [2, 3] ). In recent years, analyses from superstring theory have revealed an interesting perturbative window into nonperturbative physics with the use of the gluino condensate superfield variable [4, 5, 6, 7] . In [8], field theoretic discussion based on the model with U(N) gauge group and rigid N = 1 supersymmetry (see eq. (2.2) for its action S N =1 ) is given and this is in accord with the string theory based developments.\n\nSuperstring theory, on the other hand, insists upon maximally extended supersymmetry with no adjustable parameter. A scenario that one may draw is that this extended supersymmetry becomes spontaneously broken to N = 1. Along this vein, a field theory model with U(N) gauge group and rigid N = 2 supersymmetry spontaneously broken to N = 1 has been introduced in [9, 10, 11] (see eq. (2.1) for its action S N =2 ), generalizing the abelian counterpart of [12] . (See also [13] for N = 2 supergravity and [14] for related discussions.) Several properties of this model have been derived.\n\nIn this letter, we make a first analysis on the interplay between the effective superpotential and partially as well as spontaneously broken N = 2 supersymmetry, shedding a light upon the comparison of the two models mentioned above. A key aspect of this comparison is that the fermionic shift symmetry of S N =1 gets replaced by the second (spontaneously broken) supersymmetry of S N =2 . In fact, this is one of the original motivations/results of [9] .\n\nThe fermionic shift symmetry of S N =1 supplies the well-known formula [7, 8] constraining the form of the effective superpotential, which is originally proposed from flux compactification of string theory [15, 16] . Based on a diagrammatic analysis [17] (for a review see [18] ), we are able to state how this form undergoes modifications in the model S N =2 . After giving a few accounts of the model in the next section, we present a diagrammatic analysis of W ef f in section III. Our final understanding is summarized in eq. (3.10). This is followed by a computation of the two-loop contribution to W ef f in section IV. In the final section, we derive a set of two equations on the two generating functions R(z) and T (z) of the one-point functions, generalizing the argument based on the chiral ring and the Konishi anomaly in [8] . We observe a modification from that given in [8] here as well.\n\nII. The U (N ) gauged model with spontaneously broken N = 2 supersymmetry Let us briefly recall a few ingredients of the model, which are needed in what follows. The action [9] given in the Wess-Zumino gauge can be written as S N =2 = d 4 xd 4 θ -i 2 Tr Φe adV ∂F (Φ) ∂Φ h.c. + ξV 0\n\n+ d 4 xd 2 θ - i 4 ∂ 2 F (Φ)\n\n∂Φ a ∂Φ b W a W b + eΦ 0 + m ∂F (Φ) ∂Φ 0 + h.c. , (2.1)\n\nwhere V = V a t a and W α are the vector superfield and the gauge superfield strength respectively and Φ = Φ a t a (a = 0, 1, . . . , N 2 -1) is the chiral superfield * . There are three Fayet-Iliopoulos parameters (e, m, ξ) which are all real. For simplicity, we choose the prepotential as a single trace function of degree n + 2: F (Φ) = n+1 k=1 g k TrΦ k+1 /(k + 1)!. While this action is shown to be invariant under the N = 2 supersymmetry transformations [9, 10], the vacuum breaks half of the N = 2 supersymmetries. Extremizing the scalar potential, we obtain the condition ∂ 2 F ∂Φ 0 ∂Φ 0 = -(e ± iξ)/m, which is a polynomial of order n and this determines the expectation value of the scalar field.\n\nThe action S N =2 in (2.1) is to be compared with that of the N = 1, U(N) gauge model with a single trace tree level superpotential W (Φ):\n\nS N =1 = d 4 xd 4 θ Tr Φe adV Φ + d 4 xd 2 θ Tr (iτ WW + W (Φ)) + h.c. , (2.2)\n\nwhere τ is a complex gauge coupling τ = θ/2π + 4πi/g 2 .\n\nIn [9] , it is checked that the second supersymmetry reduces to the fermionic shift symmetry in the limit m → ∞. The action S N =2 in fact reduces to S N =1 in the limit m, e, ξ → ∞ with mg k (k ≥ 2) fixed [19] . We show that our result reduces to that of [7, 17] in this limit."},{"section_type":"OTHER","section_title":"III. Diagrammatic analysis of the effective superpotential","text":"In this letter, we consider the matter-induced part of the effective superpotential by integrating out the massive degrees of freedom Φ:\n\ne i R d 4 x(d 2 θW ef f +h.c.+d 4 θ(nonchiral terms)) = DΦD Φe iS N =2 .\n\n(3.1) * a = 0 corresponds to the overall U (1) part.\n\nLet us take W α (or V ) as the background field † . We consider the case of unbroken U(N) gauge group. For simplicity, we choose Φ = 0 by setting g 1 = -(e ± iξ)/m.\n\nWe are interested in the holomorphic superpotential which does not contain the antiholomorphic couplings ḡk . We can take ḡk = 0 for k ≥ 3 without loss of generality. Collecting the Φ dependent terms, we obtain\n\nSΦ = d 4 xd 4 θ -i 2 Tr Φe adV ∂F (Φ) ∂Φ -(ḡ 1 Φ + ḡ2 2 Φ2 )e adV Φ + d 4 xd 2 θ mḡ 2 2 Tr Φ2 = d 4 xd 4 θTr Φḡ 2 - 2m ∇ 2 + i 4 Φ Φ + i 2 ḡ1 Φ - ∂F ∂Φ Φ . (3.2)\n\nIn the last expression, we have introduced a covariantly anti-chiral superfield Φ = Φe adV , which satisfies ∇ α Φ = 0 (∇ α = e -adV D α e adV ). Eq. (3.2) is quadratic in Φ and can be integrated straightforwardly. As a result, we obtain the following terms,\n\n1 16ḡ 2 ḡ1 Φ - ∂F ∂Φ - 2m ∇ 2 + i 4 Φ -1 ḡ1 Φ - ∂F ∂Φ = (Img 1 ) 2 8m ḡ2 Φ∇ 2 Φ + . . . , (3.3)\n\nwhere . . . denotes the higher order interaction terms, which we will not consider here. Indeed, these interaction vertices are higher order in m -1 compared to the vertices which we consider below. These contribute to our main result (3.10) as higher order corrections in m -1 and do not spoil our conclusion that the effective superpotential is modified from the case of S N =1 (2.2).\n\nReplacing d 2 θ integration by -∇2 /4 and collecting the terms which are not in SΦ, we obtain an action after the Φ integration ‡ :\n\nd 4 xd 2 θTr - (Img 1 ) 2 32mḡ 2 Φ ∇2 ∇ 2 Φ + m n+1 k=2 g k k! Φ k - i 4 n+1 k=3 k-1 s=0 g k k! (WΦ s WΦ k-1-s ) . (3.4)\n\nThe first two terms are already present in the integrations with regard to the action S N =1 (2.2). The last term is new and originates from the gauge kinetic term in eq. (2.1). As we will see below, this last term does contribute to the effective superpotential and becomes responsible for the violation of the well-known relation [7, 8] between the effective superpotential of the gauge theory and the planar free energy of the matrix model having the tree level (bare) superpotential as its potential.\n\n† The simplest background is that consisting of a vanishing gauge field A µ and a constant gaugino λ α , which satisfies {λ α , λ β } = 0 [18] . This configuration implies that traces of more than two W vanish. ‡ In eq. (3.4), it is understood that the generating functional has a renormalized perturbation expansion in which a nonvanishing tadpole is always canceled by a nonvanishing value of the source coupled to Φ. This implies that the tadpole can in practice be ignored.\n\nAfter rescaling Φ → aΦ with a 2 = mḡ 2 /(Img 1 ) 2 , the quadratic part of the action (3.4) reduces to\n\n1 2 Φ -+ m ′ + 1 2 adW α D α Φ - ig ′ 3 2 (2WWΦ 2 + WΦWΦ),\n\nwhere we have used the relation ∇2 ∇ 2 Φ = 16( Φ -adW α D α Φ/2) and introduced m ′ = a 2 mg 2 and g ′ 3 = a 2 g 3 /12. The propagator in the momentum space is\n\n∆(p, π) = ∞ 0 dse -s(p 2 +m ′ + 1 2 adW α πα-ig ′ 3 M ).\n\nThe Grassmann momentum π α is Fourier transformation of superspace coordinate θ α and the matrix M is\n\nM abcd = (WW) da δ bc + (WW) bc δ da + W da W bc , (3.5)\n\nwhere we have exhibited the gauge index dependence explicitly. This matrix is not present in the propagator of [17] . Using eq. (3.5), we are able to insert W without involving the momentum π α .\n\nThe interaction terms in eq. (3.4) are divided into the following two types: type I. m g k a k k! Tr Φ k , k = 3, . . . , n + 1. type II.\n\ni 4 k-1 s=0 g k a k-1 k! Tr(WΦ s WΦ k-1-s ), k = 4, . . . , n + 1.\n\nType I vertices are already present in [17] . Type II vertices are not present in [17] . They insert two W in specific ways.\n\nBefore going on to consider loop diagrams, let us first demonstrate that we have only to consider planar diagrams in our case as well [17, 18] . For a given diagram, we denote by V the number of vertices, by P the number of propagators and by h the number of holes (or index loops). There are V sets of chiral superspace integrations from V vertices. One of them becomes the chiral superspace integration over the effective superpotential, and the number of remaining π α momentum integrations is P -V + 1. These Grassmann integrations must be saturated by 1 2 adW α π α terms in the propagators. Furthermore, we can freely insert W both from the M terms in the propagators and from the type II vertices. If we denote the number of these additional insertions by 2α, the total number of W insertions is 2(P -V + 1 + α). On the other hand, one index loop can accommodate at most two W. Thus we have h ≥ P -V + 1 + α. This implies that only the planar diagrams contribute to the effective superpotential as the Euler number of the diagram is χ = V -P + h.\n\nA planar diagram with h index loops has (h -1) loop momenta. Let us consider the (h -1)-loop planar diagrams (contributing to the (h -1)-loop vacuum amplitude) in which all vertices are type I. Let us, for a moment, ignore the M term of (3.5). The calculation is then the same as that of [17] which we briefly describe. Each diagram is a product of the bosonic part obtained by integrating over the momentum p and the fermionic one coming from the π α integrations. As we have seen in the last paragraph, we have exactly 2(h -1) W insertions in the fermion part. There are two possibilities for these W insertions. The one is to keep one of the index loops empty, filling the remaining index loops with two W. This yields NS h-1 term, where\n\nS = -1 64π 2 Tr U (N ) W α W α .\n\nThe other is to fill each of two index loops chosen with single W, which yields S h-2 w α w α terms where w α = 1 8π TrW α . After calculating the both parts, we perform the Schwinger parameter integrals. Clearly this procedure is universal to every (h -1)-loop planar diagram up to the multiplications by the symmetric factor and by the coupling constants. Therefore every such diagram is a product of these factors with the following expression\n\nP i=1 ds i e -( P s i )m ′ 1 4 h-1 {NhS h-1 + h C 2 2S h-2 w α w α } ≡ P i=1 ds i e -( P s i )m ′ A (h-1) 0 , (3.6)\n\nwhere we have introduced A (h-1) 0\n\n. The factor h of the first term comes from the choice of the empty index loop, and h C 2 of the second term is the combination of inserting two W into different index loops. The most important fact is that the dependence on Schwinger parameters of the bosonic part is cancelled by that of the fermionic part. This explains that the calculation of the effective superpotential of the gauge theory reduces to that of the matrix model [17] .\n\nThere are two types of corrections to A (h-1) 0 . The one is due to the presence of the M terms in the propagators, which we denote by A (h-1) 1 . The other is due to the type II vertices, which is obtained by replacing one of the type I vertices in A (h-1) 0 by the corresponding type II vertex and by summing over all possibilities. We denote this by A (h-1) 2\n\n. We consider them in order.\n\nLet us see the effects of the M term, namely, eq. (3.5). It plays a role of inserting two W further. Thus we will obtain terms which are proportional to S h . Note that we cannot insert more than two W because, in such case, at least one of the index loops has more than two insertions of W. For the parts contributing to NS h-1 , which have an empty index loop, we can further insert W α W α from the first two terms in (3.5). In the case in which they are inserted in the a-th index loop, we obtain S 4 h-1 ig ′ 3 ia s ia TrWW, where i a labels the propagators which form the a-th index loop. The absence of factor N is explained by the absence of an empty index loop. The factor h is not present as we have so far 6 restricted ourselves to the a-th index loop. Summing over all index loops, we obtain the first contribution to A (h-1) 1\n\n: a S 4 h-1 ig ′ 3 ia s ia TrWW = 2ig ′ 3 i s i S 4 h-1\n\nTrWW,\n\nwhere we have used that when all index loops are summed, they pass through each double line propagator exactly twice.\n\nLet us note that the parts contributing to the second term of eq. (3.6) can receive further insertions of W as well. They have two index loops with a single W insertion, for which we can exploit the last term of M. An insertion of this term requires that two index loops share a propagator. Let us define the index A = 1, . . . , h C 2 as labeling the combinations of such two index loops and the index à labeling the cases which have a common propagator in the two index loops chosen. Let us further introduce the index i à labeling the common propagator in case Ã. With these notations, we obtain the second contribution to A (h-1) 1 :\n\n2S h-2 4 h-1 ig ′ 3 i à s i à 1 64π 2 W α ab W αcd W β ba W βdc = ig ′ 3 i s i S 4 h-1 TrW α W α .\n\nPutting all these together, we obtain the contributions from the vertices of type I, P i=1\n\nds i e -( P s i )m ′ (A (h-1) 0 + A (h-1) 1 (s i )) = h m ′P S 4 h-1 N - 16π 2 iP g 3 S hmg 2 + h C 2 2m ′P S 4 h-2 w α w α . (3.7)\n\nIt is important that the above new term has Schwinger parameter dependence aside from the exponential factor. In [17] , it was pointed out that the cancellation of this dependence represents the reduction of the system to the matrix model. The appearance of this new term with Schwinger parameter dependence may spoil this reduction. Note also that this new term does not have an overall factor N, indicating the violation of the well-known relation due to Dijkgraaf-Vafa [7].\n\nWe now turn to the vertices of type II which contain two W insertions. The ℓ-th order vertex in Φ is\n\nTr(2WWΦ ℓ + WΦWΦ ℓ-1 + . . . + WΦ ℓ-1 WΦ). (3.8)\n\nwhere we have omitted the overall factors. The first term inserts two W into an index loop while the remainder insert them into two different index loops. Having done 2(h -1) π α 7 Figure 1: two-loop planar diagrams integrations, we obtain 2(h -1) W insertions. We can therefore use vertex (3.8) only once in a diagram. When this is done, insertion of the M term from the propagator is disallowed.\n\nLet us consider A (h-1) 2 and suppose that one of the type I vertices, Tr Φ ℓ , is replaced by the above vertex (3.8). The first term connects ℓ index loops and we can insert W 2 into ℓ different ways. Thus we obtain S 4 h-1 2ℓTrWW as a contribution to A (h-1) 2\n\n. For the other terms of eq. (3.8), there are in total ℓ(ℓ -1) ways of inserting two W into different index loops. These give\n\n2S h-2 4 h-1 ℓ(ℓ -1) 1 64π 2 W α ab W αcd W β ba W βdc = S 4 h-1 ℓ(ℓ -1)TrWW.\n\nSumming the above two contributions, we obtain S 4 h-1 ℓ(ℓ + 1)TrWW. Thus, in any (h -1)-loop diagram, changing a vertex from type I to type II is equivalent to considering only NS h-1 terms in eq. (3.6) and changing the coupling constant by mg ℓ → 16π 2 ig ℓ+1 S Nh , for ℓ ≥ 3. (3.9) Therefore, we obtain a formula for the contribution from the (h -1)-loop diagrams with P propagators to W ef f in (2.1),\n\nW (h-1) ef f = N ∂F (h-1) ∂S + ∂ 2 F (h-1) ∂S 2 w α w α - 16π 2 iP mg 3 hmg 2 ∂F (h-1) ∂S S m + W (h-1) 2 , (3.10) where W (h-1) 2\n\nis defined by replacing, in the first term, one coupling constant according to eq. (3.9) and summing over all possibilities. We have denoted by F (h-1) the (h -1)-loop contribution to the planar free energy of the matrix model."},{"section_type":"OTHER","section_title":"IV. Example","text":"As a sample computation, let us take the two-loop contribution to the effective superpotential. There are two two-loop planar diagrams depicted in Fig. 1 . Collecting all possible 8 insertions of W, we obtain\n\nW ( 2\n\n) ef f = - (mg 3 ) 2 32(mg 2 ) 3 NS 2 - (mg 3 ) 2 16(mg 2 ) 3 Sw α w α + π 2 i(mg 3 ) 3 2(mg 2 ) 4 S 3 m - π 2 i(mg 3 )(mg 4 ) 3(mg 2 ) 3 S 3 m . (4.1)\n\nThe first two terms are the ones which are present in the computation based on [7, 8] with S N =1 . The third one comes from the M term in the propagator and the last one from the type II vertices. Note that, in the limit m → ∞ with mg k (k ≥ 2) fixed, we reproduce the result of [17] . In an arbitrary loop amplitude, the situation is the same: new terms are of order m -1 in this limit.\n\nThe overall U(1) part does not decouple from the SU(N) part. This can be easily seen by translating S into the glueball superfield Ŝ = -1 64π 2 Tr SU (N ) W α W α and extracting the factor in front of w α w α . By the existence of the last two terms in eq. (3.10), it is nonvanishing. For example, in the two-loop example, this part in (4.1) reads\n\nπi(mg 3 )[2(mg 2 )(mg 4 ) -3(mg 3 ) 2 ] 2(mg 2 ) 4 Ŝ2 m w α w α = 0,"},{"section_type":"OTHER","section_title":"V. The chiral ring and the generalized Konishi anomaly","text":"An alternative approach to the effective superpotential is to exploit and extend the properties of the N = 1 chiral ring and the generalized Konishi anomaly equations based on reference [20, 8] . The anomalous Ward identity of our model for the general transformation δΦ = f (Φ, W) is\n\n- 1 64π 2 W α , W α , ∂f ∂Φ ij ij Φ = Trf W ′ (Φ) Φ - i 4 Tr(f F ′′′ (Φ)W α W α ) Φ , (5.1) where W ′′ (Φ) = mF ′′′ (Φ).\n\nIn terms of the two generating functions of chiral one-point functions\n\nR(z) = - 1 64π 2 TrW α W α 1 z -Φ Φ , T (z) = Tr 1 z -Φ Φ ,\n\nthe anomalous Ward identities (5.1) are\n\nR(z) 2 = W ′ (z)R(z) + 1 4 f (z), 2R(z)T (z) = W ′ (z)T (z) + 1 4 c(z) + 16π 2 iF ′′′ (z)R(z) + 1 4 c(z), 9\n\nwhere f (z) and c(z) are polynomials of degree n -1 in z and c(z) is a polynomial of degree n -2:\n\nf (z) = - 1 16π 2 Tr (W ′ (Φ) -W ′ (z))W α W α z -Φ Φ , c(z) = 4 W ′ (Φ) -W ′ (z) z -Φ Φ , c(z) = -i (F ′′′ (Φ) -F ′′′ (z))W α W α z -Φ Φ .\n\nThe last term of eq. (5.1) does not contribute to the equation for R(z) because of the chiral ring relation TrW α W α W β W β = 0. The equation for R(z) is the same as that of [8], which is the loop equation of the matrix model. On the other hand, the equation for T (z) alters from that of [8] .\n\nThe final step of this approach is to express the effective superpotential in terms of R(z) and T (z). Taking a variational derivative of (3.1) with respect to the coupling g k , we obtain\n\n∂W ef f ∂g k = m k! dzz k T (z) + 16π 2 i (k -1)! dzz k-1 R(z).\n\nHence we can determine the effective superpotential up to g k independent terms."},{"section_type":"OTHER","section_title":"Acknowledgements","text":"We thank Kazuhito Fujiwara, Yosuke Imamura, Hiroaki Kanno, Hironobu Kihara, Yasunari Kurita, Kazutoshi Ohta and Makoto Sakaguchi for useful discussions. We are grateful to Hiraku Yonemura for his collaboration at an early stage. This work is supported in part by the Grant-in-Aid for Scientific Research (18540285) from the Ministry of Education, Science and Culture, Japan. Support from the 21 century COE program \"Constitution of wide-angle mathematical basis focused on knots\" is gratefully appreciated. The preliminary version of this work was presented in YITP workshop \"Fundamental Problems and Applications of Quantum Field Theory\", YITP-W-06-16 in Yukawa Institute for Theoretical Physics, Kyoto University (December 14-16 2006) . We wish to acknowledge the participants for stimulating discussions."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. Introduction\",\n \"text\": \"For more than two decades, effective superpotential has been a central object in the nonperturbative study of N = 1 supersymmetric theories. This object is protected from perturbative corrections in the conventional sense [1] , and yet receives important nonperturbative corrections (see for example [2, 3] ). In recent years, analyses from superstring theory have revealed an interesting perturbative window into nonperturbative physics with the use of the gluino condensate superfield variable [4, 5, 6, 7] . In [8], field theoretic discussion based on the model with U(N) gauge group and rigid N = 1 supersymmetry (see eq. (2.2) for its action S N =1 ) is given and this is in accord with the string theory based developments.\\n\\nSuperstring theory, on the other hand, insists upon maximally extended supersymmetry with no adjustable parameter. A scenario that one may draw is that this extended supersymmetry becomes spontaneously broken to N = 1. Along this vein, a field theory model with U(N) gauge group and rigid N = 2 supersymmetry spontaneously broken to N = 1 has been introduced in [9, 10, 11] (see eq. (2.1) for its action S N =2 ), generalizing the abelian counterpart of [12] . (See also [13] for N = 2 supergravity and [14] for related discussions.) Several properties of this model have been derived.\\n\\nIn this letter, we make a first analysis on the interplay between the effective superpotential and partially as well as spontaneously broken N = 2 supersymmetry, shedding a light upon the comparison of the two models mentioned above. A key aspect of this comparison is that the fermionic shift symmetry of S N =1 gets replaced by the second (spontaneously broken) supersymmetry of S N =2 . In fact, this is one of the original motivations/results of [9] .\\n\\nThe fermionic shift symmetry of S N =1 supplies the well-known formula [7, 8] constraining the form of the effective superpotential, which is originally proposed from flux compactification of string theory [15, 16] . Based on a diagrammatic analysis [17] (for a review see [18] ), we are able to state how this form undergoes modifications in the model S N =2 . After giving a few accounts of the model in the next section, we present a diagrammatic analysis of W ef f in section III. Our final understanding is summarized in eq. (3.10). This is followed by a computation of the two-loop contribution to W ef f in section IV. In the final section, we derive a set of two equations on the two generating functions R(z) and T (z) of the one-point functions, generalizing the argument based on the chiral ring and the Konishi anomaly in [8] . We observe a modification from that given in [8] here as well.\\n\\nII. The U (N ) gauged model with spontaneously broken N = 2 supersymmetry Let us briefly recall a few ingredients of the model, which are needed in what follows. The action [9] given in the Wess-Zumino gauge can be written as S N =2 = d 4 xd 4 θ -i 2 Tr Φe adV ∂F (Φ) ∂Φ h.c. + ξV 0\\n\\n+ d 4 xd 2 θ - i 4 ∂ 2 F (Φ)\\n\\n∂Φ a ∂Φ b W a W b + eΦ 0 + m ∂F (Φ) ∂Φ 0 + h.c. , (2.1)\\n\\nwhere V = V a t a and W α are the vector superfield and the gauge superfield strength respectively and Φ = Φ a t a (a = 0, 1, . . . , N 2 -1) is the chiral superfield * . There are three Fayet-Iliopoulos parameters (e, m, ξ) which are all real. For simplicity, we choose the prepotential as a single trace function of degree n + 2: F (Φ) = n+1 k=1 g k TrΦ k+1 /(k + 1)!. While this action is shown to be invariant under the N = 2 supersymmetry transformations [9, 10], the vacuum breaks half of the N = 2 supersymmetries. Extremizing the scalar potential, we obtain the condition ∂ 2 F ∂Φ 0 ∂Φ 0 = -(e ± iξ)/m, which is a polynomial of order n and this determines the expectation value of the scalar field.\\n\\nThe action S N =2 in (2.1) is to be compared with that of the N = 1, U(N) gauge model with a single trace tree level superpotential W (Φ):\\n\\nS N =1 = d 4 xd 4 θ Tr Φe adV Φ + d 4 xd 2 θ Tr (iτ WW + W (Φ)) + h.c. , (2.2)\\n\\nwhere τ is a complex gauge coupling τ = θ/2π + 4πi/g 2 .\\n\\nIn [9] , it is checked that the second supersymmetry reduces to the fermionic shift symmetry in the limit m → ∞. The action S N =2 in fact reduces to S N =1 in the limit m, e, ξ → ∞ with mg k (k ≥ 2) fixed [19] . We show that our result reduces to that of [7, 17] in this limit.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. Diagrammatic analysis of the effective superpotential\",\n \"text\": \"In this letter, we consider the matter-induced part of the effective superpotential by integrating out the massive degrees of freedom Φ:\\n\\ne i R d 4 x(d 2 θW ef f +h.c.+d 4 θ(nonchiral terms)) = DΦD Φe iS N =2 .\\n\\n(3.1) * a = 0 corresponds to the overall U (1) part.\\n\\nLet us take W α (or V ) as the background field † . We consider the case of unbroken U(N) gauge group. For simplicity, we choose Φ = 0 by setting g 1 = -(e ± iξ)/m.\\n\\nWe are interested in the holomorphic superpotential which does not contain the antiholomorphic couplings ḡk . We can take ḡk = 0 for k ≥ 3 without loss of generality. Collecting the Φ dependent terms, we obtain\\n\\nSΦ = d 4 xd 4 θ -i 2 Tr Φe adV ∂F (Φ) ∂Φ -(ḡ 1 Φ + ḡ2 2 Φ2 )e adV Φ + d 4 xd 2 θ mḡ 2 2 Tr Φ2 = d 4 xd 4 θTr Φḡ 2 - 2m ∇ 2 + i 4 Φ Φ + i 2 ḡ1 Φ - ∂F ∂Φ Φ . (3.2)\\n\\nIn the last expression, we have introduced a covariantly anti-chiral superfield Φ = Φe adV , which satisfies ∇ α Φ = 0 (∇ α = e -adV D α e adV ). Eq. (3.2) is quadratic in Φ and can be integrated straightforwardly. As a result, we obtain the following terms,\\n\\n1 16ḡ 2 ḡ1 Φ - ∂F ∂Φ - 2m ∇ 2 + i 4 Φ -1 ḡ1 Φ - ∂F ∂Φ = (Img 1 ) 2 8m ḡ2 Φ∇ 2 Φ + . . . , (3.3)\\n\\nwhere . . . denotes the higher order interaction terms, which we will not consider here. Indeed, these interaction vertices are higher order in m -1 compared to the vertices which we consider below. These contribute to our main result (3.10) as higher order corrections in m -1 and do not spoil our conclusion that the effective superpotential is modified from the case of S N =1 (2.2).\\n\\nReplacing d 2 θ integration by -∇2 /4 and collecting the terms which are not in SΦ, we obtain an action after the Φ integration ‡ :\\n\\nd 4 xd 2 θTr - (Img 1 ) 2 32mḡ 2 Φ ∇2 ∇ 2 Φ + m n+1 k=2 g k k! Φ k - i 4 n+1 k=3 k-1 s=0 g k k! (WΦ s WΦ k-1-s ) . (3.4)\\n\\nThe first two terms are already present in the integrations with regard to the action S N =1 (2.2). The last term is new and originates from the gauge kinetic term in eq. (2.1). As we will see below, this last term does contribute to the effective superpotential and becomes responsible for the violation of the well-known relation [7, 8] between the effective superpotential of the gauge theory and the planar free energy of the matrix model having the tree level (bare) superpotential as its potential.\\n\\n† The simplest background is that consisting of a vanishing gauge field A µ and a constant gaugino λ α , which satisfies {λ α , λ β } = 0 [18] . This configuration implies that traces of more than two W vanish. ‡ In eq. (3.4), it is understood that the generating functional has a renormalized perturbation expansion in which a nonvanishing tadpole is always canceled by a nonvanishing value of the source coupled to Φ. This implies that the tadpole can in practice be ignored.\\n\\nAfter rescaling Φ → aΦ with a 2 = mḡ 2 /(Img 1 ) 2 , the quadratic part of the action (3.4) reduces to\\n\\n1 2 Φ -+ m ′ + 1 2 adW α D α Φ - ig ′ 3 2 (2WWΦ 2 + WΦWΦ),\\n\\nwhere we have used the relation ∇2 ∇ 2 Φ = 16( Φ -adW α D α Φ/2) and introduced m ′ = a 2 mg 2 and g ′ 3 = a 2 g 3 /12. The propagator in the momentum space is\\n\\n∆(p, π) = ∞ 0 dse -s(p 2 +m ′ + 1 2 adW α πα-ig ′ 3 M ).\\n\\nThe Grassmann momentum π α is Fourier transformation of superspace coordinate θ α and the matrix M is\\n\\nM abcd = (WW) da δ bc + (WW) bc δ da + W da W bc , (3.5)\\n\\nwhere we have exhibited the gauge index dependence explicitly. This matrix is not present in the propagator of [17] . Using eq. (3.5), we are able to insert W without involving the momentum π α .\\n\\nThe interaction terms in eq. (3.4) are divided into the following two types: type I. m g k a k k! Tr Φ k , k = 3, . . . , n + 1. type II.\\n\\ni 4 k-1 s=0 g k a k-1 k! Tr(WΦ s WΦ k-1-s ), k = 4, . . . , n + 1.\\n\\nType I vertices are already present in [17] . Type II vertices are not present in [17] . They insert two W in specific ways.\\n\\nBefore going on to consider loop diagrams, let us first demonstrate that we have only to consider planar diagrams in our case as well [17, 18] . For a given diagram, we denote by V the number of vertices, by P the number of propagators and by h the number of holes (or index loops). There are V sets of chiral superspace integrations from V vertices. One of them becomes the chiral superspace integration over the effective superpotential, and the number of remaining π α momentum integrations is P -V + 1. These Grassmann integrations must be saturated by 1 2 adW α π α terms in the propagators. Furthermore, we can freely insert W both from the M terms in the propagators and from the type II vertices. If we denote the number of these additional insertions by 2α, the total number of W insertions is 2(P -V + 1 + α). On the other hand, one index loop can accommodate at most two W. Thus we have h ≥ P -V + 1 + α. This implies that only the planar diagrams contribute to the effective superpotential as the Euler number of the diagram is χ = V -P + h.\\n\\nA planar diagram with h index loops has (h -1) loop momenta. Let us consider the (h -1)-loop planar diagrams (contributing to the (h -1)-loop vacuum amplitude) in which all vertices are type I. Let us, for a moment, ignore the M term of (3.5). The calculation is then the same as that of [17] which we briefly describe. Each diagram is a product of the bosonic part obtained by integrating over the momentum p and the fermionic one coming from the π α integrations. As we have seen in the last paragraph, we have exactly 2(h -1) W insertions in the fermion part. There are two possibilities for these W insertions. The one is to keep one of the index loops empty, filling the remaining index loops with two W. This yields NS h-1 term, where\\n\\nS = -1 64π 2 Tr U (N ) W α W α .\\n\\nThe other is to fill each of two index loops chosen with single W, which yields S h-2 w α w α terms where w α = 1 8π TrW α . After calculating the both parts, we perform the Schwinger parameter integrals. Clearly this procedure is universal to every (h -1)-loop planar diagram up to the multiplications by the symmetric factor and by the coupling constants. Therefore every such diagram is a product of these factors with the following expression\\n\\nP i=1 ds i e -( P s i )m ′ 1 4 h-1 {NhS h-1 + h C 2 2S h-2 w α w α } ≡ P i=1 ds i e -( P s i )m ′ A (h-1) 0 , (3.6)\\n\\nwhere we have introduced A (h-1) 0\\n\\n. The factor h of the first term comes from the choice of the empty index loop, and h C 2 of the second term is the combination of inserting two W into different index loops. The most important fact is that the dependence on Schwinger parameters of the bosonic part is cancelled by that of the fermionic part. This explains that the calculation of the effective superpotential of the gauge theory reduces to that of the matrix model [17] .\\n\\nThere are two types of corrections to A (h-1) 0 . The one is due to the presence of the M terms in the propagators, which we denote by A (h-1) 1 . The other is due to the type II vertices, which is obtained by replacing one of the type I vertices in A (h-1) 0 by the corresponding type II vertex and by summing over all possibilities. We denote this by A (h-1) 2\\n\\n. We consider them in order.\\n\\nLet us see the effects of the M term, namely, eq. (3.5). It plays a role of inserting two W further. Thus we will obtain terms which are proportional to S h . Note that we cannot insert more than two W because, in such case, at least one of the index loops has more than two insertions of W. For the parts contributing to NS h-1 , which have an empty index loop, we can further insert W α W α from the first two terms in (3.5). In the case in which they are inserted in the a-th index loop, we obtain S 4 h-1 ig ′ 3 ia s ia TrWW, where i a labels the propagators which form the a-th index loop. The absence of factor N is explained by the absence of an empty index loop. The factor h is not present as we have so far 6 restricted ourselves to the a-th index loop. Summing over all index loops, we obtain the first contribution to A (h-1) 1\\n\\n: a S 4 h-1 ig ′ 3 ia s ia TrWW = 2ig ′ 3 i s i S 4 h-1\\n\\nTrWW,\\n\\nwhere we have used that when all index loops are summed, they pass through each double line propagator exactly twice.\\n\\nLet us note that the parts contributing to the second term of eq. (3.6) can receive further insertions of W as well. They have two index loops with a single W insertion, for which we can exploit the last term of M. An insertion of this term requires that two index loops share a propagator. Let us define the index A = 1, . . . , h C 2 as labeling the combinations of such two index loops and the index à labeling the cases which have a common propagator in the two index loops chosen. Let us further introduce the index i à labeling the common propagator in case Ã. With these notations, we obtain the second contribution to A (h-1) 1 :\\n\\n2S h-2 4 h-1 ig ′ 3 i à s i à 1 64π 2 W α ab W αcd W β ba W βdc = ig ′ 3 i s i S 4 h-1 TrW α W α .\\n\\nPutting all these together, we obtain the contributions from the vertices of type I, P i=1\\n\\nds i e -( P s i )m ′ (A (h-1) 0 + A (h-1) 1 (s i )) = h m ′P S 4 h-1 N - 16π 2 iP g 3 S hmg 2 + h C 2 2m ′P S 4 h-2 w α w α . (3.7)\\n\\nIt is important that the above new term has Schwinger parameter dependence aside from the exponential factor. In [17] , it was pointed out that the cancellation of this dependence represents the reduction of the system to the matrix model. The appearance of this new term with Schwinger parameter dependence may spoil this reduction. Note also that this new term does not have an overall factor N, indicating the violation of the well-known relation due to Dijkgraaf-Vafa [7].\\n\\nWe now turn to the vertices of type II which contain two W insertions. The ℓ-th order vertex in Φ is\\n\\nTr(2WWΦ ℓ + WΦWΦ ℓ-1 + . . . + WΦ ℓ-1 WΦ). (3.8)\\n\\nwhere we have omitted the overall factors. The first term inserts two W into an index loop while the remainder insert them into two different index loops. Having done 2(h -1) π α 7 Figure 1: two-loop planar diagrams integrations, we obtain 2(h -1) W insertions. We can therefore use vertex (3.8) only once in a diagram. When this is done, insertion of the M term from the propagator is disallowed.\\n\\nLet us consider A (h-1) 2 and suppose that one of the type I vertices, Tr Φ ℓ , is replaced by the above vertex (3.8). The first term connects ℓ index loops and we can insert W 2 into ℓ different ways. Thus we obtain S 4 h-1 2ℓTrWW as a contribution to A (h-1) 2\\n\\n. For the other terms of eq. (3.8), there are in total ℓ(ℓ -1) ways of inserting two W into different index loops. These give\\n\\n2S h-2 4 h-1 ℓ(ℓ -1) 1 64π 2 W α ab W αcd W β ba W βdc = S 4 h-1 ℓ(ℓ -1)TrWW.\\n\\nSumming the above two contributions, we obtain S 4 h-1 ℓ(ℓ + 1)TrWW. Thus, in any (h -1)-loop diagram, changing a vertex from type I to type II is equivalent to considering only NS h-1 terms in eq. (3.6) and changing the coupling constant by mg ℓ → 16π 2 ig ℓ+1 S Nh , for ℓ ≥ 3. (3.9) Therefore, we obtain a formula for the contribution from the (h -1)-loop diagrams with P propagators to W ef f in (2.1),\\n\\nW (h-1) ef f = N ∂F (h-1) ∂S + ∂ 2 F (h-1) ∂S 2 w α w α - 16π 2 iP mg 3 hmg 2 ∂F (h-1) ∂S S m + W (h-1) 2 , (3.10) where W (h-1) 2\\n\\nis defined by replacing, in the first term, one coupling constant according to eq. (3.9) and summing over all possibilities. We have denoted by F (h-1) the (h -1)-loop contribution to the planar free energy of the matrix model.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. Example\",\n \"text\": \"As a sample computation, let us take the two-loop contribution to the effective superpotential. There are two two-loop planar diagrams depicted in Fig. 1 . Collecting all possible 8 insertions of W, we obtain\\n\\nW ( 2\\n\\n) ef f = - (mg 3 ) 2 32(mg 2 ) 3 NS 2 - (mg 3 ) 2 16(mg 2 ) 3 Sw α w α + π 2 i(mg 3 ) 3 2(mg 2 ) 4 S 3 m - π 2 i(mg 3 )(mg 4 ) 3(mg 2 ) 3 S 3 m . (4.1)\\n\\nThe first two terms are the ones which are present in the computation based on [7, 8] with S N =1 . The third one comes from the M term in the propagator and the last one from the type II vertices. Note that, in the limit m → ∞ with mg k (k ≥ 2) fixed, we reproduce the result of [17] . In an arbitrary loop amplitude, the situation is the same: new terms are of order m -1 in this limit.\\n\\nThe overall U(1) part does not decouple from the SU(N) part. This can be easily seen by translating S into the glueball superfield Ŝ = -1 64π 2 Tr SU (N ) W α W α and extracting the factor in front of w α w α . By the existence of the last two terms in eq. (3.10), it is nonvanishing. For example, in the two-loop example, this part in (4.1) reads\\n\\nπi(mg 3 )[2(mg 2 )(mg 4 ) -3(mg 3 ) 2 ] 2(mg 2 ) 4 Ŝ2 m w α w α = 0,\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"V. The chiral ring and the generalized Konishi anomaly\",\n \"text\": \"An alternative approach to the effective superpotential is to exploit and extend the properties of the N = 1 chiral ring and the generalized Konishi anomaly equations based on reference [20, 8] . The anomalous Ward identity of our model for the general transformation δΦ = f (Φ, W) is\\n\\n- 1 64π 2 W α , W α , ∂f ∂Φ ij ij Φ = Trf W ′ (Φ) Φ - i 4 Tr(f F ′′′ (Φ)W α W α ) Φ , (5.1) where W ′′ (Φ) = mF ′′′ (Φ).\\n\\nIn terms of the two generating functions of chiral one-point functions\\n\\nR(z) = - 1 64π 2 TrW α W α 1 z -Φ Φ , T (z) = Tr 1 z -Φ Φ ,\\n\\nthe anomalous Ward identities (5.1) are\\n\\nR(z) 2 = W ′ (z)R(z) + 1 4 f (z), 2R(z)T (z) = W ′ (z)T (z) + 1 4 c(z) + 16π 2 iF ′′′ (z)R(z) + 1 4 c(z), 9\\n\\nwhere f (z) and c(z) are polynomials of degree n -1 in z and c(z) is a polynomial of degree n -2:\\n\\nf (z) = - 1 16π 2 Tr (W ′ (Φ) -W ′ (z))W α W α z -Φ Φ , c(z) = 4 W ′ (Φ) -W ′ (z) z -Φ Φ , c(z) = -i (F ′′′ (Φ) -F ′′′ (z))W α W α z -Φ Φ .\\n\\nThe last term of eq. (5.1) does not contribute to the equation for R(z) because of the chiral ring relation TrW α W α W β W β = 0. The equation for R(z) is the same as that of [8], which is the loop equation of the matrix model. On the other hand, the equation for T (z) alters from that of [8] .\\n\\nThe final step of this approach is to express the effective superpotential in terms of R(z) and T (z). Taking a variational derivative of (3.1) with respect to the coupling g k , we obtain\\n\\n∂W ef f ∂g k = m k! dzz k T (z) + 16π 2 i (k -1)! dzz k-1 R(z).\\n\\nHence we can determine the effective superpotential up to g k independent terms.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"We thank Kazuhito Fujiwara, Yosuke Imamura, Hiroaki Kanno, Hironobu Kihara, Yasunari Kurita, Kazutoshi Ohta and Makoto Sakaguchi for useful discussions. We are grateful to Hiraku Yonemura for his collaboration at an early stage. This work is supported in part by the Grant-in-Aid for Scientific Research (18540285) from the Ministry of Education, Science and Culture, Japan. Support from the 21 century COE program \\\"Constitution of wide-angle mathematical basis focused on knots\\\" is gratefully appreciated. The preliminary version of this work was presented in YITP workshop \\\"Fundamental Problems and Applications of Quantum Field Theory\\\", YITP-W-06-16 in Yukawa Institute for Theoretical Physics, Kyoto University (December 14-16 2006) . We wish to acknowledge the participants for stimulating discussions.\"\n }\n]","isConversational":false}}},{"rowIdx":187,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1069"},"arxiv_id":{"kind":"string","value":"0704.1069"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1367-2630/9/7/224"},"content_hash":{"kind":"string","value":"fee712c164d1c7eafe992b56c90448f22ae70b6370c0e412bad7fab2dd2807b2"},"title":{"kind":"string","value":"Optical Zeno Gate: Bounds for Fault Tolerant Operation"},"abstract":{"kind":"string","value":"In principle the Zeno effect controlled-sign gate of Franson et al's (PRA 70,\n062302, 2004) is a deterministic two-qubit optical gate. However, when\nrealistic values of photon loss are considered its fidelity is significantly\nreduced. Here we consider the use of measurement based quantum processing\ntechniques to enhance the operation of the Zeno gate. With the help of quantum\nteleportation, we show that it is possible to achieve a Zeno CNOT gate (GC-Zeno\ngate) that gives (near) unit fidelity and moderate probability of success of\n0.76 with a one-photon to two-photon transmission ratio $\\kappa=10^4$. We\ninclude some mode-mismatch effects and estimate the bounds on the mode overlap\nand $\\kappa$ for which fault tolerant operation would be possible."},"authors":{"kind":"list like","value":["Patrick M. Leung and Timothy C. Ralph"],"string":"[\n \"Patrick M. Leung and Timothy C. Ralph\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"Photons are a natural choice for making qubits because the quantum information encoded can have a long decoherence time and is easy to manipulate and measure. Also, photonic qubits are the only type of qubits that are feasible for long distance quantum communication. Quantum information processing requires universal two-qubit entangling gates. Knill et al [1] showed that it is theoretically possible to do scalable quantum computing with linear optics by using measurement induced interactions to perform the two-qubit gates. However, despite a continuous effort in reducing the resource requirement [2, 3, 4, 5] , the resource overhead is still high for linear optical quantum computing. Franson et al[6] proposed the use of two-photon absorption non-linearity and exploiting the quantum Zeno effect to implement a control sign (CZ) gate that requires much less resources than linear optics schemes. However, the problem with the Zeno gate is that photon loss affects the performance of the Zeno gate significantly and the single photon to twophoton loss ratio requirement is very stringent. One solution couldbe to combine measurement induced quantum processing with the Zeno gate. Previously we have shown that when using the Zeno gate for qubit fusion [8], state distillation [9] with post-selection can boost the gate fidelity to unity and that for less stringent absorption ratios the gate has an advantage in success probability over linear optics gates for fusing clusters of qubits. These clusters of qubits can then be used for cluster state quantum computing [10] . In addition, Myers and Gilchrist [7] have shown that the performance of the Zeno gate may be enhanced by using error correction codes such as redundancy and parity encoding.\n\nHere we design a high fidelity Zeno CNOT gate suitable for circuit-based quantum computing. Although with the current estimate of the photon loss ratio, only a poor fidelity Zeno gate is directly achievable, here we show that it is possible to use two of these Zeno gates to do Bell measurements and implement a Gottesman-Chuang[11] teleportation type of CNOT gate (GC-Zeno gate) that, like the fusion gate, gives high fidelity via state distillation and moderate success probability via partially off-line state preparation. We include the effect of mode-mismatch and detector efficiency on the scheme and estimate lower bounds on the parameter which in principle allow fault tolerant operation.\n\nThe paper is arranged in the following way. The introduction continues with a subsection on the Zeno CZ gate, which describes the scheme and modelling of the gate and give descriptions on the modelling parameters that are also used for modelling the GC-Zeno CZ gate. In section II, we discuss the GC-Zeno gate and the effect of mode-mismatch and detector efficiency on the gate. In section III, we give estimates of the lower bounds on the photon loss ratio and mode-matching, followed by a subsection on the advantage in using state distillation. We conclude and summarize in section IV.\n\nFranson et al's control sign gate scheme consists of a pair of optical fibers weakly evanescently coupled and doped with two-photon absorbing atoms. The purpose of the two-photon absorbers is to suppress the occurrence of two photon state components in the two fibre modes via the Quantum Zeno effect. This allows the state to remain in the computational basis. After a length of fibre corresponding to a complete swap of the two fibre modes, a π phase difference is produced between the |11 term and the other three basis terms. After swapping the fibre modes by simply crossing them, a CZ gate is achieved. The gate becomes near deterministic and performs a near unitary operation when the Quantum Zeno effect is strong and photon loss is insignificant. However, with current technology, the strength of the 2 Quantum Zeno effect is a few orders of magnitude below what is required, and thus the Zeno gate has significant photon loss.\n\nIn [8], the gate is modelled as a succession of n weak beamsplitters followed by two-photon absorbers as shown in Fig. 1 . As n → ∞ the model tends to the continuous coupling limit envisaged for the physical realization. The gate operates on the single-rail encoding for which |0 L = |0 and |1 L = |1 with the kets representing photon Fock states. Fig. 2 shows how the single rail CZ can be converted into a dual rail [13] CZ with logical encoding |0 L = |H and |1 L = |V . Let L be the total length of n absorbers. Also, let γ 1 = exp( -λ nκ ) and γ 2 = exp( -λ n ) be the probability of single photon and two-photon transmission respectively for one absorber. Here the parameter λ = χL, where L is the length of the absorber and χ is the corresponding proportionality constant related to the absorption cross section. Furthermore, κ specifies the relative strength of the two transmissions and relates them by γ 2 = γ κ 1 . This CZ gate does the following operation:\n\n|00 → |00 |01 → γ n/2 1 |01 |10 → γ n/2 1 |10 |11 → -γ n 1 τ |11 + f (|02 , |20 ) ( 1\n\n)\n\nwhere the new expression for τ is given by:\n\nτ n,λ = 2 -3 2 -n d n,λ (g n,λ + d n,λ √ 2 ) n ( √ 2d n,λ -h n,λ ) +(g n,λ - d n,λ √ 2 ) n ( √ 2d n,λ + h n,λ ) d n,λ = (1 + cos 2π n )(1 + γ 2 ) + 2 √ γ 2 (cos( 2π n ) - 3\n\n)\n\ng n,λ = (cos π n )( √ γ 2 + 1) h n,λ = 2(cos π n )( √ γ 2 -1) ( 2\n\n)\n\nThe explicit form of the |02 , |20 state components are suppressed, as they lie outside the computational basis and so do not explicitly contribute to the fidelity.\n\nFIG. 1: Construction of our CZ gate.\n\nFIG. 2: CZ gate in dual rail implementation.\n\nFrom equation 1, it is clear that the amplitude of the four computational states are unequal and this lowers the gate fidelity. With the current best estimate of κ = 10 4 , the unherald fidelity is only 0.94. If the gate is used in a measurement based strategy then state distillation can be used and the fidelity of the gate can be improved by trading off some success probability. Figure 3 shows the distillated Zeno CZ gate circuit. The τ is simply an interferometer consisting of two 50-50 beam splitters with a two-photon absorber in each arm, which gives operation:\n\n|00 → |00 , |01 → γ ′ 1 |01 , |10 → γ ′ 1 |10 , and |11 → γ ′ 1 τ |11 . Here γ ′ 1 = τ 1/κ\n\nis the single photon transmission coefficient of the absorber. The distillation beam splitters labelled 1 to 3 have transmission coeffi-\n\ncient γ n 1 , γ ′ 1 , γ n 1 γ ′ 1 τ respectively.\n\nWith these distillations in place, the operation of the distillated Zeno CZ gate is:\n\n|00 → γ n 1 γ ′ 1 τ |00 |01 → γ n 1 γ ′ 1 τ |01 |10 → γ n 1 γ ′ 1 τ |10 |11 → -γ n 1 γ ′ 1 τ |11 + f (|02 , |20 ) ( 3\n\n)\n\nAfter measuring the output (detectors at output are not shown in fig. 3 ) and treating the photon bunching terms (|02 , |20 ) and the terms with photon loss as failures (excluded from Eq. 3 for clarity), renormalising the states gives unit heralded fidelity independent of λ and probability of success\n\nP s = γ 2n 1 γ ′2 1 τ 2 = e -2λ/κ τ 2+2/κ .\n\nFIG. 3: Schematic of distillated Zeno CZ gate.\n\nAs argued above, state distillation can improve the fidelity of the Zeno gate to unity by trading off success probability. However, the output of the distillated Zeno 3 FIG. 4: Schematic of GC-Zeno gate. The state |χ is\n\n((|00 + |11 )|00 + (|01 + |10 )|11 )/2\n\ngate contains terms outside the computational basis due to photon loss and photon bunching. Hence if we want the gate to have unit fidelity, it is necessary to measure the output and exclude these failure terms by post-selection. However, such post-selection means that the gate can no longer be used directly as a CNOT gate for circuit-based quantum computing. [11] showed the viability in using state teleportation and single qubit operations to implement a CNOT gate. The scheme requires the four qubit entangled state\n\n|χ = ((|00 + |11 )|00 + (|01 + |10 )|11 )/2.\n\nPreparing the entangled state requires a CZ operation, which can be done off-line with linear optics with high Bell measurements are made between the input qubits and the first and last qubits of |χ . The measurement results are fed forward for some single qubit corrections such that the circuit gives a proper CNOT operation. Here we propose using such scheme, as shown in figure 4 , to implement a GC-Zeno CNOT gate with high fidelity. Since this gate includes state distillation, post-selection and off-line state preparation, the gate has unit fidelity (under perfect mode-matching) and moderate success probability. Figure 5 plots the probability of success against the one-photon to two-photon transmission ratio κ. It shows that with κ = 10 4 (current best estimate), the probability of success is about 0.76, which is better than the break even point of 0.25 for the linear optics version of this gate [1].\n\nFrom source preparation to gate operation to detection, mode-mismatch is an unavoidable issue in optical quantum computing that causes unlocated errors which lowers the fidelity of the device [15] . Fortunately, with the help of quantum error correction, a certain amount of unlocated error rate, including but not limited to modemismatch errors, can be tolerated. A reliable quantum\n\nFIG. 5: Plot of probability of success versus log(κ) (in base 10) for GC-Zeno gate. Note that the success probability is not one at κ = 10 8 , but that the curve asymptotically approaches one for very large κ. Result is per two input qubits. Detector inefficiency is taken into account in accord with Dawson et al's bound.\n\ngate must therefore have unlocated error rates below this threshold.\n\nThe dominant source of mode-mismatch error in the GC-Zeno gate is from the CZ gate and τ gate, where two-photon interaction occurs. Here we follow Rohde et al's [14] analysis to examine the effect of such error. We take the simplest model in which the mode-mismatch is present between the photons entering the gate but remain constant through the gate. In this case, the modemismatch in two-photon interaction can be analysed as having two-photons fail to interact with some probability. This allows us to write the operations for the CZ gate as follow, where 0 < Γ < 1 quantifies the overlap of the two wavepackets. Γ 2 is the probability that the two photons successfully interacted and Γ = 0 for completely mode-mismatched and Γ = 1 for completely mode-matched. The bar in the | 11 term indicates modemismatched component of the state.\n\n|00 → |00 |01 → γ n 1 |01 |10 → γ n 1 |10 |11 → -Γγ n 1 τ |11 + 1 -Γ 2 γ n 1 | 11 +f (|02 , |20 ) ( 4\n\n)\n\nAnd similarly for the operations of τ gate:\n\n|00 → |00 |01 → γ ′ 1 |01 |10 → γ ′ 1 |10 |11 → Γγ ′ 1 τ |11 + 1 -Γ 2 γ ′ 1 | 11 +f (|02 , |20 ) ( 5\n\n)\n\nWith the equations for the CZ and τ gate[16], and given a normalized input state (α|00 + β|01 + δ|10 + ǫ|11 ), we can obtain analytical expression for the fidelity F (per qubit) and success probability P s (per qubit) of 4 the GC-Zeno gate as follow. Equation 6 and 7 show that both the fidelity and success probability are state dependent due to mode-mismatch. The worst case of fidelity occurs when the input state is the equal superposition state (|00 + |01 + |10 + |11 )/2 (i.e. α = δ = β = ǫ = 1/2) and the worst case of success probability occurs when the input state is the pure state |11 (i.e. α = β = δ = 0 and ǫ = 1).\n\nF = α * A 1 + β * A 2 + δ * A 3 + ǫ * A 4 |A 1 | 2 + |A 2 | 2 + |A 3 | 2 + |A 4 | 2 ( 6\n\n) P s = e -2λ/κ τ 2/κ 2(1 + e -λ/κ τ 2+1/κ ) × |A 1 | 2 + |A 2 | 2 + |A 3 | 2 + |A 4 | 2 ( 7\n\n)\n\nwhere\n\na 1 = (τ +τ Γ+ √ 1 -Γ 2 ), a 2 = (τ -τ Γ+ √ 1 -Γ 2 ), a 3 = (τ -τ Γ - √ 1 -Γ 2 ), a 4 = (τ + τ Γ - √ 1 -Γ 2 ), and A 1 = αa 2 1 + βa 1 a 2 + δa 1 a 2 + ǫa 2 2 , A 2 = αa 1 a 3 + βa 2 a 3 + δa 1 a 4 + ǫa 2 a 4 , A 3 = αa 1 a 3 + βa 1 a 4 + δa 2 a 3 + ǫa 2 a 4 , A 4 = αa 2 3 + βa 3 a 4 + δa 3 a 4 + ǫa 2 4\n\nIn practice, even for the most advanced photon detector, detector inefficiency is always present. The effect of this noise is to reduce the probability of success of the gate but not the fidelity because the errors are locatable.\n\nWe now wish to estimate lower bounds on the modematching, Γ, and photon loss ratio, κ, that will still allow fault-tolerant operation. We allow a small amount of detector inefficiency but assume all other parameters are ideal. To make this estimate we directly use the bounds obtained by Dawson et al [12] for a deterministic error correction protocol. For this protocol, they numerically derived one bound using the 7-qubit Steane code and another bound using the 23-qubit Golay code.\n\nIn order to use the Dawson et al's bounds we need to identify the unlocated and located error rates for our gate. In general, the unlocated error rate is less than 1 -F but here we take it to be 1 -F because in our analysis, γ is almost 1, which means the other terms involved are very small. The located error rate is simply 1 -P s (both F and P s are per qubit). Using these relationships, we convert each of the bounds into a fidelity versus success probability bound. For a gate built with two-photon absorbers that have a certain single-photon to two-photon transmission ratio κ, we can find an optimal λ (i.e. choosing an optimal absorber length) that\n\nFIG. 6: Lower bounds of amount of mode-matching Γ required for a fault tolerant GC-Zeno gate versus singlephoton to two-photon transmission ratio κ. The bounds are derived from Dawson et al's [12] results on deterministic error correction protocol. The top and bottom curves are for the 7-qubit Steane code and the 23-qubit Golay code respectively. Above the curves are the regions where the amount of mode-mismatch is tolerable. Here we have used the worst case input.\n\ngives a maximum success probability. Hence, by matching the success probability with the bound, we can determine the corresponding fidelity threshold and therefore find the least amount of mode-matching required for fault tolerant gate operation. We note that the error model used by Dawson et al is specific to their optical cluster state architecture and will differ in detail from the appropriate error model for the GC-Zeno gate. Nonetheless we assume that a comparison based on the total error rates will give a good estimate of the bounds.\n\nFigure 6 shows the lower bounds on the modematching parameter Γ for a gate with a certain κ. Since the fidelity and success probability are state dependent due to mode-mismatch, in that figure, we have plotted for the case of worst fidelity input state (i.e. the equal superposition state). The top and bottom curves are best fit curves for using the 7-qubit Steane code and the 23-qubit Golay code respectively. The curves show that highly mode-matched photons are essential for robust gate operation. With the worst fidelity input state,\n\n(|00 + |01 + |10 + |11 )/2, for\n\nthe Steane code, the lowest Γ required for fault tolerant operation is about 0.998, where κ = 10 6 , and for the Golay code, the lowest Γ required is about 0.996, where κ = 5 × 10 5 . With the worst success probability input state, |11 , for the Steane code, the lowest Γ required for fault tolerant operation is about 0.995, where κ = 10 6 , and for the Golay code, the lowest Γ required is about 0.989, where κ = 5 × 10 5 . Figure 6 also shows that under (near) perfect mode-matching, the required κ can be as low as approximately 6000 for the Steane code and 2000 for the Golay code. Two-photon absorbers with such κ values may be achievable with the best of current technology.\n\nState distillation allows us to trade off some success probability against fidelity for the GC-Zeno gate, or in other words, reducing the unlocated error rate by having a larger located error rate. Since the deterministic error correction protocol can tolerate both unlocated and located errors, therefore we should ask whether state distillation is truly advantageous? We can answer this question by comparing two GC-Zeno gates in the case of perfect mode matching, where one has complete distillation and the other has no distillation. For the case of complete distillation, the deterministic error correction protocol with the 7-qubit Steane code can tolerate errors of a GC-Zeno gate with κ = 6100, and with the 23-qubit Golay code, it can tolerate errors of a GC-Zeno gate with κ = 2100. For state distillation to be advantageous under the same protocol, these values of κ must be smaller than the values of κ for the case of no distillation [17] .\n\nFor the case of no distillation, the fidelity and success probability of the gate becomes state dependent. In the parameters space of interest, the input that gives the worst fidelity is (|00 + |01 + |10 -|11 )/2. With this input state, we find that for the protocol using the 7qubit Steane code and no distillation, the critical κ is 12000. Similarly, for the protocol using the 23-qubit Golay code and no distillation, the critical κ is 4300. With an arbitrary amount of distillation, the value of κ lies between the limit of no distillation and full distillation cases. Hence it is evident that state distillation is advantageous. Also, it should be noted it is better to have only located error, which is the case when there is full distillation, than have both located and unlocated errors, which is the case when no or some distillation is utilized.\n\nIn this paper, we have shown that it is possible to build a high fidelity Zeno CNOT gate with two distillated Zeno gates implemented in the Gottesman-Chuang teleportation CNOT scheme. For one-photon to twophoton transmission ratio κ = 10 4 (current best estimate), the gate has a success probability of 0.76 under perfect mode-matching. When including measurement noise that equals one-tenth of the gate's noise, and the effect of mode-mismatch in the CZ and τ gate, we find that with the deterministic error correction protocol using the 7-qubit Steane code, the lowest Γ required for fault tolerant gate operation is 0.998, where κ = 10 6 . For using the 23-qubit Golay code, the lowest Γ required is 0.996, where κ = 5 × 10 5 . Hence, the requirement on mode-matching is stringent for a fault tolerant GC-Zeno gate. * Electronic address: pmleung@physics.uq.edu.au [1] E. Knill, R. Laflamme, and G.J. Milburn, Nature 409, 46-52 (2001) [2] N.Yoran and B.Reznik, Phys. Rev. Lett. 91, 037903 (2003) [3] M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004) [4] A.J.F.Hayes, A.Gilchrist, C.R.Myers and T.C.Ralph, J.Opt.B 6, 533 (2004) [5] D.E.Browne and T.Rudolph, Phys. Rev. Lett. 95, 010501 (2005) [6] J.D. Franson, B.C. Jacobs, and T.B. Pittman, Phys. Rev. A 70, 062302 (2004) [7] C.R. Myers and A. Gilchrist, quant-ph 0612176 (2006) [8] P.M. Leung and T.C. Ralph, Phys. Rev. A 74, 062325 (2006) [9] R.T. Thew and W.J. Munro, Phys. Rev. A 63, 030302(R) (2001) [10] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86 5188 (2001) [11] D. Gottesman, I.L. Chuang, Nature 402, 6760 (1999) [12] C.M. Dawson, H.L. Haselgrove, and M.A. Nielsen, Phys.\n\nRev. Lett. 96, 020501 (2006) [13] A.P. Lund, T.C. Ralph, Phys. Rev. A, 66, 032307 (2002) [14] P.P. Rohde and T.C. Ralph, Phys. Rev. A 73, 062312 (2006) [15] Note that mode mismatch in CZ and τ gate causes unlocated error that lowers the fidelity, in which we find that it cannot be improved with state distillation. [16] Due to mode-mismatch, the τ gate is less effective in twophoton distillation. It is true that we can increase the two-photon absorption strength in the τ gate to make up for the inefficiency. However, here we assume that we do not know the mode-matching parameter Γ of the input wavepackets, and therefore this adjustment cannot be made. In addition, increasing the two-photon distillation will increase single-photon loss as well, which lowers the probability of success. [17] An ideal Zeno gate requires strong quantum Zeno effect and strong quantum Zeno effect corresponds to a large κ value. However such large non-linearity is difficult to engineer. Hence it is desirable to have a gate that works with modest κ."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"Photons are a natural choice for making qubits because the quantum information encoded can have a long decoherence time and is easy to manipulate and measure. Also, photonic qubits are the only type of qubits that are feasible for long distance quantum communication. Quantum information processing requires universal two-qubit entangling gates. Knill et al [1] showed that it is theoretically possible to do scalable quantum computing with linear optics by using measurement induced interactions to perform the two-qubit gates. However, despite a continuous effort in reducing the resource requirement [2, 3, 4, 5] , the resource overhead is still high for linear optical quantum computing. Franson et al[6] proposed the use of two-photon absorption non-linearity and exploiting the quantum Zeno effect to implement a control sign (CZ) gate that requires much less resources than linear optics schemes. However, the problem with the Zeno gate is that photon loss affects the performance of the Zeno gate significantly and the single photon to twophoton loss ratio requirement is very stringent. One solution couldbe to combine measurement induced quantum processing with the Zeno gate. Previously we have shown that when using the Zeno gate for qubit fusion [8], state distillation [9] with post-selection can boost the gate fidelity to unity and that for less stringent absorption ratios the gate has an advantage in success probability over linear optics gates for fusing clusters of qubits. These clusters of qubits can then be used for cluster state quantum computing [10] . In addition, Myers and Gilchrist [7] have shown that the performance of the Zeno gate may be enhanced by using error correction codes such as redundancy and parity encoding.\n\nHere we design a high fidelity Zeno CNOT gate suitable for circuit-based quantum computing. Although with the current estimate of the photon loss ratio, only a poor fidelity Zeno gate is directly achievable, here we show that it is possible to use two of these Zeno gates to do Bell measurements and implement a Gottesman-Chuang[11] teleportation type of CNOT gate (GC-Zeno gate) that, like the fusion gate, gives high fidelity via state distillation and moderate success probability via partially off-line state preparation. We include the effect of mode-mismatch and detector efficiency on the scheme and estimate lower bounds on the parameter which in principle allow fault tolerant operation.\n\nThe paper is arranged in the following way. The introduction continues with a subsection on the Zeno CZ gate, which describes the scheme and modelling of the gate and give descriptions on the modelling parameters that are also used for modelling the GC-Zeno CZ gate. In section II, we discuss the GC-Zeno gate and the effect of mode-mismatch and detector efficiency on the gate. In section III, we give estimates of the lower bounds on the photon loss ratio and mode-matching, followed by a subsection on the advantage in using state distillation. We conclude and summarize in section IV."},{"section_type":"OTHER","section_title":"Ia. Zeno CZ Gate","text":"Franson et al's control sign gate scheme consists of a pair of optical fibers weakly evanescently coupled and doped with two-photon absorbing atoms. The purpose of the two-photon absorbers is to suppress the occurrence of two photon state components in the two fibre modes via the Quantum Zeno effect. This allows the state to remain in the computational basis. After a length of fibre corresponding to a complete swap of the two fibre modes, a π phase difference is produced between the |11 term and the other three basis terms. After swapping the fibre modes by simply crossing them, a CZ gate is achieved. The gate becomes near deterministic and performs a near unitary operation when the Quantum Zeno effect is strong and photon loss is insignificant. However, with current technology, the strength of the 2 Quantum Zeno effect is a few orders of magnitude below what is required, and thus the Zeno gate has significant photon loss.\n\nIn [8], the gate is modelled as a succession of n weak beamsplitters followed by two-photon absorbers as shown in Fig. 1 . As n → ∞ the model tends to the continuous coupling limit envisaged for the physical realization. The gate operates on the single-rail encoding for which |0 L = |0 and |1 L = |1 with the kets representing photon Fock states. Fig. 2 shows how the single rail CZ can be converted into a dual rail [13] CZ with logical encoding |0 L = |H and |1 L = |V . Let L be the total length of n absorbers. Also, let γ 1 = exp( -λ nκ ) and γ 2 = exp( -λ n ) be the probability of single photon and two-photon transmission respectively for one absorber. Here the parameter λ = χL, where L is the length of the absorber and χ is the corresponding proportionality constant related to the absorption cross section. Furthermore, κ specifies the relative strength of the two transmissions and relates them by γ 2 = γ κ 1 . This CZ gate does the following operation:\n\n|00 → |00 |01 → γ n/2 1 |01 |10 → γ n/2 1 |10 |11 → -γ n 1 τ |11 + f (|02 , |20 ) ( 1\n\n)\n\nwhere the new expression for τ is given by:\n\nτ n,λ = 2 -3 2 -n d n,λ (g n,λ + d n,λ √ 2 ) n ( √ 2d n,λ -h n,λ ) +(g n,λ - d n,λ √ 2 ) n ( √ 2d n,λ + h n,λ ) d n,λ = (1 + cos 2π n )(1 + γ 2 ) + 2 √ γ 2 (cos( 2π n ) - 3\n\n)\n\ng n,λ = (cos π n )( √ γ 2 + 1) h n,λ = 2(cos π n )( √ γ 2 -1) ( 2\n\n)\n\nThe explicit form of the |02 , |20 state components are suppressed, as they lie outside the computational basis and so do not explicitly contribute to the fidelity.\n\nFIG. 1: Construction of our CZ gate.\n\nFIG. 2: CZ gate in dual rail implementation.\n\nFrom equation 1, it is clear that the amplitude of the four computational states are unequal and this lowers the gate fidelity. With the current best estimate of κ = 10 4 , the unherald fidelity is only 0.94. If the gate is used in a measurement based strategy then state distillation can be used and the fidelity of the gate can be improved by trading off some success probability. Figure 3 shows the distillated Zeno CZ gate circuit. The τ is simply an interferometer consisting of two 50-50 beam splitters with a two-photon absorber in each arm, which gives operation:\n\n|00 → |00 , |01 → γ ′ 1 |01 , |10 → γ ′ 1 |10 , and |11 → γ ′ 1 τ |11 . Here γ ′ 1 = τ 1/κ\n\nis the single photon transmission coefficient of the absorber. The distillation beam splitters labelled 1 to 3 have transmission coeffi-\n\ncient γ n 1 , γ ′ 1 , γ n 1 γ ′ 1 τ respectively.\n\nWith these distillations in place, the operation of the distillated Zeno CZ gate is:\n\n|00 → γ n 1 γ ′ 1 τ |00 |01 → γ n 1 γ ′ 1 τ |01 |10 → γ n 1 γ ′ 1 τ |10 |11 → -γ n 1 γ ′ 1 τ |11 + f (|02 , |20 ) ( 3\n\n)\n\nAfter measuring the output (detectors at output are not shown in fig. 3 ) and treating the photon bunching terms (|02 , |20 ) and the terms with photon loss as failures (excluded from Eq. 3 for clarity), renormalising the states gives unit heralded fidelity independent of λ and probability of success\n\nP s = γ 2n 1 γ ′2 1 τ 2 = e -2λ/κ τ 2+2/κ .\n\nFIG. 3: Schematic of distillated Zeno CZ gate."},{"section_type":"OTHER","section_title":"II. ZENO GATE USING GOTTESMAN-CHUANG SCHEME","text":"As argued above, state distillation can improve the fidelity of the Zeno gate to unity by trading off success probability. However, the output of the distillated Zeno 3 FIG. 4: Schematic of GC-Zeno gate. The state |χ is\n\n((|00 + |11 )|00 + (|01 + |10 )|11 )/2\n\ngate contains terms outside the computational basis due to photon loss and photon bunching. Hence if we want the gate to have unit fidelity, it is necessary to measure the output and exclude these failure terms by post-selection. However, such post-selection means that the gate can no longer be used directly as a CNOT gate for circuit-based quantum computing. [11] showed the viability in using state teleportation and single qubit operations to implement a CNOT gate. The scheme requires the four qubit entangled state"},{"section_type":"OTHER","section_title":"Gottesman and Chuang","text":"|χ = ((|00 + |11 )|00 + (|01 + |10 )|11 )/2.\n\nPreparing the entangled state requires a CZ operation, which can be done off-line with linear optics with high Bell measurements are made between the input qubits and the first and last qubits of |χ . The measurement results are fed forward for some single qubit corrections such that the circuit gives a proper CNOT operation. Here we propose using such scheme, as shown in figure 4 , to implement a GC-Zeno CNOT gate with high fidelity. Since this gate includes state distillation, post-selection and off-line state preparation, the gate has unit fidelity (under perfect mode-matching) and moderate success probability. Figure 5 plots the probability of success against the one-photon to two-photon transmission ratio κ. It shows that with κ = 10 4 (current best estimate), the probability of success is about 0.76, which is better than the break even point of 0.25 for the linear optics version of this gate [1]."},{"section_type":"OTHER","section_title":"IIa. Effect of Mode-Mismatch","text":"From source preparation to gate operation to detection, mode-mismatch is an unavoidable issue in optical quantum computing that causes unlocated errors which lowers the fidelity of the device [15] . Fortunately, with the help of quantum error correction, a certain amount of unlocated error rate, including but not limited to modemismatch errors, can be tolerated. A reliable quantum\n\nFIG. 5: Plot of probability of success versus log(κ) (in base 10) for GC-Zeno gate. Note that the success probability is not one at κ = 10 8 , but that the curve asymptotically approaches one for very large κ. Result is per two input qubits. Detector inefficiency is taken into account in accord with Dawson et al's bound.\n\ngate must therefore have unlocated error rates below this threshold.\n\nThe dominant source of mode-mismatch error in the GC-Zeno gate is from the CZ gate and τ gate, where two-photon interaction occurs. Here we follow Rohde et al's [14] analysis to examine the effect of such error. We take the simplest model in which the mode-mismatch is present between the photons entering the gate but remain constant through the gate. In this case, the modemismatch in two-photon interaction can be analysed as having two-photons fail to interact with some probability. This allows us to write the operations for the CZ gate as follow, where 0 < Γ < 1 quantifies the overlap of the two wavepackets. Γ 2 is the probability that the two photons successfully interacted and Γ = 0 for completely mode-mismatched and Γ = 1 for completely mode-matched. The bar in the | 11 term indicates modemismatched component of the state.\n\n|00 → |00 |01 → γ n 1 |01 |10 → γ n 1 |10 |11 → -Γγ n 1 τ |11 + 1 -Γ 2 γ n 1 | 11 +f (|02 , |20 ) ( 4\n\n)\n\nAnd similarly for the operations of τ gate:\n\n|00 → |00 |01 → γ ′ 1 |01 |10 → γ ′ 1 |10 |11 → Γγ ′ 1 τ |11 + 1 -Γ 2 γ ′ 1 | 11 +f (|02 , |20 ) ( 5\n\n)\n\nWith the equations for the CZ and τ gate[16], and given a normalized input state (α|00 + β|01 + δ|10 + ǫ|11 ), we can obtain analytical expression for the fidelity F (per qubit) and success probability P s (per qubit) of 4 the GC-Zeno gate as follow. Equation 6 and 7 show that both the fidelity and success probability are state dependent due to mode-mismatch. The worst case of fidelity occurs when the input state is the equal superposition state (|00 + |01 + |10 + |11 )/2 (i.e. α = δ = β = ǫ = 1/2) and the worst case of success probability occurs when the input state is the pure state |11 (i.e. α = β = δ = 0 and ǫ = 1).\n\nF = α * A 1 + β * A 2 + δ * A 3 + ǫ * A 4 |A 1 | 2 + |A 2 | 2 + |A 3 | 2 + |A 4 | 2 ( 6\n\n) P s = e -2λ/κ τ 2/κ 2(1 + e -λ/κ τ 2+1/κ ) × |A 1 | 2 + |A 2 | 2 + |A 3 | 2 + |A 4 | 2 ( 7\n\n)\n\nwhere\n\na 1 = (τ +τ Γ+ √ 1 -Γ 2 ), a 2 = (τ -τ Γ+ √ 1 -Γ 2 ), a 3 = (τ -τ Γ - √ 1 -Γ 2 ), a 4 = (τ + τ Γ - √ 1 -Γ 2 ), and A 1 = αa 2 1 + βa 1 a 2 + δa 1 a 2 + ǫa 2 2 , A 2 = αa 1 a 3 + βa 2 a 3 + δa 1 a 4 + ǫa 2 a 4 , A 3 = αa 1 a 3 + βa 1 a 4 + δa 2 a 3 + ǫa 2 a 4 , A 4 = αa 2 3 + βa 3 a 4 + δa 3 a 4 + ǫa 2 4"},{"section_type":"OTHER","section_title":"IIb. Effect of Detector Efficiency","text":"In practice, even for the most advanced photon detector, detector inefficiency is always present. The effect of this noise is to reduce the probability of success of the gate but not the fidelity because the errors are locatable."},{"section_type":"OTHER","section_title":"III. ESTIMATE OF BOUNDS FOR FAULT TOLERANCE","text":"We now wish to estimate lower bounds on the modematching, Γ, and photon loss ratio, κ, that will still allow fault-tolerant operation. We allow a small amount of detector inefficiency but assume all other parameters are ideal. To make this estimate we directly use the bounds obtained by Dawson et al [12] for a deterministic error correction protocol. For this protocol, they numerically derived one bound using the 7-qubit Steane code and another bound using the 23-qubit Golay code.\n\nIn order to use the Dawson et al's bounds we need to identify the unlocated and located error rates for our gate. In general, the unlocated error rate is less than 1 -F but here we take it to be 1 -F because in our analysis, γ is almost 1, which means the other terms involved are very small. The located error rate is simply 1 -P s (both F and P s are per qubit). Using these relationships, we convert each of the bounds into a fidelity versus success probability bound. For a gate built with two-photon absorbers that have a certain single-photon to two-photon transmission ratio κ, we can find an optimal λ (i.e. choosing an optimal absorber length) that\n\nFIG. 6: Lower bounds of amount of mode-matching Γ required for a fault tolerant GC-Zeno gate versus singlephoton to two-photon transmission ratio κ. The bounds are derived from Dawson et al's [12] results on deterministic error correction protocol. The top and bottom curves are for the 7-qubit Steane code and the 23-qubit Golay code respectively. Above the curves are the regions where the amount of mode-mismatch is tolerable. Here we have used the worst case input.\n\ngives a maximum success probability. Hence, by matching the success probability with the bound, we can determine the corresponding fidelity threshold and therefore find the least amount of mode-matching required for fault tolerant gate operation. We note that the error model used by Dawson et al is specific to their optical cluster state architecture and will differ in detail from the appropriate error model for the GC-Zeno gate. Nonetheless we assume that a comparison based on the total error rates will give a good estimate of the bounds.\n\nFigure 6 shows the lower bounds on the modematching parameter Γ for a gate with a certain κ. Since the fidelity and success probability are state dependent due to mode-mismatch, in that figure, we have plotted for the case of worst fidelity input state (i.e. the equal superposition state). The top and bottom curves are best fit curves for using the 7-qubit Steane code and the 23-qubit Golay code respectively. The curves show that highly mode-matched photons are essential for robust gate operation. With the worst fidelity input state,\n\n(|00 + |01 + |10 + |11 )/2, for\n\nthe Steane code, the lowest Γ required for fault tolerant operation is about 0.998, where κ = 10 6 , and for the Golay code, the lowest Γ required is about 0.996, where κ = 5 × 10 5 . With the worst success probability input state, |11 , for the Steane code, the lowest Γ required for fault tolerant operation is about 0.995, where κ = 10 6 , and for the Golay code, the lowest Γ required is about 0.989, where κ = 5 × 10 5 . Figure 6 also shows that under (near) perfect mode-matching, the required κ can be as low as approximately 6000 for the Steane code and 2000 for the Golay code. Two-photon absorbers with such κ values may be achievable with the best of current technology."},{"section_type":"OTHER","section_title":"IIIa. Advantage of Using State Distillation","text":"State distillation allows us to trade off some success probability against fidelity for the GC-Zeno gate, or in other words, reducing the unlocated error rate by having a larger located error rate. Since the deterministic error correction protocol can tolerate both unlocated and located errors, therefore we should ask whether state distillation is truly advantageous? We can answer this question by comparing two GC-Zeno gates in the case of perfect mode matching, where one has complete distillation and the other has no distillation. For the case of complete distillation, the deterministic error correction protocol with the 7-qubit Steane code can tolerate errors of a GC-Zeno gate with κ = 6100, and with the 23-qubit Golay code, it can tolerate errors of a GC-Zeno gate with κ = 2100. For state distillation to be advantageous under the same protocol, these values of κ must be smaller than the values of κ for the case of no distillation [17] .\n\nFor the case of no distillation, the fidelity and success probability of the gate becomes state dependent. In the parameters space of interest, the input that gives the worst fidelity is (|00 + |01 + |10 -|11 )/2. With this input state, we find that for the protocol using the 7qubit Steane code and no distillation, the critical κ is 12000. Similarly, for the protocol using the 23-qubit Golay code and no distillation, the critical κ is 4300. With an arbitrary amount of distillation, the value of κ lies between the limit of no distillation and full distillation cases. Hence it is evident that state distillation is advantageous. Also, it should be noted it is better to have only located error, which is the case when there is full distillation, than have both located and unlocated errors, which is the case when no or some distillation is utilized."},{"section_type":"CONCLUSION","section_title":"IV. CONCLUSION","text":"In this paper, we have shown that it is possible to build a high fidelity Zeno CNOT gate with two distillated Zeno gates implemented in the Gottesman-Chuang teleportation CNOT scheme. For one-photon to twophoton transmission ratio κ = 10 4 (current best estimate), the gate has a success probability of 0.76 under perfect mode-matching. When including measurement noise that equals one-tenth of the gate's noise, and the effect of mode-mismatch in the CZ and τ gate, we find that with the deterministic error correction protocol using the 7-qubit Steane code, the lowest Γ required for fault tolerant gate operation is 0.998, where κ = 10 6 . For using the 23-qubit Golay code, the lowest Γ required is 0.996, where κ = 5 × 10 5 . Hence, the requirement on mode-matching is stringent for a fault tolerant GC-Zeno gate. * Electronic address: pmleung@physics.uq.edu.au [1] E. Knill, R. Laflamme, and G.J. Milburn, Nature 409, 46-52 (2001) [2] N.Yoran and B.Reznik, Phys. Rev. Lett. 91, 037903 (2003) [3] M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004) [4] A.J.F.Hayes, A.Gilchrist, C.R.Myers and T.C.Ralph, J.Opt.B 6, 533 (2004) [5] D.E.Browne and T.Rudolph, Phys. Rev. Lett. 95, 010501 (2005) [6] J.D. Franson, B.C. Jacobs, and T.B. Pittman, Phys. Rev. A 70, 062302 (2004) [7] C.R. Myers and A. Gilchrist, quant-ph 0612176 (2006) [8] P.M. Leung and T.C. Ralph, Phys. Rev. A 74, 062325 (2006) [9] R.T. Thew and W.J. Munro, Phys. Rev. A 63, 030302(R) (2001) [10] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86 5188 (2001) [11] D. Gottesman, I.L. Chuang, Nature 402, 6760 (1999) [12] C.M. Dawson, H.L. Haselgrove, and M.A. Nielsen, Phys.\n\nRev. Lett. 96, 020501 (2006) [13] A.P. Lund, T.C. Ralph, Phys. Rev. A, 66, 032307 (2002) [14] P.P. Rohde and T.C. Ralph, Phys. Rev. A 73, 062312 (2006) [15] Note that mode mismatch in CZ and τ gate causes unlocated error that lowers the fidelity, in which we find that it cannot be improved with state distillation. [16] Due to mode-mismatch, the τ gate is less effective in twophoton distillation. It is true that we can increase the two-photon absorption strength in the τ gate to make up for the inefficiency. However, here we assume that we do not know the mode-matching parameter Γ of the input wavepackets, and therefore this adjustment cannot be made. In addition, increasing the two-photon distillation will increase single-photon loss as well, which lowers the probability of success. [17] An ideal Zeno gate requires strong quantum Zeno effect and strong quantum Zeno effect corresponds to a large κ value. However such large non-linearity is difficult to engineer. Hence it is desirable to have a gate that works with modest κ."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"Photons are a natural choice for making qubits because the quantum information encoded can have a long decoherence time and is easy to manipulate and measure. Also, photonic qubits are the only type of qubits that are feasible for long distance quantum communication. Quantum information processing requires universal two-qubit entangling gates. Knill et al [1] showed that it is theoretically possible to do scalable quantum computing with linear optics by using measurement induced interactions to perform the two-qubit gates. However, despite a continuous effort in reducing the resource requirement [2, 3, 4, 5] , the resource overhead is still high for linear optical quantum computing. Franson et al[6] proposed the use of two-photon absorption non-linearity and exploiting the quantum Zeno effect to implement a control sign (CZ) gate that requires much less resources than linear optics schemes. However, the problem with the Zeno gate is that photon loss affects the performance of the Zeno gate significantly and the single photon to twophoton loss ratio requirement is very stringent. One solution couldbe to combine measurement induced quantum processing with the Zeno gate. Previously we have shown that when using the Zeno gate for qubit fusion [8], state distillation [9] with post-selection can boost the gate fidelity to unity and that for less stringent absorption ratios the gate has an advantage in success probability over linear optics gates for fusing clusters of qubits. These clusters of qubits can then be used for cluster state quantum computing [10] . In addition, Myers and Gilchrist [7] have shown that the performance of the Zeno gate may be enhanced by using error correction codes such as redundancy and parity encoding.\\n\\nHere we design a high fidelity Zeno CNOT gate suitable for circuit-based quantum computing. Although with the current estimate of the photon loss ratio, only a poor fidelity Zeno gate is directly achievable, here we show that it is possible to use two of these Zeno gates to do Bell measurements and implement a Gottesman-Chuang[11] teleportation type of CNOT gate (GC-Zeno gate) that, like the fusion gate, gives high fidelity via state distillation and moderate success probability via partially off-line state preparation. We include the effect of mode-mismatch and detector efficiency on the scheme and estimate lower bounds on the parameter which in principle allow fault tolerant operation.\\n\\nThe paper is arranged in the following way. The introduction continues with a subsection on the Zeno CZ gate, which describes the scheme and modelling of the gate and give descriptions on the modelling parameters that are also used for modelling the GC-Zeno CZ gate. In section II, we discuss the GC-Zeno gate and the effect of mode-mismatch and detector efficiency on the gate. In section III, we give estimates of the lower bounds on the photon loss ratio and mode-matching, followed by a subsection on the advantage in using state distillation. We conclude and summarize in section IV.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Ia. Zeno CZ Gate\",\n \"text\": \"Franson et al's control sign gate scheme consists of a pair of optical fibers weakly evanescently coupled and doped with two-photon absorbing atoms. The purpose of the two-photon absorbers is to suppress the occurrence of two photon state components in the two fibre modes via the Quantum Zeno effect. This allows the state to remain in the computational basis. After a length of fibre corresponding to a complete swap of the two fibre modes, a π phase difference is produced between the |11 term and the other three basis terms. After swapping the fibre modes by simply crossing them, a CZ gate is achieved. The gate becomes near deterministic and performs a near unitary operation when the Quantum Zeno effect is strong and photon loss is insignificant. However, with current technology, the strength of the 2 Quantum Zeno effect is a few orders of magnitude below what is required, and thus the Zeno gate has significant photon loss.\\n\\nIn [8], the gate is modelled as a succession of n weak beamsplitters followed by two-photon absorbers as shown in Fig. 1 . As n → ∞ the model tends to the continuous coupling limit envisaged for the physical realization. The gate operates on the single-rail encoding for which |0 L = |0 and |1 L = |1 with the kets representing photon Fock states. Fig. 2 shows how the single rail CZ can be converted into a dual rail [13] CZ with logical encoding |0 L = |H and |1 L = |V . Let L be the total length of n absorbers. Also, let γ 1 = exp( -λ nκ ) and γ 2 = exp( -λ n ) be the probability of single photon and two-photon transmission respectively for one absorber. Here the parameter λ = χL, where L is the length of the absorber and χ is the corresponding proportionality constant related to the absorption cross section. Furthermore, κ specifies the relative strength of the two transmissions and relates them by γ 2 = γ κ 1 . This CZ gate does the following operation:\\n\\n|00 → |00 |01 → γ n/2 1 |01 |10 → γ n/2 1 |10 |11 → -γ n 1 τ |11 + f (|02 , |20 ) ( 1\\n\\n)\\n\\nwhere the new expression for τ is given by:\\n\\nτ n,λ = 2 -3 2 -n d n,λ (g n,λ + d n,λ √ 2 ) n ( √ 2d n,λ -h n,λ ) +(g n,λ - d n,λ √ 2 ) n ( √ 2d n,λ + h n,λ ) d n,λ = (1 + cos 2π n )(1 + γ 2 ) + 2 √ γ 2 (cos( 2π n ) - 3\\n\\n)\\n\\ng n,λ = (cos π n )( √ γ 2 + 1) h n,λ = 2(cos π n )( √ γ 2 -1) ( 2\\n\\n)\\n\\nThe explicit form of the |02 , |20 state components are suppressed, as they lie outside the computational basis and so do not explicitly contribute to the fidelity.\\n\\nFIG. 1: Construction of our CZ gate.\\n\\nFIG. 2: CZ gate in dual rail implementation.\\n\\nFrom equation 1, it is clear that the amplitude of the four computational states are unequal and this lowers the gate fidelity. With the current best estimate of κ = 10 4 , the unherald fidelity is only 0.94. If the gate is used in a measurement based strategy then state distillation can be used and the fidelity of the gate can be improved by trading off some success probability. Figure 3 shows the distillated Zeno CZ gate circuit. The τ is simply an interferometer consisting of two 50-50 beam splitters with a two-photon absorber in each arm, which gives operation:\\n\\n|00 → |00 , |01 → γ ′ 1 |01 , |10 → γ ′ 1 |10 , and |11 → γ ′ 1 τ |11 . Here γ ′ 1 = τ 1/κ\\n\\nis the single photon transmission coefficient of the absorber. The distillation beam splitters labelled 1 to 3 have transmission coeffi-\\n\\ncient γ n 1 , γ ′ 1 , γ n 1 γ ′ 1 τ respectively.\\n\\nWith these distillations in place, the operation of the distillated Zeno CZ gate is:\\n\\n|00 → γ n 1 γ ′ 1 τ |00 |01 → γ n 1 γ ′ 1 τ |01 |10 → γ n 1 γ ′ 1 τ |10 |11 → -γ n 1 γ ′ 1 τ |11 + f (|02 , |20 ) ( 3\\n\\n)\\n\\nAfter measuring the output (detectors at output are not shown in fig. 3 ) and treating the photon bunching terms (|02 , |20 ) and the terms with photon loss as failures (excluded from Eq. 3 for clarity), renormalising the states gives unit heralded fidelity independent of λ and probability of success\\n\\nP s = γ 2n 1 γ ′2 1 τ 2 = e -2λ/κ τ 2+2/κ .\\n\\nFIG. 3: Schematic of distillated Zeno CZ gate.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. ZENO GATE USING GOTTESMAN-CHUANG SCHEME\",\n \"text\": \"As argued above, state distillation can improve the fidelity of the Zeno gate to unity by trading off success probability. However, the output of the distillated Zeno 3 FIG. 4: Schematic of GC-Zeno gate. The state |χ is\\n\\n((|00 + |11 )|00 + (|01 + |10 )|11 )/2\\n\\ngate contains terms outside the computational basis due to photon loss and photon bunching. Hence if we want the gate to have unit fidelity, it is necessary to measure the output and exclude these failure terms by post-selection. However, such post-selection means that the gate can no longer be used directly as a CNOT gate for circuit-based quantum computing. [11] showed the viability in using state teleportation and single qubit operations to implement a CNOT gate. The scheme requires the four qubit entangled state\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Gottesman and Chuang\",\n \"text\": \"|χ = ((|00 + |11 )|00 + (|01 + |10 )|11 )/2.\\n\\nPreparing the entangled state requires a CZ operation, which can be done off-line with linear optics with high Bell measurements are made between the input qubits and the first and last qubits of |χ . The measurement results are fed forward for some single qubit corrections such that the circuit gives a proper CNOT operation. Here we propose using such scheme, as shown in figure 4 , to implement a GC-Zeno CNOT gate with high fidelity. Since this gate includes state distillation, post-selection and off-line state preparation, the gate has unit fidelity (under perfect mode-matching) and moderate success probability. Figure 5 plots the probability of success against the one-photon to two-photon transmission ratio κ. It shows that with κ = 10 4 (current best estimate), the probability of success is about 0.76, which is better than the break even point of 0.25 for the linear optics version of this gate [1].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IIa. Effect of Mode-Mismatch\",\n \"text\": \"From source preparation to gate operation to detection, mode-mismatch is an unavoidable issue in optical quantum computing that causes unlocated errors which lowers the fidelity of the device [15] . Fortunately, with the help of quantum error correction, a certain amount of unlocated error rate, including but not limited to modemismatch errors, can be tolerated. A reliable quantum\\n\\nFIG. 5: Plot of probability of success versus log(κ) (in base 10) for GC-Zeno gate. Note that the success probability is not one at κ = 10 8 , but that the curve asymptotically approaches one for very large κ. Result is per two input qubits. Detector inefficiency is taken into account in accord with Dawson et al's bound.\\n\\ngate must therefore have unlocated error rates below this threshold.\\n\\nThe dominant source of mode-mismatch error in the GC-Zeno gate is from the CZ gate and τ gate, where two-photon interaction occurs. Here we follow Rohde et al's [14] analysis to examine the effect of such error. We take the simplest model in which the mode-mismatch is present between the photons entering the gate but remain constant through the gate. In this case, the modemismatch in two-photon interaction can be analysed as having two-photons fail to interact with some probability. This allows us to write the operations for the CZ gate as follow, where 0 < Γ < 1 quantifies the overlap of the two wavepackets. Γ 2 is the probability that the two photons successfully interacted and Γ = 0 for completely mode-mismatched and Γ = 1 for completely mode-matched. The bar in the | 11 term indicates modemismatched component of the state.\\n\\n|00 → |00 |01 → γ n 1 |01 |10 → γ n 1 |10 |11 → -Γγ n 1 τ |11 + 1 -Γ 2 γ n 1 | 11 +f (|02 , |20 ) ( 4\\n\\n)\\n\\nAnd similarly for the operations of τ gate:\\n\\n|00 → |00 |01 → γ ′ 1 |01 |10 → γ ′ 1 |10 |11 → Γγ ′ 1 τ |11 + 1 -Γ 2 γ ′ 1 | 11 +f (|02 , |20 ) ( 5\\n\\n)\\n\\nWith the equations for the CZ and τ gate[16], and given a normalized input state (α|00 + β|01 + δ|10 + ǫ|11 ), we can obtain analytical expression for the fidelity F (per qubit) and success probability P s (per qubit) of 4 the GC-Zeno gate as follow. Equation 6 and 7 show that both the fidelity and success probability are state dependent due to mode-mismatch. The worst case of fidelity occurs when the input state is the equal superposition state (|00 + |01 + |10 + |11 )/2 (i.e. α = δ = β = ǫ = 1/2) and the worst case of success probability occurs when the input state is the pure state |11 (i.e. α = β = δ = 0 and ǫ = 1).\\n\\nF = α * A 1 + β * A 2 + δ * A 3 + ǫ * A 4 |A 1 | 2 + |A 2 | 2 + |A 3 | 2 + |A 4 | 2 ( 6\\n\\n) P s = e -2λ/κ τ 2/κ 2(1 + e -λ/κ τ 2+1/κ ) × |A 1 | 2 + |A 2 | 2 + |A 3 | 2 + |A 4 | 2 ( 7\\n\\n)\\n\\nwhere\\n\\na 1 = (τ +τ Γ+ √ 1 -Γ 2 ), a 2 = (τ -τ Γ+ √ 1 -Γ 2 ), a 3 = (τ -τ Γ - √ 1 -Γ 2 ), a 4 = (τ + τ Γ - √ 1 -Γ 2 ), and A 1 = αa 2 1 + βa 1 a 2 + δa 1 a 2 + ǫa 2 2 , A 2 = αa 1 a 3 + βa 2 a 3 + δa 1 a 4 + ǫa 2 a 4 , A 3 = αa 1 a 3 + βa 1 a 4 + δa 2 a 3 + ǫa 2 a 4 , A 4 = αa 2 3 + βa 3 a 4 + δa 3 a 4 + ǫa 2 4\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IIb. Effect of Detector Efficiency\",\n \"text\": \"In practice, even for the most advanced photon detector, detector inefficiency is always present. The effect of this noise is to reduce the probability of success of the gate but not the fidelity because the errors are locatable.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. ESTIMATE OF BOUNDS FOR FAULT TOLERANCE\",\n \"text\": \"We now wish to estimate lower bounds on the modematching, Γ, and photon loss ratio, κ, that will still allow fault-tolerant operation. We allow a small amount of detector inefficiency but assume all other parameters are ideal. To make this estimate we directly use the bounds obtained by Dawson et al [12] for a deterministic error correction protocol. For this protocol, they numerically derived one bound using the 7-qubit Steane code and another bound using the 23-qubit Golay code.\\n\\nIn order to use the Dawson et al's bounds we need to identify the unlocated and located error rates for our gate. In general, the unlocated error rate is less than 1 -F but here we take it to be 1 -F because in our analysis, γ is almost 1, which means the other terms involved are very small. The located error rate is simply 1 -P s (both F and P s are per qubit). Using these relationships, we convert each of the bounds into a fidelity versus success probability bound. For a gate built with two-photon absorbers that have a certain single-photon to two-photon transmission ratio κ, we can find an optimal λ (i.e. choosing an optimal absorber length) that\\n\\nFIG. 6: Lower bounds of amount of mode-matching Γ required for a fault tolerant GC-Zeno gate versus singlephoton to two-photon transmission ratio κ. The bounds are derived from Dawson et al's [12] results on deterministic error correction protocol. The top and bottom curves are for the 7-qubit Steane code and the 23-qubit Golay code respectively. Above the curves are the regions where the amount of mode-mismatch is tolerable. Here we have used the worst case input.\\n\\ngives a maximum success probability. Hence, by matching the success probability with the bound, we can determine the corresponding fidelity threshold and therefore find the least amount of mode-matching required for fault tolerant gate operation. We note that the error model used by Dawson et al is specific to their optical cluster state architecture and will differ in detail from the appropriate error model for the GC-Zeno gate. Nonetheless we assume that a comparison based on the total error rates will give a good estimate of the bounds.\\n\\nFigure 6 shows the lower bounds on the modematching parameter Γ for a gate with a certain κ. Since the fidelity and success probability are state dependent due to mode-mismatch, in that figure, we have plotted for the case of worst fidelity input state (i.e. the equal superposition state). The top and bottom curves are best fit curves for using the 7-qubit Steane code and the 23-qubit Golay code respectively. The curves show that highly mode-matched photons are essential for robust gate operation. With the worst fidelity input state,\\n\\n(|00 + |01 + |10 + |11 )/2, for\\n\\nthe Steane code, the lowest Γ required for fault tolerant operation is about 0.998, where κ = 10 6 , and for the Golay code, the lowest Γ required is about 0.996, where κ = 5 × 10 5 . With the worst success probability input state, |11 , for the Steane code, the lowest Γ required for fault tolerant operation is about 0.995, where κ = 10 6 , and for the Golay code, the lowest Γ required is about 0.989, where κ = 5 × 10 5 . Figure 6 also shows that under (near) perfect mode-matching, the required κ can be as low as approximately 6000 for the Steane code and 2000 for the Golay code. Two-photon absorbers with such κ values may be achievable with the best of current technology.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IIIa. Advantage of Using State Distillation\",\n \"text\": \"State distillation allows us to trade off some success probability against fidelity for the GC-Zeno gate, or in other words, reducing the unlocated error rate by having a larger located error rate. Since the deterministic error correction protocol can tolerate both unlocated and located errors, therefore we should ask whether state distillation is truly advantageous? We can answer this question by comparing two GC-Zeno gates in the case of perfect mode matching, where one has complete distillation and the other has no distillation. For the case of complete distillation, the deterministic error correction protocol with the 7-qubit Steane code can tolerate errors of a GC-Zeno gate with κ = 6100, and with the 23-qubit Golay code, it can tolerate errors of a GC-Zeno gate with κ = 2100. For state distillation to be advantageous under the same protocol, these values of κ must be smaller than the values of κ for the case of no distillation [17] .\\n\\nFor the case of no distillation, the fidelity and success probability of the gate becomes state dependent. In the parameters space of interest, the input that gives the worst fidelity is (|00 + |01 + |10 -|11 )/2. With this input state, we find that for the protocol using the 7qubit Steane code and no distillation, the critical κ is 12000. Similarly, for the protocol using the 23-qubit Golay code and no distillation, the critical κ is 4300. With an arbitrary amount of distillation, the value of κ lies between the limit of no distillation and full distillation cases. Hence it is evident that state distillation is advantageous. Also, it should be noted it is better to have only located error, which is the case when there is full distillation, than have both located and unlocated errors, which is the case when no or some distillation is utilized.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"IV. CONCLUSION\",\n \"text\": \"In this paper, we have shown that it is possible to build a high fidelity Zeno CNOT gate with two distillated Zeno gates implemented in the Gottesman-Chuang teleportation CNOT scheme. For one-photon to twophoton transmission ratio κ = 10 4 (current best estimate), the gate has a success probability of 0.76 under perfect mode-matching. When including measurement noise that equals one-tenth of the gate's noise, and the effect of mode-mismatch in the CZ and τ gate, we find that with the deterministic error correction protocol using the 7-qubit Steane code, the lowest Γ required for fault tolerant gate operation is 0.998, where κ = 10 6 . For using the 23-qubit Golay code, the lowest Γ required is 0.996, where κ = 5 × 10 5 . Hence, the requirement on mode-matching is stringent for a fault tolerant GC-Zeno gate. * Electronic address: pmleung@physics.uq.edu.au [1] E. Knill, R. Laflamme, and G.J. Milburn, Nature 409, 46-52 (2001) [2] N.Yoran and B.Reznik, Phys. Rev. Lett. 91, 037903 (2003) [3] M.A. Nielsen, Phys. Rev. Lett. 93, 040503 (2004) [4] A.J.F.Hayes, A.Gilchrist, C.R.Myers and T.C.Ralph, J.Opt.B 6, 533 (2004) [5] D.E.Browne and T.Rudolph, Phys. Rev. Lett. 95, 010501 (2005) [6] J.D. Franson, B.C. Jacobs, and T.B. Pittman, Phys. Rev. A 70, 062302 (2004) [7] C.R. Myers and A. Gilchrist, quant-ph 0612176 (2006) [8] P.M. Leung and T.C. Ralph, Phys. Rev. A 74, 062325 (2006) [9] R.T. Thew and W.J. Munro, Phys. Rev. A 63, 030302(R) (2001) [10] R. Raussendorf and H.J. Briegel, Phys. Rev. Lett. 86 5188 (2001) [11] D. Gottesman, I.L. Chuang, Nature 402, 6760 (1999) [12] C.M. Dawson, H.L. Haselgrove, and M.A. Nielsen, Phys.\\n\\nRev. Lett. 96, 020501 (2006) [13] A.P. Lund, T.C. Ralph, Phys. Rev. A, 66, 032307 (2002) [14] P.P. Rohde and T.C. Ralph, Phys. Rev. A 73, 062312 (2006) [15] Note that mode mismatch in CZ and τ gate causes unlocated error that lowers the fidelity, in which we find that it cannot be improved with state distillation. [16] Due to mode-mismatch, the τ gate is less effective in twophoton distillation. It is true that we can increase the two-photon absorption strength in the τ gate to make up for the inefficiency. However, here we assume that we do not know the mode-matching parameter Γ of the input wavepackets, and therefore this adjustment cannot be made. In addition, increasing the two-photon distillation will increase single-photon loss as well, which lowers the probability of success. [17] An ideal Zeno gate requires strong quantum Zeno effect and strong quantum Zeno effect corresponds to a large κ value. However such large non-linearity is difficult to engineer. Hence it is desirable to have a gate that works with modest κ.\"\n }\n]","isConversational":false}}},{"rowIdx":188,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1073"},"arxiv_id":{"kind":"string","value":"0704.1073"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.064020"},"content_hash":{"kind":"string","value":"148c809db5725248492c69a93d04e4a35fb7ed85a244bc3f944f0d0c6d4b58f3"},"title":{"kind":"string","value":"Shell-mediated tunnelling between (anti-)de Sitter vacua"},"abstract":{"kind":"string","value":"We give an extensive study of the tunnelling between arbitrary (anti-)de\nSitter spacetimes separated by an infinitesimally thin relativistic shell in\narbitrary spacetime dimensions. In particular, we find analytically an exact\nexpression for the tunnelling amplitude.\n The detailed spacetime structures that can arise are discussed, together with\nan effective \"regularization scheme\" for \"before tunnelling\" configurations."},"authors":{"kind":"list like","value":["Stefano Ansoldi and Lorenzo Sindoni"],"string":"[\n \"Stefano Ansoldi and Lorenzo Sindoni\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["gr-qc","hep-th"],"string":"[\n \"gr-qc\",\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"gr-qc"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The study of vacuum decay initiated more than 30 years ago with the work of Callan and Coleman [1, 2] ; in the following years the interest in the subject increased and the possible interplay of true vacuum bubbles with gravitation was also studied [3, 4] , together with bubbles collisions and their importance in the early universe [5, 6] . At the same time, and as opposed with the true vacuum bubbles of Coleman et al., false vacuum bubbles were also considered. In connection with gravity, the behavior of regions of false vacuum, first studied by Sato et al. [7, 8, 9, 10, 11, 12] , was for example analyzed in [13, 14, 15, 16, 17, 18, 19, 20] . For additional papers analyzing it in the context of inflation, we refer the reader to [21, 22, 23] ; interesting links with more phenomenologically oriented approaches can also be drawn, as witnessed, for instance, by the recent [24] .\n\nIn these last works, a minisuperspace approximation was adopted to quantize the system. In more detail, since general relativistic shells 1 can be used as a convenient model and since in spherical symmetry the system only has one degree of freedom, standard semiclassical methods might be suitable to analyze the decay process (see [25] for an early, in principle, discussion of this point). In connection with cosmology, spaces equipped with a cosmological constant (i.e. de Sitter space and generalizations) have been naturally considered. In this context it also worth to remark the important role that they play in connection with the problem of gravitational entropy, causal structure and the presence of horizons (see, for instance [26, 27, 28, 29, 30] as well as the suggestive [31] ).\n\nNotwithstanding many interesting results, after 30 years, and with different flavors, the problem of the stability of (the de Sitter) vacuum in connection with the dynamics of false vacuum bubbles, is still a debated one [19, 32, 33] .\n\nThe still open issues are highly non-trivial and go back to the, also long lasting, problem of formulating a consistent framework for a quantum theory of gravity [34, 35, 36, 37, 38, 39, 40] , but we will not take explicitly this point of view here. We will instead analyze a specific situation, described below, in which the nucleation rate can be computed exactly in arbitrary spacetime dimensions. We will, then, explicitly compute in closed form the nucleation rate in the semiclassical approximation, compare it with existing results, and discuss in detail the associated spacetime structures: an analysis of how quantum effects may be relevant in the context of tunnelling from nothing configurations [41] will also be given. These results, extend some results present in the literature (for instance [16, 42, 43, 44] ) and can also provide a useful limiting case of more general situations, for instance those discussed in [33] . At the same time, to consider spacetimes of higher dimensionality and negative cosmological constant is especially important in view of recent results in the context of the AdS/CFT correspondence [45] and of the braneworld cosmological scenario [46, 47] (see also [48] and references therein for a study in the context of noncommutative branes).\n\nApart from the papers already cited above, the instanton approach has also been discussed by other authors (see for instance [49, 50, 51] ) and although it will not be directly related to the present paper, we cannot avoid mentioning the suggestive relationship between the decay of the cosmological constant, membranes generated by higher rank gauge potentials and black holes, 2 which have appeared in many papers in the literature [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] .\n\nThe structure of the paper is the following: in section II we recall the formalism by which a general relativistic thin spherical brane/shell can be described; this also gives us the opportunity to fix conventions and notations and briefly describe the canonical approach to its semiclassical quantization (subsection II C); a convenient dimensionless formalism is also introduced (subsection II D). We then directly come (section III) to the main result of this paper, which is the calculation of the tunnelling rate between the classical configurations of the system, in arbitrary spacetime dimensions; the results for the cases of 3, 4 and 5 spacetime dimensions are explicitly presented with dedicated plots of the values of the action as a function of the two dimensionless parameters of the model (subsection III C). The exact results for the tunnelling amplitude/probability calculated in section III correspond to specific transitions which take place in spacetime and that will be discussed later on, in a dedicated appendix. We discuss, instead, in the main text (section IV) a proposal to regularize some spacetime configurations which appear to be singular, relating this issue to the description of the brane energy-matter content. In the concluding section V, the results of the paper are summarized; two appendices follows with the detailed description of the parameter space of the system (appendix A) and of all the Penrose diagrams associated with different values of the parameters of the problem (appendix B): they are crucial to fully grasp the physical system under consideration.\n\nIn this section we are going to briefly review some results about the dynamics of co-dimension one branes in an (N + 1)-dimensional spacetime, where, under the words co-dimension one brane, we understand an Ndimensional hypersurface Σ separating the (N + 1)dimensional manifold in two domains, M -and M + having Σ as a common part of their boundary: in brief ∂M -∩ ∂M + = Σ. In what follows we are going to use the clear notation of [63] : the formulation developed there can be readily extended to higher dimensions, just by letting the indices run on an extended set of values. We will thus quickly report this paraphrase with the purpose of recalling some notations and conventions. In particular let us choose two arbitrary systems of N + 1 independent vector fields E ±(a) in M ± , respectively, with dual forms Ω (b) ± . Denoting with g (±) the four dimensional metric tensors in the two manifolds we can write 2 , 2 In a few equations below, we will use the notation -, -to denote the scalar product. We also anticipate the notation ∇ Y X for the covariant derivative of the vector fiend X in the direction\n\nin general, g (±) = g (±)ab Ω (a) (±) ⊗ Ω (b) (±) ; in our notation Latin indices a, b (as well as all other latin indices) will vary in the set {0, 1, 2, . . . , N }.\n\nLet us first concentrate our attention on Σ: it is also a manifold, as M ± , and we will denote by e (µ) an Ndimensional system of (commuting) independent vector fields on Σ, with dual system ω (ν) ; the indices µ and ν (as well as all other Greek indices) will vary in the set {0, 1, . . . , N -1}. Because of the physical interpretation we will be interested in the case in which Σ is embedded as a timelike surface in M ± . Since Σ is timelike the normal to Σ in M ± is a spacelike vector n, which we choose normalized so that n, n = +1. Moreover, our convention is that the normal points from the \"-\" to the \"+\" domain of spacetime.\n\nThe components of n (an unambiguously defined nonnull vector) will be different, in general, when measured by an observer in M -or by one in M + , according to the following definition: a) .\n\nn a | ± = n, E (a) | ± = n, E ±(\n\nIn terms of n the extrinsic curvature of the surface Σ can be expressed as 3 K µν = n, ∇ e (µ) e (ν)\n\nand, in general, it is different on theand + side. Moreover, by the Gauss-Codazzi formalism, the geometry of spacetime around Σ can be described in terms of the intrinsic geometry of the hypersurface and of its extrinsic curvature. In the spirit of this formalism, let us denote by h the intrinsic metric of the hypersurface Σ, i.e. h = h µν ω (µ) ⊗ ω (ν) .\n\nWhen we will work with quantities defined in the bulk we will need to distinguish the ones defined in M + from the ones defined in M -, and to this end we will use, as we did above, \"±\" superscripts or subscripts. In many cases we will also be interested in the jump of these quantities across Σ: for instance, if we consider the extrinsic curvature, we may need the to consider the difference K + αβ -K - αβ : following [63, 64] we are going to rewrite this difference as [K αβ ]. Throughout this paper this will be the only meaning that we will give to the square brackets, i.e.\n\ndef.\n\n= A + -A -.\n\nTo avoid confusion no other use of the square brackets will be done.\n\nof the vector field Y . 3 We stress the normalization condition on n.\n\nThe brane Σ can be more than just a mathematical surface, i.e. we can (and, in most of the cases, we want to) equip it with a matter-energy content: it is then an infinitesimally 4 thin distribution of matter-energy. Thanks to the above mentioned Gauss-Codazzi formalism, we can rewrite Einstein equations to make explicit the contribution from the localized matter. Then, the dynamics of Σ as a surface separating M + from M -is obtained solving the following system of equations,\n\nK + αβ -K - αβ ≡ [K αβ ] = 8πG N +1 (S αβ -h αβ S/2) ; ( 1\n\n)\n\nthese are Israel's junction conditions [63, 64] which relate the jump in the extrinsic curvature [K µν ], i.e. the \"jump\" in the way the surface is embedded in each geometry, to the stress-energy tensor S µν of the matter contained on Σ. The tensor S must also satisfy a conservation equation,\n\n(N ) ∇, S µ = [T (n, e (µ) )],\n\nwhere T is the stress energy tensor describing the content of the complete spacetime manifold. Once we have specified the matter content of the bulk (and hence the geometry according to Einstein equations) the description of the dynamics of the system is obtained by solving the system of equations (1); we refer the reader to [63] and [66] for additional material and related considerations.\n\nThe set-up that is of interest for us is a simplified one, in which all the system is spherically symmetric and the surface stress-energy tensor is that of the, so called, tension model, with S = -κh,\n\nwhere κ > 0 is a constant called the tension of the brane. In what follows, it will be convenient to also define κ = 4π(N -2)G N +1 κ.\n\nThanks to the spherical symmetry, the system of equations (1) can then be reduced to a single equation [67], R ǫ -Ṙ2 + f -(R)ǫ + Ṙ2 + f + (R) = κR 2 ; (2) 4 Strictly speaking Σ is a source defined in a distributional way:\n\nfor additional material on this point, the reader is referred, for instance, to [65].\n\nin the above, f ± (r ± ) are the metric functions of the static line element adapted to the spherical symmetry, i.e., taking the four dimensional case as a convenient example, we choose in both M ± the coordinate system x a ± = (t ± , θ ± , φ ± , r ± ) corresponding to the basis vectors e t± , e θ± , e φ± and e r± , such that the metric is reduced to the form g (±)ab = diag(-f ± (r ± ), r 2 ± , r 2 ± sin 2 θ ± , 1/f ± (r ± )). Generalizations to spacetimes with higher dimensionality are just more cumbersome to write but trivial in their substance. In these coordinate systems, we denote the radius of the brane by R(τ ), where τ is the proper time of an observer comoving with the brane and an overdot denotes the derivative with respect to τ . Moreover, as anticipated in the introduction, we consider that this N -dimensional brane separates two (N + 1)-dimensional spacetimes of the de Sitter/anti-de Sitter type, in general with different cosmological constants Λ ± . We thus have f ± (r ± ) = 1 -2Λ ± N (N -1) r 2 ± .\n\nFinally ǫ ± are signs, defined as ǫ ± (R) = sign( n, e r± )| r±=R and are crucial quantities to obtain the Penrose diagrams associated with the considered brane configuration.\n\nC. The effective action/the momentum Before proceeding with the analysis of the physical system that we introduced in the previous subsection, we believe it is useful to recall some important points about the structure of the junction conditions in spherical symmetry. For a generic junction between spherically symmetric spacetimes across a (spherical) brane carrying matter described by a given, but otherwise arbitrary, equation of state, it would prove very useful to extract an effective action for the dynamics of the brane starting directly from the Einstein-Hilbert action and from a suitable action for the matter fields and the brane [67] . This is the content of a consistent literature on the subject [68, 69, 70, 71, 72, 73, 74, 75, 76] among which we would like to single out the more general, and recent, [77] : the problem is not trivial at all, because of the same subtleties that appear, for instance, in the Hamiltonian formulation of General Relativity. Among the various possible approaches we will closely follow the quantization procedure originally proposed by Farhi, Guth and Guven [16] , which has, lately, been followed also in [67] and [78] . For the Lagrangian approach to the classical dynamics we will instead follow the clear and more general exposition of [77] . We summarize here the relevant elements of these approaches referring the reader to the literature for additional details. In particular, the effective action for the shell can be obtained starting from the Einstein-Hilbert action with the Gibbons-Hawking 4 boundary terms [16, 77, 79] . Using the Gauss-Codazzi formalism [80] , it is readily seen that the Einstein-Hilbert action for a spherical co-dimension one brane (which in our case will separate two domains of (anti-)de Sitter spacetime) can be decomposed in non-dynamical bulk contributions (indeed, we consider the geometry of the bulk spacetimes M ± fixed) and a dynamical boundary contribution described in terms of the extrinsic curvature of the brane Σ. It is enlightening (and, perhaps, necessary) not to use all the freedom in fixing the coordinate systems. As an exemplification we do not use the freedom given by the reparametrization invariance with respect to the proper time of the brane and we, thus, introduce a lapse function N (s), so that the brane induced metric can be written as h (N ) µν = diag( -N (s) 2 , R 2 (s), . . .) in the coordinates (s, R, . . .) (the \". . . \" stand for the trivial spherically symmetric part). Then, following [67, 77] , the effective action for the degrees of freedom associated with the radial and time coordinates takes the form\n\nS eff ∝ R N -2 ǫ Ṙ2 + N 2 f (R) -Ṙ arctanh Ṙ ǫ Ṙ2 + N 2 f (R) sign(f ) -κN R N -1 dτ. ( 3\n\n)\n\nThus, the effective Lagrangian of the system is given by\n\nL = P (N ) Ṙ -H,\n\nwhere, following [67], we have defined the effective Hamiltonian as\n\nH = -R N -2 ǫ Ṙ2 + N 2 f (R) + κ N R N -1\n\nand the effective momentum as\n\nP (N ) (R, Ṙ) = -R N -2 arctanh Ṙ ǫ Ṙ2 + N 2 f (R) sign(f ) . ( 4\n\n)\n\nThen, the equations of motion for the R and N degrees of freedom are dH dτ = Ṅ N H and H(R, Ṙ, N ) = 0; the second equation is a first integral of the former one, and is the Hamiltonian constraint. We see that it encodes all the information about the dynamics of the system, being identical to the only remaining junction condition. The first equation is the second order equation of motion of the system, given by the total derivative with respect to the proper time of the Hamiltonian constraint.\n\nIn what follows, we will be mostly interested in the expression of the momentum evaluated on a solution of the equation of motion. In order to build a canonical structure (as it is done, for example, in [46] ) we first have to prove the existence of a well defined symplectic structure on the phase space of the Lagrangian system; in particular the Legendre transform has to be invertible, which in our case means nothing but the invertibility of the conjugate momentum as a function of the velocity [81] .\n\nAs appreciated already in [16] , for generic co-dimension one branes this cannot be proved to be always satisfied 5 . However, for the system under consideration the canonical structure always exists and P (N ) can rigorously be considered the canonical momentum conjugate to the coordinate R; thus, in the specific case of interest here, the canonical construction of the corresponding quantum theory is a well posed problem. To prove this statement, let us consider\n\n∂P (N ) ∂ Ṙ = -R N -2 1 ǫ Ṙ2 + N 2 f (R) .\n\nThis quantity can be zero if and only if two conditions are simultaneously satisfied:\n\n1. ǫ + = ǫ -;\n\n5 In these cases the quantum theory cannot be built with the canonical formalism. Alternative formulations using the path integral approach have been considered in the seminal work of Farhi, Guth and Guven [16].\n\n5 2. there is a point R in the configuration space such that f + (R) = f -(R).\n\nFor our particular choice of branes in (anti-)de Sitter spacetime, we see that in order for the second condition to hold we must have Λ -= Λ + ; but this implies 6 ǫ + = -1, ǫ -= +1 so that the first condition then fails. Thus, for the class of junctions that we are considering one can always build a canonical structure.\n\nFollowing, for instance, [82] , it is possible to determine the classical dynamics of the system by studying an equivalent one dimensional problem for the radial degree of freedom R, taking also into account the values of the signs ǫ ± (R).\n\nBefore proceeding we will set up a different system of dimensionless quantities, in order to remove the arbitrariness in the definition of the normalization of the cosmological constants and of the bubble's tension. If we define the quantities x = κR, τ = κτ,\n\nλ ± = 2Λ ± κ2 N (N -1) , α = λ -+ λ + , β = λ --λ + , equation (2) then becomes 7 -ǫ ẋ2 + 1 -λx 2 x = x 2 . ( 5\n\n)\n\nFor completeness we also report the dimensionless form of the effective momentum (4) when we impose the gauge choice N ≡ 1:\n\nP (x, ẋ) = -x N -2 ǫ arctanh ẋ ẋ2 + f (x) f |f | . ( 6\n\n)\n\nThis quantity will be central in what follows, but for the moment we keep our attention on equation (5). Despite its unusual look, it is easily proved that it is equivalent to a system of equations, which, in the notation that we are using, takes the form\n\nẋ2 + V (x) = 0 ǫ ± (x) = -sgn(β ± 1) , ( 7\n\n)\n\n6 This can be easily seen from the junction condition (2) remembering that κ > 0. 7 An overdot will denote, from now on, a derivative with respect to τ .\n\nwhere the potential V (x) has a simple parabolic form V (x) = 1 -x 2 x 2 0 (8) and x 0 , the turning point, is given by\n\nx 2 0 = 4 1 + 2α + β 2 , ( 9\n\n)\n\nwhen the right-hand side is positive; otherwise there are no nontrivial classical solutions. All the solutions of the problem can be easily classified in according to the values of α and β: we refer the reader to appendix A for details. The expressions for the signs ǫ ± relate, instead, the global geometrical structure of spacetime for this brane configuration only to the difference between the inner and outer cosmological constants, not to the details of the trajectory itself. This is a portion of the content of the junction condition, that is necessary for the description of the global spacetime structure: a careful discussion of this point can be found in appendix B. The form of equation (5) for a brane separating two (anti-)de Sitter spacetimes, which, in turn, is responsible for the simple quadratic form of the potential (8), allows exact solution of this particular case. It is, in fact, not difficult to see that (5) has: 1. the trivial solution x(τ ) ≡ 0, which always exists; 2. the solution\n\nx(τ ) = x 0 cosh τ -τ (0) x 0 , ( 10\n\n)\n\nwhich satisfies the initial condition x(τ (0) ) = x 0 ; this solution is known as the bounce solution and exists only if the condition 1 + 2α + β 2 > 0 is satisfied.\n\nWe also, incidentally, note that\n\nβ = ±1 ⇐⇒ 1 x 2 0 = λ ∓\n\ni.e. when |β| = 1 the turning point radius coincides with one of the two cosmological horizons, if they exist. We also anticipate that a more careful analysis of the trivial solution x(τ ) ≡ 0 will be required, since, although it is given to us by the mathematics of the problem, we are mostly interested in its physical role, especially when we will turn on (semiclassical) quantum effects (see section IV).\n\nUsing the notation introduced in the last section we can now describe how the semiclassical regime of the 6 brane dynamics looks like. In particular we can consider the tunnelling between the zero-radius solution towards the bouncing solution (10), and vice versa.\n\nFor example in one direction the semiclassical picture is as follows: we have a brane, of very small radius; of course, when it is very small, the quantum properties of its matter content will be non-negligible (we will further discuss this issue later on in section IV) and their interplay with gravity will be non-trivial; although we do not know the full quantum gravity description of the system, we will consider that, thanks to quantum effects, the brane will have a certain probability to tunnel under the potential barrier given by the effective potential (8) into the bounce solution. When emerging after the tunnelling quantum effects will likely become less and less important as compared with gravitational ones (and as far as the interaction with the bulk spacetime is concerned), so that the evolution of the brane will closely resemble that of the classical junction. This is not the case in the first stage of the evolution, involving the tunnelling process, and we will try to understand, at least at an effective level, the \"responsibilities\" of both the quantum and gravitational realms in this process. In particular, quantum effects will be considered in a modification at small scales of the stress-energy tensor (see section IV). Gravitational effects, at small scales, are also present, and described by the geometric character of Israel junction conditions. We think that our description might be able to take into account both these effect, at least within the limits represented by the semiclassical approximation. In this sense, although our treatment will be effective, it will give us the possibility to consistently solve the ambiguity arising at the mathematical level and represented by the x(τ ) ≡ 0 solution. We will suggest that, at the semiclassical level, a more detailed treatment of both, the quantum and gravitational aspects, is unlikely to be necessary.\n\nFor the system under consideration, tunnelling trajectory can be described by constructing the instanton in the Euclidean sector. The construction of the instanton in the general case is still an unsolved problem: we will follow the approach of [16] ; moreover, the problems left open in [16] do not affect the present case. In fact it is easy to see that in our case the instanton describing the tunnelling can always be constructed without the necessity to introduce the pseudo-manifold that in [16] was necessary to deal with multiple covering of points in the Euclidean sector. Moreover the different descriptions of the tunnelling process that were obtained in [16] using the canonical or the path-integral approaches in our case also coincide, because they are consequences of properties of the dynamical variables during the tunnelling trajectory which are not present in the system that we are considering (please, see [83, 84, 85] for a more detailed description of these issues).\n\nIn this way we know that the tunnelling can be described directly at the effective level, where, the relevant aspects of the Euclidean junction can be determined by the Wick rotated classical effective system. We thus define τe = -iτ; then, denoting with a prime the derivative with respect to τe , we have that the Euclidean system is obtained with the formal substitutions ẋ → ix ′ for the \"velocity\" and P → i Pe for the momentum (it is not difficult to prove that the above substitution rules are rigorous results and that they can be obtained performing an Euclidean junction, as done, for instance, in [16] ). In particular\n\nPe (x, x ′ ) = -x N -2 arctan x ′ ǫ f (x) -(x ′ ) 2 . ( 11\n\n)\n\nIn this way, the tunnelling process can be modelled using the effective tunnelling trajectory, which solves the Euclidean equation:\n\n-(x ′ ) 2 + 1 -(x/x 0 ) 2 = 0. ( 12\n\n)\n\nThe analytic expression of the tunnelling trajectory of a brane expanding from zero radius at Euclidean time τe = 0 to the bouncing solution is:\n\nx(τ e ) = x 0 sin(τ e /x 0 ).\n\nThe opposite process is then described by\n\nx(τ e ) = x 0 cos(τ e /x 0 ),\n\nif we assume that at Euclidean time τe = 0 the brane starts contracting from x 0 . These two processes occur with certain probabilities, whose amplitudes A can be expressed in the usual semiclassical approximation as\n\nA(0 → x 0 ) ∝ exp(-I e [x]),\n\nwhere I e [x] is the Euclidean and can be obtained as the integral of the Euclidean momentum evaluated on a solution of the Euclidean equation of motion (12). If, for simplicity, we define\n\nI e [x] = Ω N -1 16πG N +1 1 κ N -1 Īe [x],\n\nwhere Ω N -1 is the volume of S N -1 , then Īe [x] can be obtained as\n\nĪe [x] = x0 0 Pe (x)dx. ( 13\n\n)\n\nWe stress again that Pe (x) is the Euclidean momentum evaluated on a solution of the Euclidean equation of motion and can be obtained by substituting (11) . We thus have\n\nx ′ = V (x) in\n\nPe (x) = -x N -2 arctan 2 V (x) (β + ω)x , 7\n\nwhere, to make writing more compact 8 , we have introduced the quantities ω ± defined as ω ± = ±1. Note that in the Euclidean case we have some freedom in appropriately choosing one of the possible branches of the inverse tangent function. Different choices will affect, in general, the result that we obtain for the action. As in [16] we observe that non-careful choices will make the action a discontinuous function of the parameters; this can be seen without difficulties. Preliminarily, let us anticipate that with the symbol \"arctan\", we will indicate the branch of the inverse tangent function with range in [-π/2, π/2]. Let us then consider β = ω and let us integrate by parts the integral (13). We obtain\n\nĪe [x] = - x N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 + + x0 0 x N -1 N -1 d arctan 2 V (x) (β + ω)x .\n\nLet us now consider the first term above. Clearly, if we chose the branch of the arctan function to be the one with range in [-π/2, π/2], then we have lim β→-ω -\n\nx N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 = - ωπx N -1 0 2(N -1) but lim β→-ω + x N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 = + ωπx N -1 0 2(N -1) ,\n\nso that the action develops a discontinuity at β = -ω.\n\nThis discontinuity can be eliminated if we choose different branches of the inverse tangent functions (please remember that in the expression that we are considering we are using a compact notation to indicate the difference of two inverse tangent functions); in particular the two discontinuities at β = ±1 can be eliminated by choosing: 1. the branch with range [-π, 0] for the arctan function containing quantities of the \"+\" spacetime; 2. the branch with range [-π, 0] for the arctan function containing quantities of the \"-\" spacetime.\n\nSince in our notation \"arctan\" has range [-π/2, π/2], this means that in the equations above must be rewritten with the substitutions (Θ is the step function) arctan(\" + \") -→ arctan(\" + \") -πΘ(β + 1) 8 In view of the definition of the ω's, and of the meaning of the square brackets, the equation above is a shorthand for Pe(x) = -x N-2\n\n\" ǫ + arctan \" 2 √ V (x) (β+1)x « -ǫ -arctan \" 2 √ V (x) (β-1)x «« .\n\nand arctan(\" -\") -→ arctan(\" -\") -πΘ(β -1);\n\nfor the jump of the quantity which appears inside the expression of the Euclidean momentum, this implies the following substitution:\n\n[arctan] -→ [arctan] -πΘ(1 -β 2 ).\n\nIn this way the action integral is continuous also at the points β = -ω and is given by the integral Īe [x] = -\n\nx0 0 dxx N -2 × × arctan 2 V (x) (β + ω)x -πΘ(1 -β 2 ) .( 14\n\n)\n\nAfter these preliminary considerations, we can proceed to evaluate the tunnelling amplitude in the WKB approximation; as we anticipated and as we will see, in arbitrary spacetime dimensions this result can be expressed analytically in terms of known functions.\n\nTo calculate the first contribution to the integral (14) we proceed as follows. As a preliminary step, we observe that it can be written as a difference of two integrals with the same general structure. Let us then consider the two integrals containing the inverse tangent functions: small differences, which do not substantially affect the calculation, can be taken into account by properly using the ω's introduced above, as we already did in the expressions for the momentum. This said, we can perform 9 an integration by parts in (14) . The terms evaluated at the limits of integration, which appear in this process, do vanish and by changing the integration variable from x to ζ = (x/x 0 ) 2 (which also transforms the integration domain into the unit interval (0, 1)) we obtain the following expression,\n\nĪe [x] = - x N 0 4(N -1) × × (β + ω) 1 0 ζ (N -2)/2 √ 1 -ζ 1 1 -z ω ζ dζ + ( 15\n\n) + πx N -1 0 (N -1) Θ(1 -β 2 ), 9\n\nStrictly speaking this can be done only when β = -ω. The cases in which β = -ω are trivial and can be dealt separately; or, more simply, since we already know from the discussion in the previous subsection that the action is continuous, we can just extend it to β = -ω by continuity.\n\nwhere\n\nz ω = 1 -x 0 β + ω 2 1 + x 0 β + ω 2 . ( 16\n\n)\n\nWhen z ω < 1 one of the above integrals diverges since there is a pole of the integrand on the domain of integration. On the other hand, it is easy to see that the condition z ω < 1 implies\n\nx 2 0 4 (β + ω) 2 < 0\n\nand cannot be realized for any choice of the parameters if a tunnelling trajectory has to exist. The value z ω = 1 can instead be obtained if β = -ω and we know that the action can be extended by continuity to these values, although the above procedure to calculate the integral is not valid. Thus, under the conditions β = -ω, we have that z ω > 1 is satisfied and equation (15) gives\n\nĪe [x] = - x N 0 4(N -1) Γ(N/2)Γ(1/2) Γ((N + 1)/2) × × (β + ω) 2 F 1 1, N 2 , N + 1 2 , z ω + (17) + πx N -1 0 (N -1) Θ(1 -β 2 ),\n\nwhere Γ is the Euler's gamma function, 2 F 1 the hypergeometric function. Note that x 0 depends on α, β and so the first factor of the formula, depending on the number of spacetime dimensions, cannot be ignored even for a qualitative description of the tunnelling amplitude.\n\nIt is useful to specialize the result (17) to particular situations. We are going to do this by considering the cases in which spacetime is three, four and five dimensional. Below we are going to explicitly discuss these three cases. A comparative presentation of the results can be found in the contour plots of figure 1 .\n\nWe start then with the three dimensional case, which is lower-dimensional gravity; in three spacetime dimensions it seems more clear how to build a quantum theory out of the classical junction conditions [46] . Moreover three dimensional gravity is an interesting system by itself, it allows an easier visualization of some results and can be used for interesting specific toy models. Anyway, in this case we have N = 2 and we can express 2 F 1 (1, 1; 3/2; x) in terms of elementary functions as\n\n2 F 1 1, 1; 3 2 ; y = arcsin( √ y) √ 1 -y √ y . ( 18\n\n)\n\nCorrespondingly the action becomes\n\nĪe = - x 2 0 4 Γ(1)Γ(1/2) Γ(3/2) × × (β + ω) arcsin( √ z ω ) √ 1 -z ω √ z ω + ( 19\n\n) + πx 0 Θ(1 -β 2 ).\n\nThe corresponding probability is plotted as a function of β in figure 2 for some non-negative values of α and in figure 3 for some negative values of α. Not many comments are necessary, of course, for the four dimensional case, the original arena on which this calculation was performed. Again we can take advantage of a simple expression for the corresponding hypergeometric function\n\n2 F 1 (1, 3/2; 2; y) = 2 y √ 1 -y - 2 y , ( 20\n\n)\n\nwhich brings the final result in the form\n\nĪe = - πx 2 0 4 × ( 21\n\n) × sgn(β + ω) 1 -(1 -z ω ) 1/2 z ω -Θ(1 -β 2 ) .\n\nThe plots of the probability as a function of β can be found in figure 4 for some non-negative values of α and in figure 5 for some negative values of α. This result is the same as the one obtained in [86] and it also reduces to the one calculated in [3]: it, thus, represents a useful consistency check. Finally, the five dimensional case is interesting in the context of the Randall-Sundrum scenario. In this case, using\n\n2 F 1 1, 2; 5 2 ; y = 3 2y arcsin( √ y) √ 1 -yy 1/2 -1 , ( 22\n\n)\n\nthe result for the action integral can be put in the following form:\n\nĪe = - x 4 0 12 Γ(2)Γ(1/2) Γ(5/2) × × (β + ω) arcsin( √ z ω ) √ 1 -z ω (z ω ) 3/2 - 1 z ω + ( 23\n\n) + πx 3 0 3 Θ(1 -β 2 ).\n\nAgain we present two plots of the corresponding probability as a function of β; figure 6 shows the behavior for some non-negative values of α, whereas plots for some negative values of α can be found in figure 7 .\n\nUp to this point we have discussed the physics of the system rather quickly, focusing mainly on the mathematics necessary to describe the classical and the semiclassical phases. There is still a point which deserves a 9 a b c +10 +5 0 -5 -10 +10 +5 0 -5 -10 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 α α β β β β β β no tunnel zone no tunnel zone no tunnel zone\n\nFIG. 1: Contour plots of the values of the tunnelling probability in 3 (case a), 4 (case b) and 5 (case c) spacetime dimensions in the β-α plane. The area inside the crosshatched parabola in the bottom of each picture is the region in parameter space where tunnelling cannot occur since no bounce solutions exists (no tunnel zone). A better understanding of what happens close to the limiting parabola can be understood from the following figures 3, 5 and 7. Next to next contours indicate a jump of the probability of 0.1, with the lightest gray tone indicating the range (0.9, 1.0) and the darkest the range (0.0, 0.1). The same color indicates the same range of values of the probability in all the three sub-diagrams.\n\ndetailed discussion, since it involves non-trivial aspects of the dynamics of the brane. Indeed, the physical process we have considered is the tunnelling of spacetime, from a classical situation representing a spacetime containing a small brane to another classical configuration describing a bouncing brane. In our picture, the pre-tunnelling state is represented by the x ≡ 0 solution of the junction condition. Although this case might appear rather simple at first sight and one could be tempted to just state that the initial state is, for example, the full M + spacetime (a point of view which has been taken for example in [67] ), much more care has to be taken. The main reason is that when x = 0, the brane's world-volume degenerates to a curve. Thus, the formalism that we used to describe the brane breaks down and the analysis that we have made cannot be considered as rigorous as it is in the case of the bouncing solutions. As a manifestation of this fundamental problem, we observe that the equations determining the ǫ ± signs do not hold for the particular solution x ≡ 0. In particular, for this solution the junction condition as written in (5) does not provide any mean to solve this problem.\n\nA way out of this situation appears when we reflect on the fact that this process, although described semiclassically, is quantum in its true nature. Thus an approximation of the system which considers it completely classical before the tunnelling, i.e. in the degenerate configuration, might be too rough and might require a more careful consideration. During the tunnelling trajectory, and even more for the x ≡ 0 classical solution, quantum effects are supposed to be, if not dominant, at least relevant enough to modify the classical picture of the junction.\n\nWe will propose here an attempt to address the problem. Our proposal should be considered a first step further, but far from a first principle solution of the complex quantum problem; it just aims to show that the mathematical and physical aspects of the problem stated above can be dealt with by giving an effective formulation for the phenomena that might arise at small scales. At the same time, although we will just build an effective model, we will not be very demanding about its main properties: in this way, hopefully, the model, although effective, could mimic well enough effects produced by quantum gravity whatever will be their, still undiscovered, true nature.\n\nIn this respect, we would also like to point out the following: i) the main effect of our proposal is to slightly perturb the effective potential (8) in the tunnelling region; then, we will ii) show that the perturbed problem is free from ambiguities and iii) use for the unperturbed case the results obtained in the perturbed one, when the perturbation becomes smaller and smaller. To have a definite model, we will regard the small scale behavior of the matter composing the brane/shell as the physical origin for the perturbations (see below). On the other hand, the consequence of these perturbations in the effective formulation is completely generic: it is, in fact, possible to show that other physical motivations, as for instance the quantum properties of spacetime at small scales, could be reflected in a similar way in the effective 10 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.005 0.01 0.015 0.02 0.025 0.03 0.03 0.02 0.01 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 2: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of β for fixed non-negative values of α as listed above. The detailed behavior for α = 0 around β = 0, is also shown (to better see that the probability is small but non-zero).\n\nformulation. For these reasons, we can regard our conclusions model independent to a high degree, at least within the context defined by the semiclassical approximation.\n\nTo introduce our model we can naturally think the brane as sourced by some matter fields whose nature is, ultimately, quantum. We thus argue that, although at large scales the approximation for the brane stress energy tensor that we made in subsection II B might be rough but still appropriate, at small scales it will instead break down due to quantum effects. We will model these additional quantum effects by adding a term to the brane stress-energy tensor as follows:\n\nS -→ S + S q , ( 24\n\n)\n\nwhere the \"quantum\" contribution S q will be parametrized as\n\nS q = ρ q u ⊗ u + σ q h. ( 25\n\n)\n\nThe conservation equation for S q under the assumption of spherical symmetry gives (we are going to use, from now on, the dimensionless versions of the parameters ρ q and σ q , which following the notation above, are called ρq and σq )\n\ndρ q dx + N ρq x = dσ q dx . ( 26\n\n)\n\nWe will choose preliminarily\n\nρq (x) = ax q ( 27\n\n) so that σq (x) = a q + N q x q . ( 28\n\n)\n\nIn this case the junction condition becomes 10\n\n-ǫ (q) ẋ2 + 1 -λx 2 x = x 2 µ(x), ( 29\n\n)\n\n10 We remember that we are following the convention in which square brackets indicate the jump of the enclosed quantity, i.e. the difference of it when evaluated in the '+' and '-' domains. We are now using the suffix \". . . q\" or \". . . (q) \" to indicate quantities when the energy momentum tensor is modified, as described in the text, to take into account quantum effects; thus the two signs will now be ǫ (q)± .\n\n11 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00\n\nFIG. 3: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of β for fixed negative values of α as listed above. For α < -1/2, not all values of β are allowed. We have put the mark on each curve in correspondence of the minimum value of β for which (at the given value of α) the tunnelling process can exist. For smaller values, the infinitely expanding solution, in fact, is not present.\n\nwith µ(x) = 1 + āx q (30) and, for short, ā = a(q + N )/q. Now we are going to slightly restrict the parameters ā and q to make these general settings appropriate for our model. In particular, the modification to the stress energy tensor, described by ā and q was introduced to model the quantum effects at small scales; thus it should be negligible at large scales and this can be achieved if q + 1 < 0, i.e. q < -1.\n\nWe can now compute the modified effective potential Vq . The potential turns out to be\n\nVq (x) = - x 2 β 2 + 2α(1 + āx q ) 2 + (1 + āx q ) 4 4(1 + āx q ) 2 ( 31\n\n) x 0 - ā2 4x -2q-2 ( 32\n\n) ā 0 V (x) + (1 -β 2 )ā 2 2x -q-2 . ( 33\n\n)\n\nExtracting the behavior of (31) for small x we get (32), whereas for small ā the leading contribution is given by (33). These are useful results. Indeed we see from (32) that, quite generally, and certainly under the above q < -1 condition, the potential satisfies\n\nlim x→0 + Vq (x) = -∞.\n\nThis implies that the Vq (x) allows not only the bounce brane junction, as V (x) does, but also bounded solutions. Thus the addition of the S q term to the stress energy tensor, which in our picture is supposed to take into account quantum effects at small scales, in fact does his job by trading the x ≡ 0 solution for a bounded one of finite (i.e. non-vanishing) size. Moreover this solution exists under very general assumptions about the form of S q , so that we do not have to commit ourselves too much about the underlying quantum gravity physics of which S q is roughly supposed to take into account some effective semiclassical description. Looking, now at equation (33) we also see that for small ā, apart from the evident qualitative difference at small scales, the potential resembles closely the non-perturbed one (and this happens, in particular, along the tunnelling trajectory). Thus it will be a sufficiently good approximation to evaluate the tunnelling probability in the, analytically much simpler, unperturbed case. At this point, our picture of the spacetime transition will be as follows. We will take as final configurations of the tunnelling process the junctions obtained without considering the correction, which in our setup is strongly suppressed at large distances; then, the initial state, which is the x ≡ 0 solutions, will be \"regularized\" by considering the junction as the limiting case of the bounded perturbed junction when ā → 0. In particular, the sign ambiguity, which affects the x ≡ 0 solution, will be solved by choosing the signs of the bounded trajectory 12 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.002 0.004 0.006 0.008 0.01 0.01 2 3 × 0.01 1 3 × 0.01 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 4: Plots of the values of the tunnelling probability in a spacetime of dimension 4, as a function of β for fixed non-negative values of α as listed above. Again the detailed behavior for α = 0 around β = 0 is shown (to better see that the probability, although very small, is non-vanishing).\n\nof the perturbed model. These signs can be obtained in closed form considering the x → 0 limit of the junction condition, which is dominated by the correction. Therefore\n\nǫ ± = ∓1 ( 34\n\n)\n\nUsing this prescription, then, one can build the spacetime diagrams representing the tunnelling process. We refer the reader to appendices for a complete discussion of the parameter space (appendix A) and of the global spacetime diagrams representing the various physical situations (appendix B).\n\nTo conclude the discussion, we would like to remark that, despite the fact that the regularization procedure we have described changes dramatically the situation in the region near x = 0 of the configuration space of the brane, the effect on the calculation we have performed is not significant. This is due to the fact that in the tunnelling region, the potential is substantially modified only in a narrow region close to the turning point corresponding to the maximum radius of the bounded solution of the modified classical junction condition. In the tunnelling region, the perturbation to the potential is finite, and by sending ā → 0 the tunnelling amplitude of the modified problem is approximated arbitrarily well by our calculation in section III. Of course, in the real physical problem we expect that quantum effects would prevent the brane from shrinking to zero radius: thus the mathematical limit ā → 0 should be read as a more physically sound limit in which ā tends to a small but finite value related to the ultimate physical nature of the brane itself (which is presently only partially understood). As a consequence, the analytical result that we have obtained (which relies on a WKB approximated description of the quantum process) can also be consider an approximation of the full quantum result at the lowest power in the ratio between the maximum radius of the bounded solution of the modified classical junction equations and x 0 .\n\nIn this paper we have calculated analytically the tunnelling amplitude for a domain of spacetime of de Sitter/anti-de Sitter type in a background which, again, is de Sitter/anti-de Sitter. The analytical result holds in arbitrary spacetime dimensions greater than three, and generalizes already existing four dimensional calcu-13 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00\n\nFIG. 5: Plots of the values of the tunnelling probability in a spacetime of dimension 4, as a function of β for fixed negative values of α as listed above. The same observations as in the case of 3 spacetime dimensions apply.\n\nlations. It is not difficult to see that this result reduces, for appropriate values of the parameters, to the result found by Coleman and de Luccia [3] for the false vacuum to true vacuum transition in four spacetime dimensions. Also the results by Parke [86] , again in the four dimensional case, are correctly reproduced.\n\nWe have, also, discussed extensively, in the text and in appendix B, the spacetime structures that can arise for all possible values of the parameters characterizing the model. Some of these spacetime structures, already discussed in the literature, are known as tunnelling from nothing configurations. We have also exposed a possible issue of the shell formalism in the description of the pre-tunnelling state and proposed a solution which relies on the above mentioned tunnelling from nothing configurations; in this way we have been able to make what we consider to be a consistent choice for the before tunnelling configurations. This proposal, which is implemented by considering a modification to the stress-energy tensor for the matter on the shell at small states, is motivated by the observation that if quantum effects are non-negligible on scales at which the tunnelling process occurs, they should also be non-negligible at smaller scales, where the before tunnelling configurations live. By modelling the influence of these quantum effects with a quite generic modification to the form of the stress-energy tensor at small scales, we have proposed an unambiguous rule to fix the initial configuration. This approach to the problem, seems to us consistent with the level at which we are modelling quantum effects for this gravitational system, which is the semiclassical approximation for an infinitesimally thin distribution of matter and energy. At the same time, it has already been shown that, if we add a more refined matter content on the shell (as for instance a collection of gauge fields), modifications similar to the one that we have considered in this paper naturally appear [87] . Moreover, we note that it gives a very consistent picture of the tunnelling process, since all the possible types of tunnelling result in a \"sudden expansion\" of a very small region of spacetime from a small size (where quantum effects are certainly non negligible) to a much bigger size, with radius of the order of x 0 . This shows that the regularized tunnelling always models what in the literature has been called tunnelling \"from nothing\". In our case the \"nothing\" is exactly the quantum state of the spacetime junction that we model, in an effective way, using (29). It thus seems that some tunnelling configurations present in the literature and of more difficult interpretation [67] might be ruled out by the quantum properties of matter and/or of spacetime at small scales. In fact, it is suggestive to reflect about the fact that, even in our very simplified and purely effective treatment of quantum effects before and during the tunnelling, all tunnelling processes start exactly \"from nothing\" (as interpreted above), i.e. from a configuration which is consistent with the quantum properties of the following tunnelling process.\n\nTo conclude, we would like to explicitly distinguish between the analytical result for the tunnelling probability and the proposal to interpret the pre-tunnelling 14 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.0002 0.0004 0.0006 0.0008 0.001 0.001 2 3 × 0.001 1 3 × 0.001 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 6: Plots of the values of the tunnelling probability in a spacetime of dimension 5, as a function of β for fixed non-negative values of α as listed above. Again the detailed behavior for α = 0 around β = 0 is shown (to better see that the probability, although much smaller than in the previous cases, is still non-vanishing).\n\nconfigurations. Indeed the analytical result is a natural generalization of already existing calculations and it incorporates them as special cases. Its validity is completely independent from our interpretation of the before tunnelling configurations and it can represent a useful limit case to check the results of more elaborated models: for instance, a de Sitter-Schwarzschild shell configuration should reproduce, in the limit of vanishing Schwarzschild mass, our result for a tunnelling between a de Sitter space of assigned cosmological constant and a de Sitter space with vanishing cosmological constant, i.e. Minkowski space.\n\nWe would like to thank M. C. Johnson and S. Sonego for useful discussions and constructive criticism. This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818. The work of S. Ansoldi is supported in part by a grant from the Fulbright Commission.\n\nAPPENDIX A: PARAMETER SPACE\n\nIn this appendix we will elaborate about the classification of the possible junctions between two (anti-)de Sitter spacetimes. After switching to the dimensionless formulation (see subsection II D), we remain with two parameters, i.e. λ ± , or, which is the same, α and β. We will mostly use the latter quantities in the following analysis, since many relevant features of the solutions to Israel's junction condition related to the causal structure of the full spacetime manifold M are easily deducible from the comparison between the two cosmological constants. In particular, in figure 8 we give a classification of the possible solutions. The diagram shows the parameter space of the variables (α, β) and (λ + , λ -). The classification of the solutions using α and β, looks nicely symmetric with respect to the axis β = 0, for which λ + = λ -; this is the primary reason why we will mostly use the (α, β) parametrization in our discussion. The (λ + , λ -) axes can, anyway, be conveniently used to single out the de Sitter from the anti-de Sitter spacetime. In particular, we have defined four main groups of solutions:\n\ntype A) these are solutions in which both spacetimes joined across the brane are anti-de Sitter space-15 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00 0.25 0.5 0.75 1 1.25 1.5 0.002 0.004 0.006 0.008 0.01 0.0 0.5 1.0 1.5 0.01 0.006 0.002 × FIG. 7: Plots of the values of the tunnelling probability in a spacetime of dimension 5, as a function of β for fixed negative values of α as listed above. For α < -1/2, not all values of β are allowed. We have put the mark on each curve in correspondence of the minimum value of β for which (at the given value of α) the tunnelling process can exist and, the zoomed diagram, helps resolving the superposition that arises between the curves obtained for α = -0.50, -1.00, -1.50.\n\ntimes, i.e we have an AdS (-) -AdS (+) junction; this part of the parameter space is bounded by the parabola α = -(β 2 + 1)/2 (equivalently x -1 0 = 0), whose inside is the black region where no solution exists; type B) these solutions describe a junction of one part of de Sitter spacetime in the M -manifold with a part of anti-de Sitter spacetime in the M + manifold, i.e. they are dS (-) -AdS (+) junctions, and play the role of counterparts of the below discussed type D solutions; type C) these are junctions in which both spacetimes have the de Sitter geometry, i.e. we have dS (-) -dS (+) junctions; in some cases, qualitatively different diagrams may arise depending on which cosmological constant is the bigger, i.e. depending on the sign of β; type D) as anticipated above, this last type of solutions is similar to the type B, with the role of \"-\" and \"+\" interchanged; we thus have AdS (-) -dS (+) junctions.\n\nThe above information is not enough to completely characterize the classical spacetime obtained from the junction. In addition we need to know the behavior of the normal to the brane travelling in the spacetimes that we have determined from the diagram in figure (8) . According to our convention the normal to the brane has its tail-tip direction going from the \"-\" to the \"+\" parts of the full spacetime M; on the other hand in each of the two spacetimes M ± the corresponding signs ǫ ± determine if the normal to the brane points in the direction of increasing (ǫ = +1) or decreasing (ǫ = -1) radius. Using the results in (7) for ǫ ± we can subdivide the parameter\n\n16 λ + λ - α β A B C A A D C A FIG. 8:\n\nThe space of parameters is subdivided according to the different kinds of junctions, which can be between anti-de Sitter and anti-de Sitter spacetimes (type A), anti-de Sitter and de Sitter spacetimes (type B), de Sitter and anti-de Sitter spacetimes (type D) or de Sitter and de Sitter spacetimes (type C). The black part of the parameter space, identified by the condition 1 + 2α + β 2 < 0, singles out the values of the parameters for which tunnelling is not possible since the infinitely expanding solution does not exist.\n\nλ + λ - α β 1 1 1 2 2 3 3 3\n\nFIG. 9: The parameter space can be subdivided in three other regions (strips) according to the values of the signs ǫ±. In region 1 we have ǫ± = -1, in region 2 we have ǫ± = ∓1 and, finally, in region 3 we have ǫ± = +1.\n\nλ + λ - α β A1 B1 C1 A2 B2 C2 D2 C2 D3 C3 A3 FIG. 10: In principle we have to study 10 different cases, corresponding to as many different regions in the parameter space. These regions are obtained combining the classifications in figures 8 and 9, so that in each region there is a well defined type of junction with an unique choice for the signs.\n\nAs we will see each of the pairs A1 and A3, B1 and D3, B2 and D2, C1 and C3 will correspond to a distinct processes in spacetime; this matches well with the analytical result, in which the action turns out to be an even function of β, and gives only a total of 6 different possible processes.\n\nspace in three main regions, as in figure 9. These regions correspond to the following situations:\n\nregion 1) ǫ -= ǫ + = -1, so that in both spacetimes M ± the normal n a | ± to\n\nthe brane trajectory points in the direction of decreasing r;\n\nregion 2) ǫ -= +1 but ǫ + = -1, and in M -normal to the brane trajectory n a | -points in the direction of increasing r -but in M + the normal n a | + to the\n\nbrane trajectory points in the direction of decreasing r + ;\n\nregion 3) ǫ -= ǫ + = +1; thus in both spacetimes M ± the normals n a | ± to the brane trajectory point in the direction of increasing r ± .\n\nWe thus have various combinations of the geometries of the two spacetimes M ± according to the classification in figure 8 ; moreover we have to combine them choosing the part of spacetime on the correct side of the brane trajectory following the classification in figure 9 . This gives a total of ten subcases, which are summarized in figure 10 . The naming convention is the most natural, so that, for instance, the junction named B2 is a junction of type B, i.e. a dS (-) -AdS (+) junction, with the signs as in region 2, i.e. ǫ -= +1 and ǫ + = -1. All other names 17 follow the same convention and we thus obtain solutions of the following types A1, A2, A3, B1, B2, C1, C2, C3, D2, D3; their causal structure is detailed in the following appendix.\n\nAPPENDIX B: PENROSE DIAGRAMS & TUNNELLING\n\nHere, following the classification given in the previous appendix, we show the corresponding Penrose diagrams for the classical solutions, and a pictorial representation of the corresponding tunnelling processes. For each case we give four diagrams: 1. the complete spacetime from which we have to \"cut\" the M -part of the bulk (with the trajectory of the bubble and the associated normal) in the top left diagram; 2. the complete spacetime from which we have to \"cut\" the M + part of the bulk (with the trajectory of the bubble and the associated normal) in the top right diagram; 3. the junction, i.e. the full manifold M (again with the shell trajectory and the corresponding normal) in the bottom left diagram; 4. a pictorial representation of the creation of the brane via a tunnelling process, in the bottom right diagram; in this case we have used time translation invariance to set the \"tunnelling time\" at the coordinate time t = 0 so that the top half of the diagram represents the final state (i.e. the top half of the junction in the bottom left diagram) whereas the bottom half of the diagram is a representation (detailed below) of the pre-tunnelling configuration.\n\nWe would like to discuss preliminarily in more detail the way in which the pre-tunnelling configuration is constructed. We remember that we are considering non negligible quantum effects in the regime of the dynamics before the tunnelling. We will thus use for the signs the results coming from the modified junction (29), which are those in (34) . This gives a spacetime structure consisting of two regions (which can be of the de Sitter or of the anti-de Sitter type depending on the values of α and β) both bounded by r = 0 and by the shell radius r = R(τ ): this is the junction corresponding to the bounded solution of the modified potential (31), which is a brane starting from zero radius and expanding to a maximum radius (much smaller than any other length scale present in the problem) before recollapsing to r = 0. In the limit in which ā → 0, as discussed in section IV, this maximum radius tends, in fact, to zero. Nevertheless, in our representation of the spacetime before the tunnelling we have kept an arbitrarily finite size for the maximum radius, to make the diagram more readable. At the same time, we have slightly \"blurred\" it to make pictorially explicit that we are not dealing with a purely classical configuration but with a somehow \"heuristic\" representation of a spacetime where quantum effects are highly non trivial and certainly non negligible. We would like, anyway, to stress again that these quantum configurations will be the initial state of tunnelling processes that correspond to what in the literature has also been called \"tunnelling from nothing\" for instance [67] and references therein).\n\nWe will now present in detail the various kinds of junctions.\n\nAs discussed in main text (and with reference to figure 10 ), this class of junctions consists of the matching of two anti-de Sitter spacetimes with different cosmological constants. There are three possibilities corresponding to type A1, A2 and A3 junctions, which are shown, respectively, in figures 11, 12 and 13. r = 0 r = 0 AdS -, ǫ-= -1 AdS+, ǫ+ = -1 r = 0 r = 0 AdS + /AdS-Tunnelling FIG. 11: Case A1: AdS/AdS, ǫ-= ǫ+ = -1\n\nThe junction of type A1 (figure 11 ) has ǫ ± = -1, so in both anti-de Sitter spacetimes the normal to the brane points in the direction of decreasing radii. For this junction (and for all the ones in the 1 sector), β > 1, so that λ -is always greater than λ + . The classical junction is seen (by an observer travelling toward increasing values of the radius) as a transition from a more negative to a less negative value of the cosmological constant, when he/she crosses the brane. The global picture of the tunnelling process is then obtained according to the 18 prescription discussed in section IV. The pre-tunnelling spacetime has been discussed above and now consists of two \"small\" parts of anti-de Sitter spacetime with different cosmological constant. The tunnelling process shows the transition between a compact spacetime composed by two anti-de Sitter regions to a spacetime similar to anti-de Sitter, except for the fact that at some radius the cosmological constant changes its value. r = 0 r = 0 AdS -, ǫ-= +1 AdS+, ǫ+ = -1 r = 0 r = 0 r = 0 r = 0 AdS + /AdS-Tunnelling FIG. 12: Case A2: AdS/AdS, ǫ-= +1; ǫ+ = -1\n\nThe junction of type A2, is instead a junction where effectively a tiny junction of two anti-de Sitter spacetimes is \"inflated\" (figure 12 ). In this case the signs are given by ǫ ± = ∓1. Thus the normal to the brane points in the direction of increasing radii in the anti-de Sitter spacetime with cosmological constant λ -, but it points in the direction of decreasing radii in the anti-de Sitter spacetime with cosmological constant λ + . The net effect of the tunnelling process, which starts from the already discussed initial state, is thus to inflate the pre-tunnelling compact spacetime into a similar, but much larger, one (note that for graphical reasons the scales in the pretunnelling and post-tunnelling parts of the diagram are different; we remember that the maximum radius of the pre-tunnelling state will be much smaller than any other length scale in the problem). The various diagrams are in figure 12 .\n\nThe final type A diagram is the A3 one. This is very much as the A1 type, only that now ǫ ± = +1; in both spacetime the normal to the shell trajectory points in the direction of increasing radii. Thus (apparently) the role of M ± is interchanged, as shown in figure 13 where the dark and light gray parts looks complementary to r = 0 r = 0 AdS-, ǫ-= +1 AdS+, ǫ+ = +1 r = 0 r = 0 AdS+/AdS-Tunnelling FIG. 13: Case A3: AdS/AdS, ǫ-= ǫ+ = +1\n\nthose in figure 11 ; on the other hand, now we have β < -1 so that λ + > λ -. The global spacetime structure in the bottom left corner of figure 13 implies that an observer crossing the brane in the direction of increasing values of the radius, perceives again a transition from a more negative to a less negative value of the cosmological constant. So this diagram describes exactly the same process described by type A1, as it is possible to see from the diagram in figure 13.\n\n2. Type B\n\nThe type B solutions, B1 and B2, correspond to the junction of a part of a de Sitter spacetime, M -, with a part of anti-de Sitter spacetime, M + . Although the pre-tunnelling picture does not change very much, the configurations before the tunnelling are now de Sitteranti-de Sitter junctions.\n\nLet us first discuss the case B1, shown in figure 14 . In both spacetimes, the normal to the shell points in the direction of decreasing radii, since we have ǫ ± = -1.\n\nThe M -part of spacetime corresponds to the region of de Sitter complementary to the region between r = 0 and the brane's world-volume, whereas the anti-de Sitter part M + is the bounded region in the upper right picture of figure 14 . Thus the spacetime M shows a transition from a negative to a positive value of the cosmological constant for an observer crossing the shell in the direction of increasing radius. The configuration before the tunnelling has been described above; the tunnelling pro-19 r = 0 r = 0\n\nI - I + r = H - r = H - r = 0 dS-, ǫ-= -1 AdS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H - r = H - I + r = H - r = H - r = 0 r = 0\n\nAdS+/dS-Tunnelling FIG. 14: Case B1: dS/AdS, ǫ-= ǫ+ = -1 cess again \"inflates\" an anti-de Sitter-de Sitter junction with small volume to a much larger one, as shown in the bottom right diagram of figure 14 .\n\nr = 0 r = 0 I - I + r = H - r = H - r = 0 dS-, ǫ-= +1 AdS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H - r = H - r = 0 r = 0 I + r = H -\n\nAdS+/dS-Tunnelling FIG. 15: Case B2: dS/AdS, ǫ-= +1; ǫ+ = -1\n\nWe then come to the case B2, shown in figure 15 . Since in this case the signs which fix the orientation of the normals are ǫ ± = ∓1, the situation for M + is as in the previous case, but the M -part of spacetime changes because of the change in the sign of ǫ + . This gives for the junction the diagram in the bottom left part of figure 15 . The configuration before and after the tunnelling is shown in the bottom right corner of figure 15 . The situation is very similar to the one in the previous case.\n\n3. Type C Type C solutions are the \"de Sitter counterpart\" of type A solution. The main difference (not related to the spacetime structure) is that whereas for anti-de Sitter junctions the values of the cosmological constant are restricted (i.e. given two arbitrary values a junction might not exists) de Sitter junctions exists for all values of the cosmological constants λ ± (this is clear from figure 10 ). Please, also remember that now the pre-tunnelling diagram will consist of a junction of two de Sitter spacetime with different cosmological constants and with all the other properties described above. r = 0 r = 0\n\nI - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS-, ǫ-= -1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H + r = H + r = H - r = H - r = 0 r = 0 I + r = H + r = H -\n\ndS+/dS-Tunnelling FIG. 16: Case C1: dS/dS, ǫ-= ǫ+ = -1\n\nLet us start with case C1, shown in figure 16 . For this junction, the signs are ǫ ± = -1, so in both de Sitter spacetimes the normals to the brane point in the direction of decreasing radii. Moreover, since in this case β > 1, the cosmological constant in M -will be bigger than the cosmological constant in M + , so that the cosmological horizon H -will be smaller than H + . The diagram for the junction is shown in the bottom left part of figure 16 . It is 20 interesting to see some effects of the brane on the spacetime structure. There are observers that crossing the brane can move behind the cosmological horizon of M - without going through H -. They can also come out, by crossing the brane in the opposite direction. At the same time observers, by moving accurately and with proper timing, might end up behind the cosmological horizon of M + without crossing any horizon but using the shell as a \"gate\" 11 . As usual no additional considerations are required for the before tunnelling configuration. We note, instead, that the semiclassical transition has again the effect of \"inflating\" a small quantum spacetime region, as clearly shown in figure 16 .\n\nr = 0 r = 0 I - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS -, ǫ-= +1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + I - I + r = H - r = H - r = H + r = H + r = 0 r = 0 I + I + r = H - r = H + r = H + dS + /dS- Tunnelling\n\nFIG. 17: Case C2: dS/dS, ǫ-= +1; ǫ+ = -1\n\nWe now discuss case C2, in which the signs are ǫ ± = ∓1. Then the normal to the brane in the de Sitter space with cosmological constant λ -points in the direction of increasing radii, whereas the normal to the brane in the de Sitter space with cosmological constant λ + points in the direction of decreasing radii. We have again two regions of spacetime of the de Sitter type but with a different cosmological constant joined across the brane (bottom right part of figure 17 ). The full picture of the tunnelling process requires the consideration of the same initial configuration used in case C1 above. We also point 11 These considerations might be modified if we consider the influence of the observer on spacetime. To neglect this influence, as a first approximation, is fairly common in the literature to obtain some hints about the global spacetime structure.\n\nout that in this case, the sign of β is not fixed, so either of the cosmological constants λ ± may be the bigger. In particular, the junction in the bottom left corner of figure 17 shows the situation in which λ + < λ -, so that H + > H -.\n\nIn the bottom right part of figure 17 a representation of the tunnelling process is shown. r = 0 r = 0\n\nI - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS-, ǫ-= +1 dS+, ǫ+ = +1 r = 0 r = 0 I - I + r = H + r = H + r = H - r = H - r = 0 r = 0 I + r = H + r = H -\n\ndS+/dS-Tunnelling FIG. 18: Case C3: dS/dS, ǫ-= ǫ+ = +1\n\nThe last junction of the C type is C3, for which the usual set of diagrams is shown in figure 18 . As all C type ones, this is a junction between two de Sitter spacetimes and since β < -1 the relation between the cosmological constants is λ + > λ -, so that H + < H -. At this stage we might wonder if this junction is the same as C1 (mirroring what happens between the junctions of the A1 and A3 type). We will see that this is indeed the case. The normals now point in both spacetimes toward the direction of increasing radii (since ǫ ± = +1). Thus the junction is as in the bottom left part of figure 18 . When we consider also the configuration before tunnelling, we see that the diagram in the bottom right part of figure 16 is the \"switched color and reflected\" version of the one in the bottom right part of figure 18 : we remember now that the in the C1 case λ -in the light gray part was bigger than λ + in the dark side, but now exactly the opposite happens. Thus, despite the different coloring, case C3 represents the same physical situation of case C1. This observation makes interesting to complete the analysis for the remaining two cases, which is done in the following subsection. 21 4. Type D\n\nTo conclude the analysis of the global spacetime structure we have to analyze what happens for the last kind of junctions, those of type D. These are junctions between de Sitter and anti-de Sitter spacetimes, so that the pretunnelling configuration will also change accordingly, as we have discussed for the previous cases.\n\nr = 0 r = 0 r = 0 I - I + r = H + r = H + AdS-, ǫ-= +1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H + r = H + r = 0 r = 0 I + r = H + r = H +\n\ndS + /AdS-Tunnelling FIG. 19: Case D2: AdS/dS, ǫ-= +1;ǫ+ = -1\n\nThe case of the junction of type D2 is characterized by the following signs: ǫ ± = ∓1. This means that in the anti-de Sitter side the normal points in the direction of increasing radii, whereas in the de Sitter part, it points in the direction of decreasing radii. Thus the junction, shown in the bottom left corner of figure 19 , is the mirror image of the B2 case. Again, the considerations in section IV determine the geometry before the tunnelling. This brings, in complete analogy with the B2 case, to the picture of the tunnelling process given in the bottom right part of figure 19 ; again, apart from the different colors, this case is the same as B2 also in connection with the semiclassical tunnelling process.\n\nWe have thus only one case left, which is quickly dealt with, namely D3. It is now not difficult to anticipate that this process will turn out to be identical to B1, that appears in figure 14 . Indeed the signs are now ǫ ± = +1, so that in both spacetime the normals point in the direction of increasing radii. By the usual procedure we are thus led to the spacetime diagram shown in the bottom left part of figure 20 . Part of this diagram will constitute the final state of the tunnelling process. The initial state is obtained as usual and the picture of the tunnelling r = 0 r = 0 r = 0\n\nI - I + r = H + r = H + AdS-, ǫ-= +1 dS+, ǫ+ = +1 r = 0 r = 0 I - I + r = H + r = H + I + r = H + r = H + r = 0 r = 0\n\ndS+/AdS-Tunnelling FIG. 20: Case D3: AdS/dS, ǫ-= ǫ+ = +1\n\nprocess is as in the bottom right part of figure 20.\n\nFrom the analysis developed so far we have seen that, although in principle we have ten different tunnelling processes, in fact process A1 is the same as A3, process B1 is the same as D3, process B2 is the same as A2 and process C1 is the same as C3. We are thus left with only six distinct processes. From \"symmetry consideration\", it is natural to notice that |β|, not β itself, is the relevant quantity (together with α) to classify the possible tunnelling configurations. We will thus not be surprised if the tunnelling amplitude will be a function of |β|, or, which is the same, an even function of β. We would also like to note already here that all the tunnelling processes can be interpreted as a very large expansion of a mall quantum region of spacetime.\n\n[1] S. Coleman, Phys. Rev. D 15, 2929 (1977) [Erratum-ibid. D 16, 1248 (1977)]. [2] J. C. G. Callan and S. Coleman, Phys. Rev. D 16, 1762 (1977).\n\n22 [3] S. Coleman and F. D. Luccia, Phys. Rev. D 21, 3305 (1980). [4] K. Lee and E. J. Weinberg, Phys. Rev. D 36, 1088 (1987). [5] S. W. Hawking, I. G. Moss, and J. M. Stewart, Phys. Rev. D 26, 2681 (1982). [6] W. Z. Chao, Phys. Rev. D 28, 1898 (1983). [7] K. Sato, M. Sasaki, H. Kodama, and K.-I. Maeda, Progr.\n\nTheor. Phys. 65, 1443 (1981). [8] H. Kodama, M. Sasaki, K. Sato, and K.-I. Maeda, Progr.\n\nTheor. Phys. 66, 2052 (1981). [9] K. Sato, Progr. Theor. Phys. 66, 2287 (1981). [10] K.-I. Maeda, K. Sato, M. Sasaki, and H. Kodama, Phys.\n\nLett. B 108, 98 (1982). [11] K. Sato, H. Kodama, M. Sasaki, and K.-I. Maeda, Phys.\n\nLett. B 108, 103 (1982). [12] H. Kodama, M. Sasaki, and K. Sato, Progr. Theor. Phys.\n\n68, 1979 (1982). [13] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys.\n\nRev. D 36, 2919 (1987). [14] S. K. Blau, E. I. Guendelman, and A. H. Guth, Phys.\n\nRev. D 35, 1747 (1987). [15] A. Aurilia, R. S. Kissack, R. Mann, and E. Spallucci, Phys. Rev. D 35, 2961 (1987). [16] E. Farhi, A. H. Guth, and J. Guven, Nucl. Phys. B 339, 417 (1990). [17] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys.\n\nRev. D 43, R3112 (1991). [18] M. Sasaki, T. Tanaka, K. Yamamoto, and J. Yokoyama, Phys. Lett. B 317, 510 (1993). [19] T. Tanaka, Nucl. Phys. B 556, 373 (1999). [20] A. Khvedelidze, G. V. Lavrelashvili, and T. Tanaka, Phys. Rev. D 62, 083501 (2000). [21] A. Vilenkin, Phys. Rev. D 27, 2848 (1983). [22] A. D. Linde, Lett. Nuovo Cim. 39, 401 (1984). [23] A. D. Linde, Sov. Phys. JETP 60, 211 (1984). [24] D. Stojkovic, G. D. Starkman, and R. Matsuo, arXiv:hepph/0703246 (2007). [25] K. V. Kuchar and M. P. J. Ryan, Phys. Rev. D 40, 3982 (1989). [26] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). [27] R. Bousso and S. W. Hawking, Phys. Rev. D 52, 5659 (1995). [28] R. Bousso and S. W. Hawking, Phys. Rev. D 54, 6312 (1996). [29] R. R. Caldwell, H. A. Chamblin, and G. W. Gibbons, Phys. Rev. D 53, 7103 (1996). [30] T. Vachaspati and M. Trodden, Phys. Rev. D 61, 023502 (2000). [31] J. Gariel and G. L. Denmat, Class. Quantum Grav. 16, 149 (1999). [32] P. Wang and X.-H. Meng, Class. Quantum Grav. 22, 283 (2005). [33] A. Aguirre and M. C. Johnson, Phys. Rev. D 72, 103525 (2005). [34] S. W. Hawking, Phys. Rev. D 18, 1747 (1978). [35] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983). [36] A. Vilenkin, Phys. Rev. D 30, 509 (1984). [37] S. Carlip, Phys. Rev. D 46, 4387 (1992). [38] A. Vilenkin, Phys. Rev. D 58, 067301 (1998). [39] R. Bousso and A. Chamblin, Phys. Rev. D 59, 084004 (1999). [40] G. W. Gibbons, Class. Quantum Grav. 15, 2605 (1998).\n\n[41] A. Vilenkin, Phys. Lett. B 117, 25 (1982). [42] W. Fischler, D. Morgan, and J. Polchinski, Phys. Rev. D 41, 2638 (1990). [43] W. Fischler, D. Morgan, and J. Polchinski, Phys. Rev. D 42, 4042 (1990). [44] J. Garriga, Phys. Rev. D 49, 6327 (1994). [45] B. Freivogel, V. E. Hubeny, A. Maloney, R. C. Myers, M. Rangamani, and S. Shenker, JHEP 03, 007 (2006). [46] L. Anchordoqui, C. Nunez, and K. Olsen, JHEP 10, 50 (2000). [47] S. Kar, JHEP 10, 052 (2006). [48] S. Kar, Phys. Rev. D74, 126002 (2006). [49] S. J. Kolitch and D. M. Eardley, Phys. Rev. D 56, 4651 (1997). [50] S. J. Kolitch and D. M. Eardley, Phys. Rev. D 56, 4663 (1997). [51] Y. Nutku, M. B. Sheftel, and A. A. Malykh, Class. Quantum Grav. 14, L59 (1997). [52] A. Aurilia, H. Nicolai, and P. K. Townsend, Nucl. Phys. B 176, 509 (1980). [53] M. Henneaux and C. Teitelboim, Phys. Lett. B 143, 415 (1984). [54] A. Aurilia, G. Denardo, F. Legovini, and E. Spallucci, Phys. Lett. B 147, 258 (1984). [55] A. Aurilia, G. Denardo, F. Legovini, and E. Spallucci, Nucl. Phys. B 252, 523 (1985). [56] J. D. Brown and C. Teitelboim, Phys. Lett. B 195, 177 (1987). [57] J. D. Brown and C. Teitelboim, Nucl. Phys. B 297, 787 (1988). [58] F. Mellor and I. Moss, Class. Quantum Grav. 6, 1379 (1989). [59] J. L. Feng, J. March-Russel, S. Sethi, and F. Wilczek, Nucl. Phys. B 602, 307 (2001). [60] S. Ansoldi, A. Aurilia, and E. Spallucci, Phys. Rev. D 64, 025008 (2001). [61] A. Gomberoff, M. Henneaux, C. Teitelboim, and F. Wilczek, Phys. Rev. D 69, 083520 (2004). [62] J. Garriga and A. Megevand, Int. J. Th. Phys. 43, 883 (2004). [63] C. Barrabes and W. Israel, Phys. Rev. D 43, 1129 (1991). [64] W. Israel, Nuovo Cimento B44, 1 (1966) [Erratum-ibid.\n\nB48, 463 (1967)]. [65] R. Geroch and J. Traschen, Phys. Rev. D 36, 1017 (1987). [66] R. Maartens, Living. Rev. Relativity 7, 7 (2004); http://www.livingreviews.org/lrr-2004-7 . [67] S. Ansoldi, A. Aurilia, R. Balbinot, and E. Spallucci, Class. Quantum. Grav. 14, 2727 (1997). [68] P. Hajicek and J. Bicak, Phys. Rev. D 56, 4706 (1997). [69] J. L. Friedman, J. Louko, and S. N. Winters-Hilt, Phys.\n\nRev. D 56, 7674 (1997). [70] P. Hajicek and J. Kijowski, Phys. Rev. D 57, 914 (1998). [71] P. Hajicek, Phys. Rev. D 57, 936 (1998). [72] J. L. Friedman, J. Louko, and B. F. Whiting, Phys. Rev. D 57, 2279 (1998). [73] P. Hajicek, Phys. Rev. D 58, 084005 (1998). [74] P. Hajicek and J. Kijowski, Phys. Rev. D 61, 129901(E) (2000). [75] S. Mukohyama, Phys. Rev. D 65, 024028 (2002). [76] R. Capovilla, J. Guven, and E. Rojas, Class. Quantum Grav. 21, 5563 (2004). [77] C. Barrabes and W. Israel, Phys. Rev. D71, 064008 (2005), gr-qc/0502108.\n\n[78] G. L. Alberghi, R. Casadio, and G. Venturi, Phys. Rev. D 60, 124018 (1999). [79] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977). [80] Misner, Thorne, and Wheeler, Gravitation (Freeman, 1973). [81] V. I. Arnold, Mathematical methods of classical mechanics (Springer Verlag, New York, 1978). [82] A. Aurilia, M. Palmer, and E. Spallucci, Phys. Rev. D 40, 2511 (1989). [83] S. Ansoldi and L. Sindoni (2004), gr-qc/0411042 (In the proceedings of the 6th International Symposium on Frontiers of Fundamental Physics (FFP6), Udine, Italy, 26-29 Sep 2004, published in Udine 2004, Frontiers of funda-mental physics 69-73). [84] S. Ansoldi (2007), gr-qc/0701117 (In the proceedings of the 16th Workshop on General Relativity and Gravitation (JGRG16), Niigata, Japan, 27 Nov -1 Dec 2006, Ken-ichi Oohara, Testuya Shiromizu, Kei-ichi Maeda and Misao Sasaki editors, Kyoto, (2007), page 114). [85] S. Ansoldi (2007), in preparation (to appear in the proceedings of \"From Quantum to Emergent Gravity: Theory and Phenomenology\", SISSA, Trieste, Italy, 11th-15th June 2007; http://pos.sissa.it ). [86] S. Parke, Phys. Lett. B 121, 313 (1983). [87] E. I. Guendelman and J. Portnoy, Class. Quantum Grav.\n\n16, 3315 (1999)."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"The study of vacuum decay initiated more than 30 years ago with the work of Callan and Coleman [1, 2] ; in the following years the interest in the subject increased and the possible interplay of true vacuum bubbles with gravitation was also studied [3, 4] , together with bubbles collisions and their importance in the early universe [5, 6] . At the same time, and as opposed with the true vacuum bubbles of Coleman et al., false vacuum bubbles were also considered. In connection with gravity, the behavior of regions of false vacuum, first studied by Sato et al. [7, 8, 9, 10, 11, 12] , was for example analyzed in [13, 14, 15, 16, 17, 18, 19, 20] . For additional papers analyzing it in the context of inflation, we refer the reader to [21, 22, 23] ; interesting links with more phenomenologically oriented approaches can also be drawn, as witnessed, for instance, by the recent [24] .\n\nIn these last works, a minisuperspace approximation was adopted to quantize the system. In more detail, since general relativistic shells 1 can be used as a convenient model and since in spherical symmetry the system only has one degree of freedom, standard semiclassical methods might be suitable to analyze the decay process (see [25] for an early, in principle, discussion of this point). In connection with cosmology, spaces equipped with a cosmological constant (i.e. de Sitter space and generalizations) have been naturally considered. In this context it also worth to remark the important role that they play in connection with the problem of gravitational entropy, causal structure and the presence of horizons (see, for instance [26, 27, 28, 29, 30] as well as the suggestive [31] ).\n\nNotwithstanding many interesting results, after 30 years, and with different flavors, the problem of the stability of (the de Sitter) vacuum in connection with the dynamics of false vacuum bubbles, is still a debated one [19, 32, 33] .\n\nThe still open issues are highly non-trivial and go back to the, also long lasting, problem of formulating a consistent framework for a quantum theory of gravity [34, 35, 36, 37, 38, 39, 40] , but we will not take explicitly this point of view here. We will instead analyze a specific situation, described below, in which the nucleation rate can be computed exactly in arbitrary spacetime dimensions. We will, then, explicitly compute in closed form the nucleation rate in the semiclassical approximation, compare it with existing results, and discuss in detail the associated spacetime structures: an analysis of how quantum effects may be relevant in the context of tunnelling from nothing configurations [41] will also be given. These results, extend some results present in the literature (for instance [16, 42, 43, 44] ) and can also provide a useful limiting case of more general situations, for instance those discussed in [33] . At the same time, to consider spacetimes of higher dimensionality and negative cosmological constant is especially important in view of recent results in the context of the AdS/CFT correspondence [45] and of the braneworld cosmological scenario [46, 47] (see also [48] and references therein for a study in the context of noncommutative branes).\n\nApart from the papers already cited above, the instanton approach has also been discussed by other authors (see for instance [49, 50, 51] ) and although it will not be directly related to the present paper, we cannot avoid mentioning the suggestive relationship between the decay of the cosmological constant, membranes generated by higher rank gauge potentials and black holes, 2 which have appeared in many papers in the literature [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] .\n\nThe structure of the paper is the following: in section II we recall the formalism by which a general relativistic thin spherical brane/shell can be described; this also gives us the opportunity to fix conventions and notations and briefly describe the canonical approach to its semiclassical quantization (subsection II C); a convenient dimensionless formalism is also introduced (subsection II D). We then directly come (section III) to the main result of this paper, which is the calculation of the tunnelling rate between the classical configurations of the system, in arbitrary spacetime dimensions; the results for the cases of 3, 4 and 5 spacetime dimensions are explicitly presented with dedicated plots of the values of the action as a function of the two dimensionless parameters of the model (subsection III C). The exact results for the tunnelling amplitude/probability calculated in section III correspond to specific transitions which take place in spacetime and that will be discussed later on, in a dedicated appendix. We discuss, instead, in the main text (section IV) a proposal to regularize some spacetime configurations which appear to be singular, relating this issue to the description of the brane energy-matter content. In the concluding section V, the results of the paper are summarized; two appendices follows with the detailed description of the parameter space of the system (appendix A) and of all the Penrose diagrams associated with different values of the parameters of the problem (appendix B): they are crucial to fully grasp the physical system under consideration."},{"section_type":"OTHER","section_title":"II. SHELL IN N + 1 DIMENSIONS","text":"In this section we are going to briefly review some results about the dynamics of co-dimension one branes in an (N + 1)-dimensional spacetime, where, under the words co-dimension one brane, we understand an Ndimensional hypersurface Σ separating the (N + 1)dimensional manifold in two domains, M -and M + having Σ as a common part of their boundary: in brief ∂M -∩ ∂M + = Σ. In what follows we are going to use the clear notation of [63] : the formulation developed there can be readily extended to higher dimensions, just by letting the indices run on an extended set of values. We will thus quickly report this paraphrase with the purpose of recalling some notations and conventions. In particular let us choose two arbitrary systems of N + 1 independent vector fields E ±(a) in M ± , respectively, with dual forms Ω (b) ± . Denoting with g (±) the four dimensional metric tensors in the two manifolds we can write 2 , 2 In a few equations below, we will use the notation -, -to denote the scalar product. We also anticipate the notation ∇ Y X for the covariant derivative of the vector fiend X in the direction\n\nin general, g (±) = g (±)ab Ω (a) (±) ⊗ Ω (b) (±) ; in our notation Latin indices a, b (as well as all other latin indices) will vary in the set {0, 1, 2, . . . , N }.\n\nLet us first concentrate our attention on Σ: it is also a manifold, as M ± , and we will denote by e (µ) an Ndimensional system of (commuting) independent vector fields on Σ, with dual system ω (ν) ; the indices µ and ν (as well as all other Greek indices) will vary in the set {0, 1, . . . , N -1}. Because of the physical interpretation we will be interested in the case in which Σ is embedded as a timelike surface in M ± . Since Σ is timelike the normal to Σ in M ± is a spacelike vector n, which we choose normalized so that n, n = +1. Moreover, our convention is that the normal points from the \"-\" to the \"+\" domain of spacetime.\n\nThe components of n (an unambiguously defined nonnull vector) will be different, in general, when measured by an observer in M -or by one in M + , according to the following definition: a) .\n\nn a | ± = n, E (a) | ± = n, E ±(\n\nIn terms of n the extrinsic curvature of the surface Σ can be expressed as 3 K µν = n, ∇ e (µ) e (ν)\n\nand, in general, it is different on theand + side. Moreover, by the Gauss-Codazzi formalism, the geometry of spacetime around Σ can be described in terms of the intrinsic geometry of the hypersurface and of its extrinsic curvature. In the spirit of this formalism, let us denote by h the intrinsic metric of the hypersurface Σ, i.e. h = h µν ω (µ) ⊗ ω (ν) .\n\nWhen we will work with quantities defined in the bulk we will need to distinguish the ones defined in M + from the ones defined in M -, and to this end we will use, as we did above, \"±\" superscripts or subscripts. In many cases we will also be interested in the jump of these quantities across Σ: for instance, if we consider the extrinsic curvature, we may need the to consider the difference K + αβ -K - αβ : following [63, 64] we are going to rewrite this difference as [K αβ ]. Throughout this paper this will be the only meaning that we will give to the square brackets, i.e.\n\ndef.\n\n= A + -A -.\n\nTo avoid confusion no other use of the square brackets will be done.\n\nof the vector field Y . 3 We stress the normalization condition on n."},{"section_type":"OTHER","section_title":"A. Junction conditions","text":"The brane Σ can be more than just a mathematical surface, i.e. we can (and, in most of the cases, we want to) equip it with a matter-energy content: it is then an infinitesimally 4 thin distribution of matter-energy. Thanks to the above mentioned Gauss-Codazzi formalism, we can rewrite Einstein equations to make explicit the contribution from the localized matter. Then, the dynamics of Σ as a surface separating M + from M -is obtained solving the following system of equations,\n\nK + αβ -K - αβ ≡ [K αβ ] = 8πG N +1 (S αβ -h αβ S/2) ; ( 1\n\n)\n\nthese are Israel's junction conditions [63, 64] which relate the jump in the extrinsic curvature [K µν ], i.e. the \"jump\" in the way the surface is embedded in each geometry, to the stress-energy tensor S µν of the matter contained on Σ. The tensor S must also satisfy a conservation equation,\n\n(N ) ∇, S µ = [T (n, e (µ) )],\n\nwhere T is the stress energy tensor describing the content of the complete spacetime manifold. Once we have specified the matter content of the bulk (and hence the geometry according to Einstein equations) the description of the dynamics of the system is obtained by solving the system of equations (1); we refer the reader to [63] and [66] for additional material and related considerations."},{"section_type":"OTHER","section_title":"B. Spherical symmetry","text":"The set-up that is of interest for us is a simplified one, in which all the system is spherically symmetric and the surface stress-energy tensor is that of the, so called, tension model, with S = -κh,\n\nwhere κ > 0 is a constant called the tension of the brane. In what follows, it will be convenient to also define κ = 4π(N -2)G N +1 κ.\n\nThanks to the spherical symmetry, the system of equations (1) can then be reduced to a single equation [67], R ǫ -Ṙ2 + f -(R)ǫ + Ṙ2 + f + (R) = κR 2 ; (2) 4 Strictly speaking Σ is a source defined in a distributional way:\n\nfor additional material on this point, the reader is referred, for instance, to [65].\n\nin the above, f ± (r ± ) are the metric functions of the static line element adapted to the spherical symmetry, i.e., taking the four dimensional case as a convenient example, we choose in both M ± the coordinate system x a ± = (t ± , θ ± , φ ± , r ± ) corresponding to the basis vectors e t± , e θ± , e φ± and e r± , such that the metric is reduced to the form g (±)ab = diag(-f ± (r ± ), r 2 ± , r 2 ± sin 2 θ ± , 1/f ± (r ± )). Generalizations to spacetimes with higher dimensionality are just more cumbersome to write but trivial in their substance. In these coordinate systems, we denote the radius of the brane by R(τ ), where τ is the proper time of an observer comoving with the brane and an overdot denotes the derivative with respect to τ . Moreover, as anticipated in the introduction, we consider that this N -dimensional brane separates two (N + 1)-dimensional spacetimes of the de Sitter/anti-de Sitter type, in general with different cosmological constants Λ ± . We thus have f ± (r ± ) = 1 -2Λ ± N (N -1) r 2 ± .\n\nFinally ǫ ± are signs, defined as ǫ ± (R) = sign( n, e r± )| r±=R and are crucial quantities to obtain the Penrose diagrams associated with the considered brane configuration.\n\nC. The effective action/the momentum Before proceeding with the analysis of the physical system that we introduced in the previous subsection, we believe it is useful to recall some important points about the structure of the junction conditions in spherical symmetry. For a generic junction between spherically symmetric spacetimes across a (spherical) brane carrying matter described by a given, but otherwise arbitrary, equation of state, it would prove very useful to extract an effective action for the dynamics of the brane starting directly from the Einstein-Hilbert action and from a suitable action for the matter fields and the brane [67] . This is the content of a consistent literature on the subject [68, 69, 70, 71, 72, 73, 74, 75, 76] among which we would like to single out the more general, and recent, [77] : the problem is not trivial at all, because of the same subtleties that appear, for instance, in the Hamiltonian formulation of General Relativity. Among the various possible approaches we will closely follow the quantization procedure originally proposed by Farhi, Guth and Guven [16] , which has, lately, been followed also in [67] and [78] . For the Lagrangian approach to the classical dynamics we will instead follow the clear and more general exposition of [77] . We summarize here the relevant elements of these approaches referring the reader to the literature for additional details. In particular, the effective action for the shell can be obtained starting from the Einstein-Hilbert action with the Gibbons-Hawking 4 boundary terms [16, 77, 79] . Using the Gauss-Codazzi formalism [80] , it is readily seen that the Einstein-Hilbert action for a spherical co-dimension one brane (which in our case will separate two domains of (anti-)de Sitter spacetime) can be decomposed in non-dynamical bulk contributions (indeed, we consider the geometry of the bulk spacetimes M ± fixed) and a dynamical boundary contribution described in terms of the extrinsic curvature of the brane Σ. It is enlightening (and, perhaps, necessary) not to use all the freedom in fixing the coordinate systems. As an exemplification we do not use the freedom given by the reparametrization invariance with respect to the proper time of the brane and we, thus, introduce a lapse function N (s), so that the brane induced metric can be written as h (N ) µν = diag( -N (s) 2 , R 2 (s), . . .) in the coordinates (s, R, . . .) (the \". . . \" stand for the trivial spherically symmetric part). Then, following [67, 77] , the effective action for the degrees of freedom associated with the radial and time coordinates takes the form\n\nS eff ∝ R N -2 ǫ Ṙ2 + N 2 f (R) -Ṙ arctanh Ṙ ǫ Ṙ2 + N 2 f (R) sign(f ) -κN R N -1 dτ. ( 3\n\n)\n\nThus, the effective Lagrangian of the system is given by\n\nL = P (N ) Ṙ -H,\n\nwhere, following [67], we have defined the effective Hamiltonian as\n\nH = -R N -2 ǫ Ṙ2 + N 2 f (R) + κ N R N -1\n\nand the effective momentum as\n\nP (N ) (R, Ṙ) = -R N -2 arctanh Ṙ ǫ Ṙ2 + N 2 f (R) sign(f ) . ( 4\n\n)\n\nThen, the equations of motion for the R and N degrees of freedom are dH dτ = Ṅ N H and H(R, Ṙ, N ) = 0; the second equation is a first integral of the former one, and is the Hamiltonian constraint. We see that it encodes all the information about the dynamics of the system, being identical to the only remaining junction condition. The first equation is the second order equation of motion of the system, given by the total derivative with respect to the proper time of the Hamiltonian constraint.\n\nIn what follows, we will be mostly interested in the expression of the momentum evaluated on a solution of the equation of motion. In order to build a canonical structure (as it is done, for example, in [46] ) we first have to prove the existence of a well defined symplectic structure on the phase space of the Lagrangian system; in particular the Legendre transform has to be invertible, which in our case means nothing but the invertibility of the conjugate momentum as a function of the velocity [81] .\n\nAs appreciated already in [16] , for generic co-dimension one branes this cannot be proved to be always satisfied 5 . However, for the system under consideration the canonical structure always exists and P (N ) can rigorously be considered the canonical momentum conjugate to the coordinate R; thus, in the specific case of interest here, the canonical construction of the corresponding quantum theory is a well posed problem. To prove this statement, let us consider\n\n∂P (N ) ∂ Ṙ = -R N -2 1 ǫ Ṙ2 + N 2 f (R) .\n\nThis quantity can be zero if and only if two conditions are simultaneously satisfied:\n\n1. ǫ + = ǫ -;\n\n5 In these cases the quantum theory cannot be built with the canonical formalism. Alternative formulations using the path integral approach have been considered in the seminal work of Farhi, Guth and Guven [16].\n\n5 2. there is a point R in the configuration space such that f + (R) = f -(R).\n\nFor our particular choice of branes in (anti-)de Sitter spacetime, we see that in order for the second condition to hold we must have Λ -= Λ + ; but this implies 6 ǫ + = -1, ǫ -= +1 so that the first condition then fails. Thus, for the class of junctions that we are considering one can always build a canonical structure."},{"section_type":"OTHER","section_title":"D. Dimensionless formalism","text":"Following, for instance, [82] , it is possible to determine the classical dynamics of the system by studying an equivalent one dimensional problem for the radial degree of freedom R, taking also into account the values of the signs ǫ ± (R).\n\nBefore proceeding we will set up a different system of dimensionless quantities, in order to remove the arbitrariness in the definition of the normalization of the cosmological constants and of the bubble's tension. If we define the quantities x = κR, τ = κτ,\n\nλ ± = 2Λ ± κ2 N (N -1) , α = λ -+ λ + , β = λ --λ + , equation (2) then becomes 7 -ǫ ẋ2 + 1 -λx 2 x = x 2 . ( 5\n\n)\n\nFor completeness we also report the dimensionless form of the effective momentum (4) when we impose the gauge choice N ≡ 1:\n\nP (x, ẋ) = -x N -2 ǫ arctanh ẋ ẋ2 + f (x) f |f | . ( 6\n\n)\n\nThis quantity will be central in what follows, but for the moment we keep our attention on equation (5). Despite its unusual look, it is easily proved that it is equivalent to a system of equations, which, in the notation that we are using, takes the form\n\nẋ2 + V (x) = 0 ǫ ± (x) = -sgn(β ± 1) , ( 7\n\n)\n\n6 This can be easily seen from the junction condition (2) remembering that κ > 0. 7 An overdot will denote, from now on, a derivative with respect to τ .\n\nwhere the potential V (x) has a simple parabolic form V (x) = 1 -x 2 x 2 0 (8) and x 0 , the turning point, is given by\n\nx 2 0 = 4 1 + 2α + β 2 , ( 9\n\n)\n\nwhen the right-hand side is positive; otherwise there are no nontrivial classical solutions. All the solutions of the problem can be easily classified in according to the values of α and β: we refer the reader to appendix A for details. The expressions for the signs ǫ ± relate, instead, the global geometrical structure of spacetime for this brane configuration only to the difference between the inner and outer cosmological constants, not to the details of the trajectory itself. This is a portion of the content of the junction condition, that is necessary for the description of the global spacetime structure: a careful discussion of this point can be found in appendix B. The form of equation (5) for a brane separating two (anti-)de Sitter spacetimes, which, in turn, is responsible for the simple quadratic form of the potential (8), allows exact solution of this particular case. It is, in fact, not difficult to see that (5) has: 1. the trivial solution x(τ ) ≡ 0, which always exists; 2. the solution\n\nx(τ ) = x 0 cosh τ -τ (0) x 0 , ( 10\n\n)\n\nwhich satisfies the initial condition x(τ (0) ) = x 0 ; this solution is known as the bounce solution and exists only if the condition 1 + 2α + β 2 > 0 is satisfied.\n\nWe also, incidentally, note that\n\nβ = ±1 ⇐⇒ 1 x 2 0 = λ ∓\n\ni.e. when |β| = 1 the turning point radius coincides with one of the two cosmological horizons, if they exist. We also anticipate that a more careful analysis of the trivial solution x(τ ) ≡ 0 will be required, since, although it is given to us by the mathematics of the problem, we are mostly interested in its physical role, especially when we will turn on (semiclassical) quantum effects (see section IV)."},{"section_type":"OTHER","section_title":"III. TUNNELLING","text":"Using the notation introduced in the last section we can now describe how the semiclassical regime of the 6 brane dynamics looks like. In particular we can consider the tunnelling between the zero-radius solution towards the bouncing solution (10), and vice versa.\n\nFor example in one direction the semiclassical picture is as follows: we have a brane, of very small radius; of course, when it is very small, the quantum properties of its matter content will be non-negligible (we will further discuss this issue later on in section IV) and their interplay with gravity will be non-trivial; although we do not know the full quantum gravity description of the system, we will consider that, thanks to quantum effects, the brane will have a certain probability to tunnel under the potential barrier given by the effective potential (8) into the bounce solution. When emerging after the tunnelling quantum effects will likely become less and less important as compared with gravitational ones (and as far as the interaction with the bulk spacetime is concerned), so that the evolution of the brane will closely resemble that of the classical junction. This is not the case in the first stage of the evolution, involving the tunnelling process, and we will try to understand, at least at an effective level, the \"responsibilities\" of both the quantum and gravitational realms in this process. In particular, quantum effects will be considered in a modification at small scales of the stress-energy tensor (see section IV). Gravitational effects, at small scales, are also present, and described by the geometric character of Israel junction conditions. We think that our description might be able to take into account both these effect, at least within the limits represented by the semiclassical approximation. In this sense, although our treatment will be effective, it will give us the possibility to consistently solve the ambiguity arising at the mathematical level and represented by the x(τ ) ≡ 0 solution. We will suggest that, at the semiclassical level, a more detailed treatment of both, the quantum and gravitational aspects, is unlikely to be necessary."},{"section_type":"OTHER","section_title":"A. Tunnelling trajectories","text":"For the system under consideration, tunnelling trajectory can be described by constructing the instanton in the Euclidean sector. The construction of the instanton in the general case is still an unsolved problem: we will follow the approach of [16] ; moreover, the problems left open in [16] do not affect the present case. In fact it is easy to see that in our case the instanton describing the tunnelling can always be constructed without the necessity to introduce the pseudo-manifold that in [16] was necessary to deal with multiple covering of points in the Euclidean sector. Moreover the different descriptions of the tunnelling process that were obtained in [16] using the canonical or the path-integral approaches in our case also coincide, because they are consequences of properties of the dynamical variables during the tunnelling trajectory which are not present in the system that we are considering (please, see [83, 84, 85] for a more detailed description of these issues).\n\nIn this way we know that the tunnelling can be described directly at the effective level, where, the relevant aspects of the Euclidean junction can be determined by the Wick rotated classical effective system. We thus define τe = -iτ; then, denoting with a prime the derivative with respect to τe , we have that the Euclidean system is obtained with the formal substitutions ẋ → ix ′ for the \"velocity\" and P → i Pe for the momentum (it is not difficult to prove that the above substitution rules are rigorous results and that they can be obtained performing an Euclidean junction, as done, for instance, in [16] ). In particular\n\nPe (x, x ′ ) = -x N -2 arctan x ′ ǫ f (x) -(x ′ ) 2 . ( 11\n\n)\n\nIn this way, the tunnelling process can be modelled using the effective tunnelling trajectory, which solves the Euclidean equation:\n\n-(x ′ ) 2 + 1 -(x/x 0 ) 2 = 0. ( 12\n\n)\n\nThe analytic expression of the tunnelling trajectory of a brane expanding from zero radius at Euclidean time τe = 0 to the bouncing solution is:\n\nx(τ e ) = x 0 sin(τ e /x 0 ).\n\nThe opposite process is then described by\n\nx(τ e ) = x 0 cos(τ e /x 0 ),\n\nif we assume that at Euclidean time τe = 0 the brane starts contracting from x 0 . These two processes occur with certain probabilities, whose amplitudes A can be expressed in the usual semiclassical approximation as\n\nA(0 → x 0 ) ∝ exp(-I e [x]),\n\nwhere I e [x] is the Euclidean and can be obtained as the integral of the Euclidean momentum evaluated on a solution of the Euclidean equation of motion (12). If, for simplicity, we define\n\nI e [x] = Ω N -1 16πG N +1 1 κ N -1 Īe [x],\n\nwhere Ω N -1 is the volume of S N -1 , then Īe [x] can be obtained as\n\nĪe [x] = x0 0 Pe (x)dx. ( 13\n\n)\n\nWe stress again that Pe (x) is the Euclidean momentum evaluated on a solution of the Euclidean equation of motion and can be obtained by substituting (11) . We thus have\n\nx ′ = V (x) in\n\nPe (x) = -x N -2 arctan 2 V (x) (β + ω)x , 7\n\nwhere, to make writing more compact 8 , we have introduced the quantities ω ± defined as ω ± = ±1. Note that in the Euclidean case we have some freedom in appropriately choosing one of the possible branches of the inverse tangent function. Different choices will affect, in general, the result that we obtain for the action. As in [16] we observe that non-careful choices will make the action a discontinuous function of the parameters; this can be seen without difficulties. Preliminarily, let us anticipate that with the symbol \"arctan\", we will indicate the branch of the inverse tangent function with range in [-π/2, π/2]. Let us then consider β = ω and let us integrate by parts the integral (13). We obtain\n\nĪe [x] = - x N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 + + x0 0 x N -1 N -1 d arctan 2 V (x) (β + ω)x .\n\nLet us now consider the first term above. Clearly, if we chose the branch of the arctan function to be the one with range in [-π/2, π/2], then we have lim β→-ω -\n\nx N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 = - ωπx N -1 0 2(N -1) but lim β→-ω + x N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 = + ωπx N -1 0 2(N -1) ,\n\nso that the action develops a discontinuity at β = -ω.\n\nThis discontinuity can be eliminated if we choose different branches of the inverse tangent functions (please remember that in the expression that we are considering we are using a compact notation to indicate the difference of two inverse tangent functions); in particular the two discontinuities at β = ±1 can be eliminated by choosing: 1. the branch with range [-π, 0] for the arctan function containing quantities of the \"+\" spacetime; 2. the branch with range [-π, 0] for the arctan function containing quantities of the \"-\" spacetime.\n\nSince in our notation \"arctan\" has range [-π/2, π/2], this means that in the equations above must be rewritten with the substitutions (Θ is the step function) arctan(\" + \") -→ arctan(\" + \") -πΘ(β + 1) 8 In view of the definition of the ω's, and of the meaning of the square brackets, the equation above is a shorthand for Pe(x) = -x N-2\n\n\" ǫ + arctan \" 2 √ V (x) (β+1)x « -ǫ -arctan \" 2 √ V (x) (β-1)x «« .\n\nand arctan(\" -\") -→ arctan(\" -\") -πΘ(β -1);\n\nfor the jump of the quantity which appears inside the expression of the Euclidean momentum, this implies the following substitution:\n\n[arctan] -→ [arctan] -πΘ(1 -β 2 ).\n\nIn this way the action integral is continuous also at the points β = -ω and is given by the integral Īe [x] = -\n\nx0 0 dxx N -2 × × arctan 2 V (x) (β + ω)x -πΘ(1 -β 2 ) .( 14\n\n)\n\nAfter these preliminary considerations, we can proceed to evaluate the tunnelling amplitude in the WKB approximation; as we anticipated and as we will see, in arbitrary spacetime dimensions this result can be expressed analytically in terms of known functions."},{"section_type":"RESULTS","section_title":"B. General result for the tunnelling amplitude","text":"To calculate the first contribution to the integral (14) we proceed as follows. As a preliminary step, we observe that it can be written as a difference of two integrals with the same general structure. Let us then consider the two integrals containing the inverse tangent functions: small differences, which do not substantially affect the calculation, can be taken into account by properly using the ω's introduced above, as we already did in the expressions for the momentum. This said, we can perform 9 an integration by parts in (14) . The terms evaluated at the limits of integration, which appear in this process, do vanish and by changing the integration variable from x to ζ = (x/x 0 ) 2 (which also transforms the integration domain into the unit interval (0, 1)) we obtain the following expression,\n\nĪe [x] = - x N 0 4(N -1) × × (β + ω) 1 0 ζ (N -2)/2 √ 1 -ζ 1 1 -z ω ζ dζ + ( 15\n\n) + πx N -1 0 (N -1) Θ(1 -β 2 ), 9\n\nStrictly speaking this can be done only when β = -ω. The cases in which β = -ω are trivial and can be dealt separately; or, more simply, since we already know from the discussion in the previous subsection that the action is continuous, we can just extend it to β = -ω by continuity.\n\nwhere\n\nz ω = 1 -x 0 β + ω 2 1 + x 0 β + ω 2 . ( 16\n\n)\n\nWhen z ω < 1 one of the above integrals diverges since there is a pole of the integrand on the domain of integration. On the other hand, it is easy to see that the condition z ω < 1 implies\n\nx 2 0 4 (β + ω) 2 < 0\n\nand cannot be realized for any choice of the parameters if a tunnelling trajectory has to exist. The value z ω = 1 can instead be obtained if β = -ω and we know that the action can be extended by continuity to these values, although the above procedure to calculate the integral is not valid. Thus, under the conditions β = -ω, we have that z ω > 1 is satisfied and equation (15) gives\n\nĪe [x] = - x N 0 4(N -1) Γ(N/2)Γ(1/2) Γ((N + 1)/2) × × (β + ω) 2 F 1 1, N 2 , N + 1 2 , z ω + (17) + πx N -1 0 (N -1) Θ(1 -β 2 ),\n\nwhere Γ is the Euler's gamma function, 2 F 1 the hypergeometric function. Note that x 0 depends on α, β and so the first factor of the formula, depending on the number of spacetime dimensions, cannot be ignored even for a qualitative description of the tunnelling amplitude."},{"section_type":"OTHER","section_title":"C. Some cases of interest","text":"It is useful to specialize the result (17) to particular situations. We are going to do this by considering the cases in which spacetime is three, four and five dimensional. Below we are going to explicitly discuss these three cases. A comparative presentation of the results can be found in the contour plots of figure 1 .\n\nWe start then with the three dimensional case, which is lower-dimensional gravity; in three spacetime dimensions it seems more clear how to build a quantum theory out of the classical junction conditions [46] . Moreover three dimensional gravity is an interesting system by itself, it allows an easier visualization of some results and can be used for interesting specific toy models. Anyway, in this case we have N = 2 and we can express 2 F 1 (1, 1; 3/2; x) in terms of elementary functions as\n\n2 F 1 1, 1; 3 2 ; y = arcsin( √ y) √ 1 -y √ y . ( 18\n\n)\n\nCorrespondingly the action becomes\n\nĪe = - x 2 0 4 Γ(1)Γ(1/2) Γ(3/2) × × (β + ω) arcsin( √ z ω ) √ 1 -z ω √ z ω + ( 19\n\n) + πx 0 Θ(1 -β 2 ).\n\nThe corresponding probability is plotted as a function of β in figure 2 for some non-negative values of α and in figure 3 for some negative values of α. Not many comments are necessary, of course, for the four dimensional case, the original arena on which this calculation was performed. Again we can take advantage of a simple expression for the corresponding hypergeometric function\n\n2 F 1 (1, 3/2; 2; y) = 2 y √ 1 -y - 2 y , ( 20\n\n)\n\nwhich brings the final result in the form\n\nĪe = - πx 2 0 4 × ( 21\n\n) × sgn(β + ω) 1 -(1 -z ω ) 1/2 z ω -Θ(1 -β 2 ) .\n\nThe plots of the probability as a function of β can be found in figure 4 for some non-negative values of α and in figure 5 for some negative values of α. This result is the same as the one obtained in [86] and it also reduces to the one calculated in [3]: it, thus, represents a useful consistency check. Finally, the five dimensional case is interesting in the context of the Randall-Sundrum scenario. In this case, using\n\n2 F 1 1, 2; 5 2 ; y = 3 2y arcsin( √ y) √ 1 -yy 1/2 -1 , ( 22\n\n)\n\nthe result for the action integral can be put in the following form:\n\nĪe = - x 4 0 12 Γ(2)Γ(1/2) Γ(5/2) × × (β + ω) arcsin( √ z ω ) √ 1 -z ω (z ω ) 3/2 - 1 z ω + ( 23\n\n) + πx 3 0 3 Θ(1 -β 2 ).\n\nAgain we present two plots of the corresponding probability as a function of β; figure 6 shows the behavior for some non-negative values of α, whereas plots for some negative values of α can be found in figure 7 ."},{"section_type":"DISCUSSION","section_title":"IV. DISCUSSION","text":"Up to this point we have discussed the physics of the system rather quickly, focusing mainly on the mathematics necessary to describe the classical and the semiclassical phases. There is still a point which deserves a 9 a b c +10 +5 0 -5 -10 +10 +5 0 -5 -10 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 α α β β β β β β no tunnel zone no tunnel zone no tunnel zone\n\nFIG. 1: Contour plots of the values of the tunnelling probability in 3 (case a), 4 (case b) and 5 (case c) spacetime dimensions in the β-α plane. The area inside the crosshatched parabola in the bottom of each picture is the region in parameter space where tunnelling cannot occur since no bounce solutions exists (no tunnel zone). A better understanding of what happens close to the limiting parabola can be understood from the following figures 3, 5 and 7. Next to next contours indicate a jump of the probability of 0.1, with the lightest gray tone indicating the range (0.9, 1.0) and the darkest the range (0.0, 0.1). The same color indicates the same range of values of the probability in all the three sub-diagrams.\n\ndetailed discussion, since it involves non-trivial aspects of the dynamics of the brane. Indeed, the physical process we have considered is the tunnelling of spacetime, from a classical situation representing a spacetime containing a small brane to another classical configuration describing a bouncing brane. In our picture, the pre-tunnelling state is represented by the x ≡ 0 solution of the junction condition. Although this case might appear rather simple at first sight and one could be tempted to just state that the initial state is, for example, the full M + spacetime (a point of view which has been taken for example in [67] ), much more care has to be taken. The main reason is that when x = 0, the brane's world-volume degenerates to a curve. Thus, the formalism that we used to describe the brane breaks down and the analysis that we have made cannot be considered as rigorous as it is in the case of the bouncing solutions. As a manifestation of this fundamental problem, we observe that the equations determining the ǫ ± signs do not hold for the particular solution x ≡ 0. In particular, for this solution the junction condition as written in (5) does not provide any mean to solve this problem.\n\nA way out of this situation appears when we reflect on the fact that this process, although described semiclassically, is quantum in its true nature. Thus an approximation of the system which considers it completely classical before the tunnelling, i.e. in the degenerate configuration, might be too rough and might require a more careful consideration. During the tunnelling trajectory, and even more for the x ≡ 0 classical solution, quantum effects are supposed to be, if not dominant, at least relevant enough to modify the classical picture of the junction.\n\nWe will propose here an attempt to address the problem. Our proposal should be considered a first step further, but far from a first principle solution of the complex quantum problem; it just aims to show that the mathematical and physical aspects of the problem stated above can be dealt with by giving an effective formulation for the phenomena that might arise at small scales. At the same time, although we will just build an effective model, we will not be very demanding about its main properties: in this way, hopefully, the model, although effective, could mimic well enough effects produced by quantum gravity whatever will be their, still undiscovered, true nature.\n\nIn this respect, we would also like to point out the following: i) the main effect of our proposal is to slightly perturb the effective potential (8) in the tunnelling region; then, we will ii) show that the perturbed problem is free from ambiguities and iii) use for the unperturbed case the results obtained in the perturbed one, when the perturbation becomes smaller and smaller. To have a definite model, we will regard the small scale behavior of the matter composing the brane/shell as the physical origin for the perturbations (see below). On the other hand, the consequence of these perturbations in the effective formulation is completely generic: it is, in fact, possible to show that other physical motivations, as for instance the quantum properties of spacetime at small scales, could be reflected in a similar way in the effective 10 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.005 0.01 0.015 0.02 0.025 0.03 0.03 0.02 0.01 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 2: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of β for fixed non-negative values of α as listed above. The detailed behavior for α = 0 around β = 0, is also shown (to better see that the probability is small but non-zero).\n\nformulation. For these reasons, we can regard our conclusions model independent to a high degree, at least within the context defined by the semiclassical approximation.\n\nTo introduce our model we can naturally think the brane as sourced by some matter fields whose nature is, ultimately, quantum. We thus argue that, although at large scales the approximation for the brane stress energy tensor that we made in subsection II B might be rough but still appropriate, at small scales it will instead break down due to quantum effects. We will model these additional quantum effects by adding a term to the brane stress-energy tensor as follows:\n\nS -→ S + S q , ( 24\n\n)\n\nwhere the \"quantum\" contribution S q will be parametrized as\n\nS q = ρ q u ⊗ u + σ q h. ( 25\n\n)\n\nThe conservation equation for S q under the assumption of spherical symmetry gives (we are going to use, from now on, the dimensionless versions of the parameters ρ q and σ q , which following the notation above, are called ρq and σq )\n\ndρ q dx + N ρq x = dσ q dx . ( 26\n\n)\n\nWe will choose preliminarily\n\nρq (x) = ax q ( 27\n\n) so that σq (x) = a q + N q x q . ( 28\n\n)\n\nIn this case the junction condition becomes 10\n\n-ǫ (q) ẋ2 + 1 -λx 2 x = x 2 µ(x), ( 29\n\n)\n\n10 We remember that we are following the convention in which square brackets indicate the jump of the enclosed quantity, i.e. the difference of it when evaluated in the '+' and '-' domains. We are now using the suffix \". . . q\" or \". . . (q) \" to indicate quantities when the energy momentum tensor is modified, as described in the text, to take into account quantum effects; thus the two signs will now be ǫ (q)± .\n\n11 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00\n\nFIG. 3: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of β for fixed negative values of α as listed above. For α < -1/2, not all values of β are allowed. We have put the mark on each curve in correspondence of the minimum value of β for which (at the given value of α) the tunnelling process can exist. For smaller values, the infinitely expanding solution, in fact, is not present.\n\nwith µ(x) = 1 + āx q (30) and, for short, ā = a(q + N )/q. Now we are going to slightly restrict the parameters ā and q to make these general settings appropriate for our model. In particular, the modification to the stress energy tensor, described by ā and q was introduced to model the quantum effects at small scales; thus it should be negligible at large scales and this can be achieved if q + 1 < 0, i.e. q < -1.\n\nWe can now compute the modified effective potential Vq . The potential turns out to be\n\nVq (x) = - x 2 β 2 + 2α(1 + āx q ) 2 + (1 + āx q ) 4 4(1 + āx q ) 2 ( 31\n\n) x 0 - ā2 4x -2q-2 ( 32\n\n) ā 0 V (x) + (1 -β 2 )ā 2 2x -q-2 . ( 33\n\n)\n\nExtracting the behavior of (31) for small x we get (32), whereas for small ā the leading contribution is given by (33). These are useful results. Indeed we see from (32) that, quite generally, and certainly under the above q < -1 condition, the potential satisfies\n\nlim x→0 + Vq (x) = -∞.\n\nThis implies that the Vq (x) allows not only the bounce brane junction, as V (x) does, but also bounded solutions. Thus the addition of the S q term to the stress energy tensor, which in our picture is supposed to take into account quantum effects at small scales, in fact does his job by trading the x ≡ 0 solution for a bounded one of finite (i.e. non-vanishing) size. Moreover this solution exists under very general assumptions about the form of S q , so that we do not have to commit ourselves too much about the underlying quantum gravity physics of which S q is roughly supposed to take into account some effective semiclassical description. Looking, now at equation (33) we also see that for small ā, apart from the evident qualitative difference at small scales, the potential resembles closely the non-perturbed one (and this happens, in particular, along the tunnelling trajectory). Thus it will be a sufficiently good approximation to evaluate the tunnelling probability in the, analytically much simpler, unperturbed case. At this point, our picture of the spacetime transition will be as follows. We will take as final configurations of the tunnelling process the junctions obtained without considering the correction, which in our setup is strongly suppressed at large distances; then, the initial state, which is the x ≡ 0 solutions, will be \"regularized\" by considering the junction as the limiting case of the bounded perturbed junction when ā → 0. In particular, the sign ambiguity, which affects the x ≡ 0 solution, will be solved by choosing the signs of the bounded trajectory 12 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.002 0.004 0.006 0.008 0.01 0.01 2 3 × 0.01 1 3 × 0.01 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 4: Plots of the values of the tunnelling probability in a spacetime of dimension 4, as a function of β for fixed non-negative values of α as listed above. Again the detailed behavior for α = 0 around β = 0 is shown (to better see that the probability, although very small, is non-vanishing).\n\nof the perturbed model. These signs can be obtained in closed form considering the x → 0 limit of the junction condition, which is dominated by the correction. Therefore\n\nǫ ± = ∓1 ( 34\n\n)\n\nUsing this prescription, then, one can build the spacetime diagrams representing the tunnelling process. We refer the reader to appendices for a complete discussion of the parameter space (appendix A) and of the global spacetime diagrams representing the various physical situations (appendix B).\n\nTo conclude the discussion, we would like to remark that, despite the fact that the regularization procedure we have described changes dramatically the situation in the region near x = 0 of the configuration space of the brane, the effect on the calculation we have performed is not significant. This is due to the fact that in the tunnelling region, the potential is substantially modified only in a narrow region close to the turning point corresponding to the maximum radius of the bounded solution of the modified classical junction condition. In the tunnelling region, the perturbation to the potential is finite, and by sending ā → 0 the tunnelling amplitude of the modified problem is approximated arbitrarily well by our calculation in section III. Of course, in the real physical problem we expect that quantum effects would prevent the brane from shrinking to zero radius: thus the mathematical limit ā → 0 should be read as a more physically sound limit in which ā tends to a small but finite value related to the ultimate physical nature of the brane itself (which is presently only partially understood). As a consequence, the analytical result that we have obtained (which relies on a WKB approximated description of the quantum process) can also be consider an approximation of the full quantum result at the lowest power in the ratio between the maximum radius of the bounded solution of the modified classical junction equations and x 0 ."},{"section_type":"CONCLUSION","section_title":"V. CONCLUSIONS","text":"In this paper we have calculated analytically the tunnelling amplitude for a domain of spacetime of de Sitter/anti-de Sitter type in a background which, again, is de Sitter/anti-de Sitter. The analytical result holds in arbitrary spacetime dimensions greater than three, and generalizes already existing four dimensional calcu-13 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00\n\nFIG. 5: Plots of the values of the tunnelling probability in a spacetime of dimension 4, as a function of β for fixed negative values of α as listed above. The same observations as in the case of 3 spacetime dimensions apply.\n\nlations. It is not difficult to see that this result reduces, for appropriate values of the parameters, to the result found by Coleman and de Luccia [3] for the false vacuum to true vacuum transition in four spacetime dimensions. Also the results by Parke [86] , again in the four dimensional case, are correctly reproduced.\n\nWe have, also, discussed extensively, in the text and in appendix B, the spacetime structures that can arise for all possible values of the parameters characterizing the model. Some of these spacetime structures, already discussed in the literature, are known as tunnelling from nothing configurations. We have also exposed a possible issue of the shell formalism in the description of the pre-tunnelling state and proposed a solution which relies on the above mentioned tunnelling from nothing configurations; in this way we have been able to make what we consider to be a consistent choice for the before tunnelling configurations. This proposal, which is implemented by considering a modification to the stress-energy tensor for the matter on the shell at small states, is motivated by the observation that if quantum effects are non-negligible on scales at which the tunnelling process occurs, they should also be non-negligible at smaller scales, where the before tunnelling configurations live. By modelling the influence of these quantum effects with a quite generic modification to the form of the stress-energy tensor at small scales, we have proposed an unambiguous rule to fix the initial configuration. This approach to the problem, seems to us consistent with the level at which we are modelling quantum effects for this gravitational system, which is the semiclassical approximation for an infinitesimally thin distribution of matter and energy. At the same time, it has already been shown that, if we add a more refined matter content on the shell (as for instance a collection of gauge fields), modifications similar to the one that we have considered in this paper naturally appear [87] . Moreover, we note that it gives a very consistent picture of the tunnelling process, since all the possible types of tunnelling result in a \"sudden expansion\" of a very small region of spacetime from a small size (where quantum effects are certainly non negligible) to a much bigger size, with radius of the order of x 0 . This shows that the regularized tunnelling always models what in the literature has been called tunnelling \"from nothing\". In our case the \"nothing\" is exactly the quantum state of the spacetime junction that we model, in an effective way, using (29). It thus seems that some tunnelling configurations present in the literature and of more difficult interpretation [67] might be ruled out by the quantum properties of matter and/or of spacetime at small scales. In fact, it is suggestive to reflect about the fact that, even in our very simplified and purely effective treatment of quantum effects before and during the tunnelling, all tunnelling processes start exactly \"from nothing\" (as interpreted above), i.e. from a configuration which is consistent with the quantum properties of the following tunnelling process.\n\nTo conclude, we would like to explicitly distinguish between the analytical result for the tunnelling probability and the proposal to interpret the pre-tunnelling 14 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.0002 0.0004 0.0006 0.0008 0.001 0.001 2 3 × 0.001 1 3 × 0.001 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 6: Plots of the values of the tunnelling probability in a spacetime of dimension 5, as a function of β for fixed non-negative values of α as listed above. Again the detailed behavior for α = 0 around β = 0 is shown (to better see that the probability, although much smaller than in the previous cases, is still non-vanishing).\n\nconfigurations. Indeed the analytical result is a natural generalization of already existing calculations and it incorporates them as special cases. Its validity is completely independent from our interpretation of the before tunnelling configurations and it can represent a useful limit case to check the results of more elaborated models: for instance, a de Sitter-Schwarzschild shell configuration should reproduce, in the limit of vanishing Schwarzschild mass, our result for a tunnelling between a de Sitter space of assigned cosmological constant and a de Sitter space with vanishing cosmological constant, i.e. Minkowski space."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"We would like to thank M. C. Johnson and S. Sonego for useful discussions and constructive criticism. This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818. The work of S. Ansoldi is supported in part by a grant from the Fulbright Commission.\n\nAPPENDIX A: PARAMETER SPACE\n\nIn this appendix we will elaborate about the classification of the possible junctions between two (anti-)de Sitter spacetimes. After switching to the dimensionless formulation (see subsection II D), we remain with two parameters, i.e. λ ± , or, which is the same, α and β. We will mostly use the latter quantities in the following analysis, since many relevant features of the solutions to Israel's junction condition related to the causal structure of the full spacetime manifold M are easily deducible from the comparison between the two cosmological constants. In particular, in figure 8 we give a classification of the possible solutions. The diagram shows the parameter space of the variables (α, β) and (λ + , λ -). The classification of the solutions using α and β, looks nicely symmetric with respect to the axis β = 0, for which λ + = λ -; this is the primary reason why we will mostly use the (α, β) parametrization in our discussion. The (λ + , λ -) axes can, anyway, be conveniently used to single out the de Sitter from the anti-de Sitter spacetime. In particular, we have defined four main groups of solutions:\n\ntype A) these are solutions in which both spacetimes joined across the brane are anti-de Sitter space-15 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00 0.25 0.5 0.75 1 1.25 1.5 0.002 0.004 0.006 0.008 0.01 0.0 0.5 1.0 1.5 0.01 0.006 0.002 × FIG. 7: Plots of the values of the tunnelling probability in a spacetime of dimension 5, as a function of β for fixed negative values of α as listed above. For α < -1/2, not all values of β are allowed. We have put the mark on each curve in correspondence of the minimum value of β for which (at the given value of α) the tunnelling process can exist and, the zoomed diagram, helps resolving the superposition that arises between the curves obtained for α = -0.50, -1.00, -1.50.\n\ntimes, i.e we have an AdS (-) -AdS (+) junction; this part of the parameter space is bounded by the parabola α = -(β 2 + 1)/2 (equivalently x -1 0 = 0), whose inside is the black region where no solution exists; type B) these solutions describe a junction of one part of de Sitter spacetime in the M -manifold with a part of anti-de Sitter spacetime in the M + manifold, i.e. they are dS (-) -AdS (+) junctions, and play the role of counterparts of the below discussed type D solutions; type C) these are junctions in which both spacetimes have the de Sitter geometry, i.e. we have dS (-) -dS (+) junctions; in some cases, qualitatively different diagrams may arise depending on which cosmological constant is the bigger, i.e. depending on the sign of β; type D) as anticipated above, this last type of solutions is similar to the type B, with the role of \"-\" and \"+\" interchanged; we thus have AdS (-) -dS (+) junctions.\n\nThe above information is not enough to completely characterize the classical spacetime obtained from the junction. In addition we need to know the behavior of the normal to the brane travelling in the spacetimes that we have determined from the diagram in figure (8) . According to our convention the normal to the brane has its tail-tip direction going from the \"-\" to the \"+\" parts of the full spacetime M; on the other hand in each of the two spacetimes M ± the corresponding signs ǫ ± determine if the normal to the brane points in the direction of increasing (ǫ = +1) or decreasing (ǫ = -1) radius. Using the results in (7) for ǫ ± we can subdivide the parameter\n\n16 λ + λ - α β A B C A A D C A FIG. 8:\n\nThe space of parameters is subdivided according to the different kinds of junctions, which can be between anti-de Sitter and anti-de Sitter spacetimes (type A), anti-de Sitter and de Sitter spacetimes (type B), de Sitter and anti-de Sitter spacetimes (type D) or de Sitter and de Sitter spacetimes (type C). The black part of the parameter space, identified by the condition 1 + 2α + β 2 < 0, singles out the values of the parameters for which tunnelling is not possible since the infinitely expanding solution does not exist.\n\nλ + λ - α β 1 1 1 2 2 3 3 3\n\nFIG. 9: The parameter space can be subdivided in three other regions (strips) according to the values of the signs ǫ±. In region 1 we have ǫ± = -1, in region 2 we have ǫ± = ∓1 and, finally, in region 3 we have ǫ± = +1.\n\nλ + λ - α β A1 B1 C1 A2 B2 C2 D2 C2 D3 C3 A3 FIG. 10: In principle we have to study 10 different cases, corresponding to as many different regions in the parameter space. These regions are obtained combining the classifications in figures 8 and 9, so that in each region there is a well defined type of junction with an unique choice for the signs.\n\nAs we will see each of the pairs A1 and A3, B1 and D3, B2 and D2, C1 and C3 will correspond to a distinct processes in spacetime; this matches well with the analytical result, in which the action turns out to be an even function of β, and gives only a total of 6 different possible processes.\n\nspace in three main regions, as in figure 9. These regions correspond to the following situations:\n\nregion 1) ǫ -= ǫ + = -1, so that in both spacetimes M ± the normal n a | ± to\n\nthe brane trajectory points in the direction of decreasing r;\n\nregion 2) ǫ -= +1 but ǫ + = -1, and in M -normal to the brane trajectory n a | -points in the direction of increasing r -but in M + the normal n a | + to the\n\nbrane trajectory points in the direction of decreasing r + ;\n\nregion 3) ǫ -= ǫ + = +1; thus in both spacetimes M ± the normals n a | ± to the brane trajectory point in the direction of increasing r ± .\n\nWe thus have various combinations of the geometries of the two spacetimes M ± according to the classification in figure 8 ; moreover we have to combine them choosing the part of spacetime on the correct side of the brane trajectory following the classification in figure 9 . This gives a total of ten subcases, which are summarized in figure 10 . The naming convention is the most natural, so that, for instance, the junction named B2 is a junction of type B, i.e. a dS (-) -AdS (+) junction, with the signs as in region 2, i.e. ǫ -= +1 and ǫ + = -1. All other names 17 follow the same convention and we thus obtain solutions of the following types A1, A2, A3, B1, B2, C1, C2, C3, D2, D3; their causal structure is detailed in the following appendix.\n\nAPPENDIX B: PENROSE DIAGRAMS & TUNNELLING\n\nHere, following the classification given in the previous appendix, we show the corresponding Penrose diagrams for the classical solutions, and a pictorial representation of the corresponding tunnelling processes. For each case we give four diagrams: 1. the complete spacetime from which we have to \"cut\" the M -part of the bulk (with the trajectory of the bubble and the associated normal) in the top left diagram; 2. the complete spacetime from which we have to \"cut\" the M + part of the bulk (with the trajectory of the bubble and the associated normal) in the top right diagram; 3. the junction, i.e. the full manifold M (again with the shell trajectory and the corresponding normal) in the bottom left diagram; 4. a pictorial representation of the creation of the brane via a tunnelling process, in the bottom right diagram; in this case we have used time translation invariance to set the \"tunnelling time\" at the coordinate time t = 0 so that the top half of the diagram represents the final state (i.e. the top half of the junction in the bottom left diagram) whereas the bottom half of the diagram is a representation (detailed below) of the pre-tunnelling configuration.\n\nWe would like to discuss preliminarily in more detail the way in which the pre-tunnelling configuration is constructed. We remember that we are considering non negligible quantum effects in the regime of the dynamics before the tunnelling. We will thus use for the signs the results coming from the modified junction (29), which are those in (34) . This gives a spacetime structure consisting of two regions (which can be of the de Sitter or of the anti-de Sitter type depending on the values of α and β) both bounded by r = 0 and by the shell radius r = R(τ ): this is the junction corresponding to the bounded solution of the modified potential (31), which is a brane starting from zero radius and expanding to a maximum radius (much smaller than any other length scale present in the problem) before recollapsing to r = 0. In the limit in which ā → 0, as discussed in section IV, this maximum radius tends, in fact, to zero. Nevertheless, in our representation of the spacetime before the tunnelling we have kept an arbitrarily finite size for the maximum radius, to make the diagram more readable. At the same time, we have slightly \"blurred\" it to make pictorially explicit that we are not dealing with a purely classical configuration but with a somehow \"heuristic\" representation of a spacetime where quantum effects are highly non trivial and certainly non negligible. We would like, anyway, to stress again that these quantum configurations will be the initial state of tunnelling processes that correspond to what in the literature has also been called \"tunnelling from nothing\" for instance [67] and references therein).\n\nWe will now present in detail the various kinds of junctions."},{"section_type":"OTHER","section_title":"Type A","text":"As discussed in main text (and with reference to figure 10 ), this class of junctions consists of the matching of two anti-de Sitter spacetimes with different cosmological constants. There are three possibilities corresponding to type A1, A2 and A3 junctions, which are shown, respectively, in figures 11, 12 and 13. r = 0 r = 0 AdS -, ǫ-= -1 AdS+, ǫ+ = -1 r = 0 r = 0 AdS + /AdS-Tunnelling FIG. 11: Case A1: AdS/AdS, ǫ-= ǫ+ = -1\n\nThe junction of type A1 (figure 11 ) has ǫ ± = -1, so in both anti-de Sitter spacetimes the normal to the brane points in the direction of decreasing radii. For this junction (and for all the ones in the 1 sector), β > 1, so that λ -is always greater than λ + . The classical junction is seen (by an observer travelling toward increasing values of the radius) as a transition from a more negative to a less negative value of the cosmological constant, when he/she crosses the brane. The global picture of the tunnelling process is then obtained according to the 18 prescription discussed in section IV. The pre-tunnelling spacetime has been discussed above and now consists of two \"small\" parts of anti-de Sitter spacetime with different cosmological constant. The tunnelling process shows the transition between a compact spacetime composed by two anti-de Sitter regions to a spacetime similar to anti-de Sitter, except for the fact that at some radius the cosmological constant changes its value. r = 0 r = 0 AdS -, ǫ-= +1 AdS+, ǫ+ = -1 r = 0 r = 0 r = 0 r = 0 AdS + /AdS-Tunnelling FIG. 12: Case A2: AdS/AdS, ǫ-= +1; ǫ+ = -1\n\nThe junction of type A2, is instead a junction where effectively a tiny junction of two anti-de Sitter spacetimes is \"inflated\" (figure 12 ). In this case the signs are given by ǫ ± = ∓1. Thus the normal to the brane points in the direction of increasing radii in the anti-de Sitter spacetime with cosmological constant λ -, but it points in the direction of decreasing radii in the anti-de Sitter spacetime with cosmological constant λ + . The net effect of the tunnelling process, which starts from the already discussed initial state, is thus to inflate the pre-tunnelling compact spacetime into a similar, but much larger, one (note that for graphical reasons the scales in the pretunnelling and post-tunnelling parts of the diagram are different; we remember that the maximum radius of the pre-tunnelling state will be much smaller than any other length scale in the problem). The various diagrams are in figure 12 .\n\nThe final type A diagram is the A3 one. This is very much as the A1 type, only that now ǫ ± = +1; in both spacetime the normal to the shell trajectory points in the direction of increasing radii. Thus (apparently) the role of M ± is interchanged, as shown in figure 13 where the dark and light gray parts looks complementary to r = 0 r = 0 AdS-, ǫ-= +1 AdS+, ǫ+ = +1 r = 0 r = 0 AdS+/AdS-Tunnelling FIG. 13: Case A3: AdS/AdS, ǫ-= ǫ+ = +1\n\nthose in figure 11 ; on the other hand, now we have β < -1 so that λ + > λ -. The global spacetime structure in the bottom left corner of figure 13 implies that an observer crossing the brane in the direction of increasing values of the radius, perceives again a transition from a more negative to a less negative value of the cosmological constant. So this diagram describes exactly the same process described by type A1, as it is possible to see from the diagram in figure 13.\n\n2. Type B\n\nThe type B solutions, B1 and B2, correspond to the junction of a part of a de Sitter spacetime, M -, with a part of anti-de Sitter spacetime, M + . Although the pre-tunnelling picture does not change very much, the configurations before the tunnelling are now de Sitteranti-de Sitter junctions.\n\nLet us first discuss the case B1, shown in figure 14 . In both spacetimes, the normal to the shell points in the direction of decreasing radii, since we have ǫ ± = -1.\n\nThe M -part of spacetime corresponds to the region of de Sitter complementary to the region between r = 0 and the brane's world-volume, whereas the anti-de Sitter part M + is the bounded region in the upper right picture of figure 14 . Thus the spacetime M shows a transition from a negative to a positive value of the cosmological constant for an observer crossing the shell in the direction of increasing radius. The configuration before the tunnelling has been described above; the tunnelling pro-19 r = 0 r = 0\n\nI - I + r = H - r = H - r = 0 dS-, ǫ-= -1 AdS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H - r = H - I + r = H - r = H - r = 0 r = 0\n\nAdS+/dS-Tunnelling FIG. 14: Case B1: dS/AdS, ǫ-= ǫ+ = -1 cess again \"inflates\" an anti-de Sitter-de Sitter junction with small volume to a much larger one, as shown in the bottom right diagram of figure 14 .\n\nr = 0 r = 0 I - I + r = H - r = H - r = 0 dS-, ǫ-= +1 AdS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H - r = H - r = 0 r = 0 I + r = H -\n\nAdS+/dS-Tunnelling FIG. 15: Case B2: dS/AdS, ǫ-= +1; ǫ+ = -1\n\nWe then come to the case B2, shown in figure 15 . Since in this case the signs which fix the orientation of the normals are ǫ ± = ∓1, the situation for M + is as in the previous case, but the M -part of spacetime changes because of the change in the sign of ǫ + . This gives for the junction the diagram in the bottom left part of figure 15 . The configuration before and after the tunnelling is shown in the bottom right corner of figure 15 . The situation is very similar to the one in the previous case.\n\n3. Type C Type C solutions are the \"de Sitter counterpart\" of type A solution. The main difference (not related to the spacetime structure) is that whereas for anti-de Sitter junctions the values of the cosmological constant are restricted (i.e. given two arbitrary values a junction might not exists) de Sitter junctions exists for all values of the cosmological constants λ ± (this is clear from figure 10 ). Please, also remember that now the pre-tunnelling diagram will consist of a junction of two de Sitter spacetime with different cosmological constants and with all the other properties described above. r = 0 r = 0\n\nI - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS-, ǫ-= -1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H + r = H + r = H - r = H - r = 0 r = 0 I + r = H + r = H -\n\ndS+/dS-Tunnelling FIG. 16: Case C1: dS/dS, ǫ-= ǫ+ = -1\n\nLet us start with case C1, shown in figure 16 . For this junction, the signs are ǫ ± = -1, so in both de Sitter spacetimes the normals to the brane point in the direction of decreasing radii. Moreover, since in this case β > 1, the cosmological constant in M -will be bigger than the cosmological constant in M + , so that the cosmological horizon H -will be smaller than H + . The diagram for the junction is shown in the bottom left part of figure 16 . It is 20 interesting to see some effects of the brane on the spacetime structure. There are observers that crossing the brane can move behind the cosmological horizon of M - without going through H -. They can also come out, by crossing the brane in the opposite direction. At the same time observers, by moving accurately and with proper timing, might end up behind the cosmological horizon of M + without crossing any horizon but using the shell as a \"gate\" 11 . As usual no additional considerations are required for the before tunnelling configuration. We note, instead, that the semiclassical transition has again the effect of \"inflating\" a small quantum spacetime region, as clearly shown in figure 16 .\n\nr = 0 r = 0 I - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS -, ǫ-= +1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + I - I + r = H - r = H - r = H + r = H + r = 0 r = 0 I + I + r = H - r = H + r = H + dS + /dS- Tunnelling\n\nFIG. 17: Case C2: dS/dS, ǫ-= +1; ǫ+ = -1\n\nWe now discuss case C2, in which the signs are ǫ ± = ∓1. Then the normal to the brane in the de Sitter space with cosmological constant λ -points in the direction of increasing radii, whereas the normal to the brane in the de Sitter space with cosmological constant λ + points in the direction of decreasing radii. We have again two regions of spacetime of the de Sitter type but with a different cosmological constant joined across the brane (bottom right part of figure 17 ). The full picture of the tunnelling process requires the consideration of the same initial configuration used in case C1 above. We also point 11 These considerations might be modified if we consider the influence of the observer on spacetime. To neglect this influence, as a first approximation, is fairly common in the literature to obtain some hints about the global spacetime structure.\n\nout that in this case, the sign of β is not fixed, so either of the cosmological constants λ ± may be the bigger. In particular, the junction in the bottom left corner of figure 17 shows the situation in which λ + < λ -, so that H + > H -.\n\nIn the bottom right part of figure 17 a representation of the tunnelling process is shown. r = 0 r = 0\n\nI - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS-, ǫ-= +1 dS+, ǫ+ = +1 r = 0 r = 0 I - I + r = H + r = H + r = H - r = H - r = 0 r = 0 I + r = H + r = H -\n\ndS+/dS-Tunnelling FIG. 18: Case C3: dS/dS, ǫ-= ǫ+ = +1\n\nThe last junction of the C type is C3, for which the usual set of diagrams is shown in figure 18 . As all C type ones, this is a junction between two de Sitter spacetimes and since β < -1 the relation between the cosmological constants is λ + > λ -, so that H + < H -. At this stage we might wonder if this junction is the same as C1 (mirroring what happens between the junctions of the A1 and A3 type). We will see that this is indeed the case. The normals now point in both spacetimes toward the direction of increasing radii (since ǫ ± = +1). Thus the junction is as in the bottom left part of figure 18 . When we consider also the configuration before tunnelling, we see that the diagram in the bottom right part of figure 16 is the \"switched color and reflected\" version of the one in the bottom right part of figure 18 : we remember now that the in the C1 case λ -in the light gray part was bigger than λ + in the dark side, but now exactly the opposite happens. Thus, despite the different coloring, case C3 represents the same physical situation of case C1. This observation makes interesting to complete the analysis for the remaining two cases, which is done in the following subsection. 21 4. Type D\n\nTo conclude the analysis of the global spacetime structure we have to analyze what happens for the last kind of junctions, those of type D. These are junctions between de Sitter and anti-de Sitter spacetimes, so that the pretunnelling configuration will also change accordingly, as we have discussed for the previous cases.\n\nr = 0 r = 0 r = 0 I - I + r = H + r = H + AdS-, ǫ-= +1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H + r = H + r = 0 r = 0 I + r = H + r = H +\n\ndS + /AdS-Tunnelling FIG. 19: Case D2: AdS/dS, ǫ-= +1;ǫ+ = -1\n\nThe case of the junction of type D2 is characterized by the following signs: ǫ ± = ∓1. This means that in the anti-de Sitter side the normal points in the direction of increasing radii, whereas in the de Sitter part, it points in the direction of decreasing radii. Thus the junction, shown in the bottom left corner of figure 19 , is the mirror image of the B2 case. Again, the considerations in section IV determine the geometry before the tunnelling. This brings, in complete analogy with the B2 case, to the picture of the tunnelling process given in the bottom right part of figure 19 ; again, apart from the different colors, this case is the same as B2 also in connection with the semiclassical tunnelling process.\n\nWe have thus only one case left, which is quickly dealt with, namely D3. It is now not difficult to anticipate that this process will turn out to be identical to B1, that appears in figure 14 . Indeed the signs are now ǫ ± = +1, so that in both spacetime the normals point in the direction of increasing radii. By the usual procedure we are thus led to the spacetime diagram shown in the bottom left part of figure 20 . Part of this diagram will constitute the final state of the tunnelling process. The initial state is obtained as usual and the picture of the tunnelling r = 0 r = 0 r = 0\n\nI - I + r = H + r = H + AdS-, ǫ-= +1 dS+, ǫ+ = +1 r = 0 r = 0 I - I + r = H + r = H + I + r = H + r = H + r = 0 r = 0\n\ndS+/AdS-Tunnelling FIG. 20: Case D3: AdS/dS, ǫ-= ǫ+ = +1\n\nprocess is as in the bottom right part of figure 20."},{"section_type":"OTHER","section_title":"Comments","text":"From the analysis developed so far we have seen that, although in principle we have ten different tunnelling processes, in fact process A1 is the same as A3, process B1 is the same as D3, process B2 is the same as A2 and process C1 is the same as C3. We are thus left with only six distinct processes. From \"symmetry consideration\", it is natural to notice that |β|, not β itself, is the relevant quantity (together with α) to classify the possible tunnelling configurations. We will thus not be surprised if the tunnelling amplitude will be a function of |β|, or, which is the same, an even function of β. We would also like to note already here that all the tunnelling processes can be interpreted as a very large expansion of a mall quantum region of spacetime.\n\n[1] S. Coleman, Phys. Rev. D 15, 2929 (1977) [Erratum-ibid. D 16, 1248 (1977)]. [2] J. C. G. Callan and S. Coleman, Phys. Rev. D 16, 1762 (1977).\n\n22 [3] S. Coleman and F. D. Luccia, Phys. Rev. D 21, 3305 (1980). [4] K. Lee and E. J. Weinberg, Phys. Rev. D 36, 1088 (1987). [5] S. W. Hawking, I. G. Moss, and J. M. Stewart, Phys. Rev. D 26, 2681 (1982). [6] W. Z. Chao, Phys. Rev. D 28, 1898 (1983). [7] K. Sato, M. Sasaki, H. Kodama, and K.-I. Maeda, Progr.\n\nTheor. Phys. 65, 1443 (1981). [8] H. Kodama, M. Sasaki, K. Sato, and K.-I. Maeda, Progr.\n\nTheor. Phys. 66, 2052 (1981). [9] K. Sato, Progr. Theor. Phys. 66, 2287 (1981). [10] K.-I. Maeda, K. Sato, M. Sasaki, and H. Kodama, Phys.\n\nLett. B 108, 98 (1982). [11] K. Sato, H. Kodama, M. Sasaki, and K.-I. Maeda, Phys.\n\nLett. B 108, 103 (1982). [12] H. Kodama, M. Sasaki, and K. Sato, Progr. Theor. Phys.\n\n68, 1979 (1982). [13] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys.\n\nRev. D 36, 2919 (1987). [14] S. K. Blau, E. I. Guendelman, and A. H. Guth, Phys.\n\nRev. D 35, 1747 (1987). [15] A. Aurilia, R. S. Kissack, R. Mann, and E. Spallucci, Phys. Rev. D 35, 2961 (1987). [16] E. Farhi, A. H. Guth, and J. Guven, Nucl. Phys. B 339, 417 (1990). [17] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys.\n\nRev. D 43, R3112 (1991). [18] M. Sasaki, T. Tanaka, K. Yamamoto, and J. Yokoyama, Phys. Lett. B 317, 510 (1993). [19] T. Tanaka, Nucl. Phys. B 556, 373 (1999). [20] A. Khvedelidze, G. V. Lavrelashvili, and T. Tanaka, Phys. Rev. D 62, 083501 (2000). [21] A. Vilenkin, Phys. Rev. D 27, 2848 (1983). [22] A. D. Linde, Lett. Nuovo Cim. 39, 401 (1984). [23] A. D. Linde, Sov. Phys. JETP 60, 211 (1984). [24] D. Stojkovic, G. D. Starkman, and R. Matsuo, arXiv:hepph/0703246 (2007). [25] K. V. Kuchar and M. P. J. Ryan, Phys. Rev. D 40, 3982 (1989). [26] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). [27] R. Bousso and S. W. Hawking, Phys. Rev. D 52, 5659 (1995). [28] R. Bousso and S. W. Hawking, Phys. Rev. D 54, 6312 (1996). [29] R. R. Caldwell, H. A. Chamblin, and G. W. Gibbons, Phys. Rev. D 53, 7103 (1996). [30] T. Vachaspati and M. Trodden, Phys. Rev. D 61, 023502 (2000). [31] J. Gariel and G. L. Denmat, Class. Quantum Grav. 16, 149 (1999). [32] P. Wang and X.-H. Meng, Class. Quantum Grav. 22, 283 (2005). [33] A. Aguirre and M. C. Johnson, Phys. Rev. D 72, 103525 (2005). [34] S. W. Hawking, Phys. Rev. D 18, 1747 (1978). [35] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983). [36] A. Vilenkin, Phys. Rev. D 30, 509 (1984). [37] S. Carlip, Phys. Rev. D 46, 4387 (1992). [38] A. Vilenkin, Phys. Rev. D 58, 067301 (1998). [39] R. Bousso and A. Chamblin, Phys. Rev. D 59, 084004 (1999). [40] G. W. Gibbons, Class. Quantum Grav. 15, 2605 (1998).\n\n[41] A. Vilenkin, Phys. Lett. B 117, 25 (1982). [42] W. Fischler, D. Morgan, and J. Polchinski, Phys. Rev. D 41, 2638 (1990). [43] W. Fischler, D. Morgan, and J. Polchinski, Phys. Rev. D 42, 4042 (1990). [44] J. Garriga, Phys. Rev. D 49, 6327 (1994). [45] B. Freivogel, V. E. Hubeny, A. Maloney, R. C. Myers, M. Rangamani, and S. Shenker, JHEP 03, 007 (2006). [46] L. Anchordoqui, C. Nunez, and K. Olsen, JHEP 10, 50 (2000). [47] S. Kar, JHEP 10, 052 (2006). [48] S. Kar, Phys. Rev. D74, 126002 (2006). [49] S. J. Kolitch and D. M. Eardley, Phys. Rev. D 56, 4651 (1997). [50] S. J. Kolitch and D. M. Eardley, Phys. Rev. D 56, 4663 (1997). [51] Y. Nutku, M. B. Sheftel, and A. A. Malykh, Class. Quantum Grav. 14, L59 (1997). [52] A. Aurilia, H. Nicolai, and P. K. Townsend, Nucl. Phys. B 176, 509 (1980). [53] M. Henneaux and C. Teitelboim, Phys. Lett. B 143, 415 (1984). [54] A. Aurilia, G. Denardo, F. Legovini, and E. Spallucci, Phys. Lett. B 147, 258 (1984). [55] A. Aurilia, G. Denardo, F. Legovini, and E. Spallucci, Nucl. Phys. B 252, 523 (1985). [56] J. D. Brown and C. Teitelboim, Phys. Lett. B 195, 177 (1987). [57] J. D. Brown and C. Teitelboim, Nucl. Phys. B 297, 787 (1988). [58] F. Mellor and I. Moss, Class. Quantum Grav. 6, 1379 (1989). [59] J. L. Feng, J. March-Russel, S. Sethi, and F. Wilczek, Nucl. Phys. B 602, 307 (2001). [60] S. Ansoldi, A. Aurilia, and E. Spallucci, Phys. Rev. D 64, 025008 (2001). [61] A. Gomberoff, M. Henneaux, C. Teitelboim, and F. Wilczek, Phys. Rev. D 69, 083520 (2004). [62] J. Garriga and A. Megevand, Int. J. Th. Phys. 43, 883 (2004). [63] C. Barrabes and W. Israel, Phys. Rev. D 43, 1129 (1991). [64] W. Israel, Nuovo Cimento B44, 1 (1966) [Erratum-ibid.\n\nB48, 463 (1967)]. [65] R. Geroch and J. Traschen, Phys. Rev. D 36, 1017 (1987). [66] R. Maartens, Living. Rev. Relativity 7, 7 (2004); http://www.livingreviews.org/lrr-2004-7 . [67] S. Ansoldi, A. Aurilia, R. Balbinot, and E. Spallucci, Class. Quantum. Grav. 14, 2727 (1997). [68] P. Hajicek and J. Bicak, Phys. Rev. D 56, 4706 (1997). [69] J. L. Friedman, J. Louko, and S. N. Winters-Hilt, Phys.\n\nRev. D 56, 7674 (1997). [70] P. Hajicek and J. Kijowski, Phys. Rev. D 57, 914 (1998). [71] P. Hajicek, Phys. Rev. D 57, 936 (1998). [72] J. L. Friedman, J. Louko, and B. F. Whiting, Phys. Rev. D 57, 2279 (1998). [73] P. Hajicek, Phys. Rev. D 58, 084005 (1998). [74] P. Hajicek and J. Kijowski, Phys. Rev. D 61, 129901(E) (2000). [75] S. Mukohyama, Phys. Rev. D 65, 024028 (2002). [76] R. Capovilla, J. Guven, and E. Rojas, Class. Quantum Grav. 21, 5563 (2004). [77] C. Barrabes and W. Israel, Phys. Rev. D71, 064008 (2005), gr-qc/0502108.\n\n[78] G. L. Alberghi, R. Casadio, and G. Venturi, Phys. Rev. D 60, 124018 (1999). [79] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977). [80] Misner, Thorne, and Wheeler, Gravitation (Freeman, 1973). [81] V. I. Arnold, Mathematical methods of classical mechanics (Springer Verlag, New York, 1978). [82] A. Aurilia, M. Palmer, and E. Spallucci, Phys. Rev. D 40, 2511 (1989). [83] S. Ansoldi and L. Sindoni (2004), gr-qc/0411042 (In the proceedings of the 6th International Symposium on Frontiers of Fundamental Physics (FFP6), Udine, Italy, 26-29 Sep 2004, published in Udine 2004, Frontiers of funda-mental physics 69-73). [84] S. Ansoldi (2007), gr-qc/0701117 (In the proceedings of the 16th Workshop on General Relativity and Gravitation (JGRG16), Niigata, Japan, 27 Nov -1 Dec 2006, Ken-ichi Oohara, Testuya Shiromizu, Kei-ichi Maeda and Misao Sasaki editors, Kyoto, (2007), page 114). [85] S. Ansoldi (2007), in preparation (to appear in the proceedings of \"From Quantum to Emergent Gravity: Theory and Phenomenology\", SISSA, Trieste, Italy, 11th-15th June 2007; http://pos.sissa.it ). [86] S. Parke, Phys. Lett. B 121, 313 (1983). [87] E. I. Guendelman and J. Portnoy, Class. Quantum Grav.\n\n16, 3315 (1999)."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"The study of vacuum decay initiated more than 30 years ago with the work of Callan and Coleman [1, 2] ; in the following years the interest in the subject increased and the possible interplay of true vacuum bubbles with gravitation was also studied [3, 4] , together with bubbles collisions and their importance in the early universe [5, 6] . At the same time, and as opposed with the true vacuum bubbles of Coleman et al., false vacuum bubbles were also considered. In connection with gravity, the behavior of regions of false vacuum, first studied by Sato et al. [7, 8, 9, 10, 11, 12] , was for example analyzed in [13, 14, 15, 16, 17, 18, 19, 20] . For additional papers analyzing it in the context of inflation, we refer the reader to [21, 22, 23] ; interesting links with more phenomenologically oriented approaches can also be drawn, as witnessed, for instance, by the recent [24] .\\n\\nIn these last works, a minisuperspace approximation was adopted to quantize the system. In more detail, since general relativistic shells 1 can be used as a convenient model and since in spherical symmetry the system only has one degree of freedom, standard semiclassical methods might be suitable to analyze the decay process (see [25] for an early, in principle, discussion of this point). In connection with cosmology, spaces equipped with a cosmological constant (i.e. de Sitter space and generalizations) have been naturally considered. In this context it also worth to remark the important role that they play in connection with the problem of gravitational entropy, causal structure and the presence of horizons (see, for instance [26, 27, 28, 29, 30] as well as the suggestive [31] ).\\n\\nNotwithstanding many interesting results, after 30 years, and with different flavors, the problem of the stability of (the de Sitter) vacuum in connection with the dynamics of false vacuum bubbles, is still a debated one [19, 32, 33] .\\n\\nThe still open issues are highly non-trivial and go back to the, also long lasting, problem of formulating a consistent framework for a quantum theory of gravity [34, 35, 36, 37, 38, 39, 40] , but we will not take explicitly this point of view here. We will instead analyze a specific situation, described below, in which the nucleation rate can be computed exactly in arbitrary spacetime dimensions. We will, then, explicitly compute in closed form the nucleation rate in the semiclassical approximation, compare it with existing results, and discuss in detail the associated spacetime structures: an analysis of how quantum effects may be relevant in the context of tunnelling from nothing configurations [41] will also be given. These results, extend some results present in the literature (for instance [16, 42, 43, 44] ) and can also provide a useful limiting case of more general situations, for instance those discussed in [33] . At the same time, to consider spacetimes of higher dimensionality and negative cosmological constant is especially important in view of recent results in the context of the AdS/CFT correspondence [45] and of the braneworld cosmological scenario [46, 47] (see also [48] and references therein for a study in the context of noncommutative branes).\\n\\nApart from the papers already cited above, the instanton approach has also been discussed by other authors (see for instance [49, 50, 51] ) and although it will not be directly related to the present paper, we cannot avoid mentioning the suggestive relationship between the decay of the cosmological constant, membranes generated by higher rank gauge potentials and black holes, 2 which have appeared in many papers in the literature [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62] .\\n\\nThe structure of the paper is the following: in section II we recall the formalism by which a general relativistic thin spherical brane/shell can be described; this also gives us the opportunity to fix conventions and notations and briefly describe the canonical approach to its semiclassical quantization (subsection II C); a convenient dimensionless formalism is also introduced (subsection II D). We then directly come (section III) to the main result of this paper, which is the calculation of the tunnelling rate between the classical configurations of the system, in arbitrary spacetime dimensions; the results for the cases of 3, 4 and 5 spacetime dimensions are explicitly presented with dedicated plots of the values of the action as a function of the two dimensionless parameters of the model (subsection III C). The exact results for the tunnelling amplitude/probability calculated in section III correspond to specific transitions which take place in spacetime and that will be discussed later on, in a dedicated appendix. We discuss, instead, in the main text (section IV) a proposal to regularize some spacetime configurations which appear to be singular, relating this issue to the description of the brane energy-matter content. In the concluding section V, the results of the paper are summarized; two appendices follows with the detailed description of the parameter space of the system (appendix A) and of all the Penrose diagrams associated with different values of the parameters of the problem (appendix B): they are crucial to fully grasp the physical system under consideration.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. SHELL IN N + 1 DIMENSIONS\",\n \"text\": \"In this section we are going to briefly review some results about the dynamics of co-dimension one branes in an (N + 1)-dimensional spacetime, where, under the words co-dimension one brane, we understand an Ndimensional hypersurface Σ separating the (N + 1)dimensional manifold in two domains, M -and M + having Σ as a common part of their boundary: in brief ∂M -∩ ∂M + = Σ. In what follows we are going to use the clear notation of [63] : the formulation developed there can be readily extended to higher dimensions, just by letting the indices run on an extended set of values. We will thus quickly report this paraphrase with the purpose of recalling some notations and conventions. In particular let us choose two arbitrary systems of N + 1 independent vector fields E ±(a) in M ± , respectively, with dual forms Ω (b) ± . Denoting with g (±) the four dimensional metric tensors in the two manifolds we can write 2 , 2 In a few equations below, we will use the notation -, -to denote the scalar product. We also anticipate the notation ∇ Y X for the covariant derivative of the vector fiend X in the direction\\n\\nin general, g (±) = g (±)ab Ω (a) (±) ⊗ Ω (b) (±) ; in our notation Latin indices a, b (as well as all other latin indices) will vary in the set {0, 1, 2, . . . , N }.\\n\\nLet us first concentrate our attention on Σ: it is also a manifold, as M ± , and we will denote by e (µ) an Ndimensional system of (commuting) independent vector fields on Σ, with dual system ω (ν) ; the indices µ and ν (as well as all other Greek indices) will vary in the set {0, 1, . . . , N -1}. Because of the physical interpretation we will be interested in the case in which Σ is embedded as a timelike surface in M ± . Since Σ is timelike the normal to Σ in M ± is a spacelike vector n, which we choose normalized so that n, n = +1. Moreover, our convention is that the normal points from the \\\"-\\\" to the \\\"+\\\" domain of spacetime.\\n\\nThe components of n (an unambiguously defined nonnull vector) will be different, in general, when measured by an observer in M -or by one in M + , according to the following definition: a) .\\n\\nn a | ± = n, E (a) | ± = n, E ±(\\n\\nIn terms of n the extrinsic curvature of the surface Σ can be expressed as 3 K µν = n, ∇ e (µ) e (ν)\\n\\nand, in general, it is different on theand + side. Moreover, by the Gauss-Codazzi formalism, the geometry of spacetime around Σ can be described in terms of the intrinsic geometry of the hypersurface and of its extrinsic curvature. In the spirit of this formalism, let us denote by h the intrinsic metric of the hypersurface Σ, i.e. h = h µν ω (µ) ⊗ ω (ν) .\\n\\nWhen we will work with quantities defined in the bulk we will need to distinguish the ones defined in M + from the ones defined in M -, and to this end we will use, as we did above, \\\"±\\\" superscripts or subscripts. In many cases we will also be interested in the jump of these quantities across Σ: for instance, if we consider the extrinsic curvature, we may need the to consider the difference K + αβ -K - αβ : following [63, 64] we are going to rewrite this difference as [K αβ ]. Throughout this paper this will be the only meaning that we will give to the square brackets, i.e.\\n\\ndef.\\n\\n= A + -A -.\\n\\nTo avoid confusion no other use of the square brackets will be done.\\n\\nof the vector field Y . 3 We stress the normalization condition on n.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Junction conditions\",\n \"text\": \"The brane Σ can be more than just a mathematical surface, i.e. we can (and, in most of the cases, we want to) equip it with a matter-energy content: it is then an infinitesimally 4 thin distribution of matter-energy. Thanks to the above mentioned Gauss-Codazzi formalism, we can rewrite Einstein equations to make explicit the contribution from the localized matter. Then, the dynamics of Σ as a surface separating M + from M -is obtained solving the following system of equations,\\n\\nK + αβ -K - αβ ≡ [K αβ ] = 8πG N +1 (S αβ -h αβ S/2) ; ( 1\\n\\n)\\n\\nthese are Israel's junction conditions [63, 64] which relate the jump in the extrinsic curvature [K µν ], i.e. the \\\"jump\\\" in the way the surface is embedded in each geometry, to the stress-energy tensor S µν of the matter contained on Σ. The tensor S must also satisfy a conservation equation,\\n\\n(N ) ∇, S µ = [T (n, e (µ) )],\\n\\nwhere T is the stress energy tensor describing the content of the complete spacetime manifold. Once we have specified the matter content of the bulk (and hence the geometry according to Einstein equations) the description of the dynamics of the system is obtained by solving the system of equations (1); we refer the reader to [63] and [66] for additional material and related considerations.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Spherical symmetry\",\n \"text\": \"The set-up that is of interest for us is a simplified one, in which all the system is spherically symmetric and the surface stress-energy tensor is that of the, so called, tension model, with S = -κh,\\n\\nwhere κ > 0 is a constant called the tension of the brane. In what follows, it will be convenient to also define κ = 4π(N -2)G N +1 κ.\\n\\nThanks to the spherical symmetry, the system of equations (1) can then be reduced to a single equation [67], R ǫ -Ṙ2 + f -(R)ǫ + Ṙ2 + f + (R) = κR 2 ; (2) 4 Strictly speaking Σ is a source defined in a distributional way:\\n\\nfor additional material on this point, the reader is referred, for instance, to [65].\\n\\nin the above, f ± (r ± ) are the metric functions of the static line element adapted to the spherical symmetry, i.e., taking the four dimensional case as a convenient example, we choose in both M ± the coordinate system x a ± = (t ± , θ ± , φ ± , r ± ) corresponding to the basis vectors e t± , e θ± , e φ± and e r± , such that the metric is reduced to the form g (±)ab = diag(-f ± (r ± ), r 2 ± , r 2 ± sin 2 θ ± , 1/f ± (r ± )). Generalizations to spacetimes with higher dimensionality are just more cumbersome to write but trivial in their substance. In these coordinate systems, we denote the radius of the brane by R(τ ), where τ is the proper time of an observer comoving with the brane and an overdot denotes the derivative with respect to τ . Moreover, as anticipated in the introduction, we consider that this N -dimensional brane separates two (N + 1)-dimensional spacetimes of the de Sitter/anti-de Sitter type, in general with different cosmological constants Λ ± . We thus have f ± (r ± ) = 1 -2Λ ± N (N -1) r 2 ± .\\n\\nFinally ǫ ± are signs, defined as ǫ ± (R) = sign( n, e r± )| r±=R and are crucial quantities to obtain the Penrose diagrams associated with the considered brane configuration.\\n\\nC. The effective action/the momentum Before proceeding with the analysis of the physical system that we introduced in the previous subsection, we believe it is useful to recall some important points about the structure of the junction conditions in spherical symmetry. For a generic junction between spherically symmetric spacetimes across a (spherical) brane carrying matter described by a given, but otherwise arbitrary, equation of state, it would prove very useful to extract an effective action for the dynamics of the brane starting directly from the Einstein-Hilbert action and from a suitable action for the matter fields and the brane [67] . This is the content of a consistent literature on the subject [68, 69, 70, 71, 72, 73, 74, 75, 76] among which we would like to single out the more general, and recent, [77] : the problem is not trivial at all, because of the same subtleties that appear, for instance, in the Hamiltonian formulation of General Relativity. Among the various possible approaches we will closely follow the quantization procedure originally proposed by Farhi, Guth and Guven [16] , which has, lately, been followed also in [67] and [78] . For the Lagrangian approach to the classical dynamics we will instead follow the clear and more general exposition of [77] . We summarize here the relevant elements of these approaches referring the reader to the literature for additional details. In particular, the effective action for the shell can be obtained starting from the Einstein-Hilbert action with the Gibbons-Hawking 4 boundary terms [16, 77, 79] . Using the Gauss-Codazzi formalism [80] , it is readily seen that the Einstein-Hilbert action for a spherical co-dimension one brane (which in our case will separate two domains of (anti-)de Sitter spacetime) can be decomposed in non-dynamical bulk contributions (indeed, we consider the geometry of the bulk spacetimes M ± fixed) and a dynamical boundary contribution described in terms of the extrinsic curvature of the brane Σ. It is enlightening (and, perhaps, necessary) not to use all the freedom in fixing the coordinate systems. As an exemplification we do not use the freedom given by the reparametrization invariance with respect to the proper time of the brane and we, thus, introduce a lapse function N (s), so that the brane induced metric can be written as h (N ) µν = diag( -N (s) 2 , R 2 (s), . . .) in the coordinates (s, R, . . .) (the \\\". . . \\\" stand for the trivial spherically symmetric part). Then, following [67, 77] , the effective action for the degrees of freedom associated with the radial and time coordinates takes the form\\n\\nS eff ∝ R N -2 ǫ Ṙ2 + N 2 f (R) -Ṙ arctanh Ṙ ǫ Ṙ2 + N 2 f (R) sign(f ) -κN R N -1 dτ. ( 3\\n\\n)\\n\\nThus, the effective Lagrangian of the system is given by\\n\\nL = P (N ) Ṙ -H,\\n\\nwhere, following [67], we have defined the effective Hamiltonian as\\n\\nH = -R N -2 ǫ Ṙ2 + N 2 f (R) + κ N R N -1\\n\\nand the effective momentum as\\n\\nP (N ) (R, Ṙ) = -R N -2 arctanh Ṙ ǫ Ṙ2 + N 2 f (R) sign(f ) . ( 4\\n\\n)\\n\\nThen, the equations of motion for the R and N degrees of freedom are dH dτ = Ṅ N H and H(R, Ṙ, N ) = 0; the second equation is a first integral of the former one, and is the Hamiltonian constraint. We see that it encodes all the information about the dynamics of the system, being identical to the only remaining junction condition. The first equation is the second order equation of motion of the system, given by the total derivative with respect to the proper time of the Hamiltonian constraint.\\n\\nIn what follows, we will be mostly interested in the expression of the momentum evaluated on a solution of the equation of motion. In order to build a canonical structure (as it is done, for example, in [46] ) we first have to prove the existence of a well defined symplectic structure on the phase space of the Lagrangian system; in particular the Legendre transform has to be invertible, which in our case means nothing but the invertibility of the conjugate momentum as a function of the velocity [81] .\\n\\nAs appreciated already in [16] , for generic co-dimension one branes this cannot be proved to be always satisfied 5 . However, for the system under consideration the canonical structure always exists and P (N ) can rigorously be considered the canonical momentum conjugate to the coordinate R; thus, in the specific case of interest here, the canonical construction of the corresponding quantum theory is a well posed problem. To prove this statement, let us consider\\n\\n∂P (N ) ∂ Ṙ = -R N -2 1 ǫ Ṙ2 + N 2 f (R) .\\n\\nThis quantity can be zero if and only if two conditions are simultaneously satisfied:\\n\\n1. ǫ + = ǫ -;\\n\\n5 In these cases the quantum theory cannot be built with the canonical formalism. Alternative formulations using the path integral approach have been considered in the seminal work of Farhi, Guth and Guven [16].\\n\\n5 2. there is a point R in the configuration space such that f + (R) = f -(R).\\n\\nFor our particular choice of branes in (anti-)de Sitter spacetime, we see that in order for the second condition to hold we must have Λ -= Λ + ; but this implies 6 ǫ + = -1, ǫ -= +1 so that the first condition then fails. Thus, for the class of junctions that we are considering one can always build a canonical structure.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D. Dimensionless formalism\",\n \"text\": \"Following, for instance, [82] , it is possible to determine the classical dynamics of the system by studying an equivalent one dimensional problem for the radial degree of freedom R, taking also into account the values of the signs ǫ ± (R).\\n\\nBefore proceeding we will set up a different system of dimensionless quantities, in order to remove the arbitrariness in the definition of the normalization of the cosmological constants and of the bubble's tension. If we define the quantities x = κR, τ = κτ,\\n\\nλ ± = 2Λ ± κ2 N (N -1) , α = λ -+ λ + , β = λ --λ + , equation (2) then becomes 7 -ǫ ẋ2 + 1 -λx 2 x = x 2 . ( 5\\n\\n)\\n\\nFor completeness we also report the dimensionless form of the effective momentum (4) when we impose the gauge choice N ≡ 1:\\n\\nP (x, ẋ) = -x N -2 ǫ arctanh ẋ ẋ2 + f (x) f |f | . ( 6\\n\\n)\\n\\nThis quantity will be central in what follows, but for the moment we keep our attention on equation (5). Despite its unusual look, it is easily proved that it is equivalent to a system of equations, which, in the notation that we are using, takes the form\\n\\nẋ2 + V (x) = 0 ǫ ± (x) = -sgn(β ± 1) , ( 7\\n\\n)\\n\\n6 This can be easily seen from the junction condition (2) remembering that κ > 0. 7 An overdot will denote, from now on, a derivative with respect to τ .\\n\\nwhere the potential V (x) has a simple parabolic form V (x) = 1 -x 2 x 2 0 (8) and x 0 , the turning point, is given by\\n\\nx 2 0 = 4 1 + 2α + β 2 , ( 9\\n\\n)\\n\\nwhen the right-hand side is positive; otherwise there are no nontrivial classical solutions. All the solutions of the problem can be easily classified in according to the values of α and β: we refer the reader to appendix A for details. The expressions for the signs ǫ ± relate, instead, the global geometrical structure of spacetime for this brane configuration only to the difference between the inner and outer cosmological constants, not to the details of the trajectory itself. This is a portion of the content of the junction condition, that is necessary for the description of the global spacetime structure: a careful discussion of this point can be found in appendix B. The form of equation (5) for a brane separating two (anti-)de Sitter spacetimes, which, in turn, is responsible for the simple quadratic form of the potential (8), allows exact solution of this particular case. It is, in fact, not difficult to see that (5) has: 1. the trivial solution x(τ ) ≡ 0, which always exists; 2. the solution\\n\\nx(τ ) = x 0 cosh τ -τ (0) x 0 , ( 10\\n\\n)\\n\\nwhich satisfies the initial condition x(τ (0) ) = x 0 ; this solution is known as the bounce solution and exists only if the condition 1 + 2α + β 2 > 0 is satisfied.\\n\\nWe also, incidentally, note that\\n\\nβ = ±1 ⇐⇒ 1 x 2 0 = λ ∓\\n\\ni.e. when |β| = 1 the turning point radius coincides with one of the two cosmological horizons, if they exist. We also anticipate that a more careful analysis of the trivial solution x(τ ) ≡ 0 will be required, since, although it is given to us by the mathematics of the problem, we are mostly interested in its physical role, especially when we will turn on (semiclassical) quantum effects (see section IV).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. TUNNELLING\",\n \"text\": \"Using the notation introduced in the last section we can now describe how the semiclassical regime of the 6 brane dynamics looks like. In particular we can consider the tunnelling between the zero-radius solution towards the bouncing solution (10), and vice versa.\\n\\nFor example in one direction the semiclassical picture is as follows: we have a brane, of very small radius; of course, when it is very small, the quantum properties of its matter content will be non-negligible (we will further discuss this issue later on in section IV) and their interplay with gravity will be non-trivial; although we do not know the full quantum gravity description of the system, we will consider that, thanks to quantum effects, the brane will have a certain probability to tunnel under the potential barrier given by the effective potential (8) into the bounce solution. When emerging after the tunnelling quantum effects will likely become less and less important as compared with gravitational ones (and as far as the interaction with the bulk spacetime is concerned), so that the evolution of the brane will closely resemble that of the classical junction. This is not the case in the first stage of the evolution, involving the tunnelling process, and we will try to understand, at least at an effective level, the \\\"responsibilities\\\" of both the quantum and gravitational realms in this process. In particular, quantum effects will be considered in a modification at small scales of the stress-energy tensor (see section IV). Gravitational effects, at small scales, are also present, and described by the geometric character of Israel junction conditions. We think that our description might be able to take into account both these effect, at least within the limits represented by the semiclassical approximation. In this sense, although our treatment will be effective, it will give us the possibility to consistently solve the ambiguity arising at the mathematical level and represented by the x(τ ) ≡ 0 solution. We will suggest that, at the semiclassical level, a more detailed treatment of both, the quantum and gravitational aspects, is unlikely to be necessary.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Tunnelling trajectories\",\n \"text\": \"For the system under consideration, tunnelling trajectory can be described by constructing the instanton in the Euclidean sector. The construction of the instanton in the general case is still an unsolved problem: we will follow the approach of [16] ; moreover, the problems left open in [16] do not affect the present case. In fact it is easy to see that in our case the instanton describing the tunnelling can always be constructed without the necessity to introduce the pseudo-manifold that in [16] was necessary to deal with multiple covering of points in the Euclidean sector. Moreover the different descriptions of the tunnelling process that were obtained in [16] using the canonical or the path-integral approaches in our case also coincide, because they are consequences of properties of the dynamical variables during the tunnelling trajectory which are not present in the system that we are considering (please, see [83, 84, 85] for a more detailed description of these issues).\\n\\nIn this way we know that the tunnelling can be described directly at the effective level, where, the relevant aspects of the Euclidean junction can be determined by the Wick rotated classical effective system. We thus define τe = -iτ; then, denoting with a prime the derivative with respect to τe , we have that the Euclidean system is obtained with the formal substitutions ẋ → ix ′ for the \\\"velocity\\\" and P → i Pe for the momentum (it is not difficult to prove that the above substitution rules are rigorous results and that they can be obtained performing an Euclidean junction, as done, for instance, in [16] ). In particular\\n\\nPe (x, x ′ ) = -x N -2 arctan x ′ ǫ f (x) -(x ′ ) 2 . ( 11\\n\\n)\\n\\nIn this way, the tunnelling process can be modelled using the effective tunnelling trajectory, which solves the Euclidean equation:\\n\\n-(x ′ ) 2 + 1 -(x/x 0 ) 2 = 0. ( 12\\n\\n)\\n\\nThe analytic expression of the tunnelling trajectory of a brane expanding from zero radius at Euclidean time τe = 0 to the bouncing solution is:\\n\\nx(τ e ) = x 0 sin(τ e /x 0 ).\\n\\nThe opposite process is then described by\\n\\nx(τ e ) = x 0 cos(τ e /x 0 ),\\n\\nif we assume that at Euclidean time τe = 0 the brane starts contracting from x 0 . These two processes occur with certain probabilities, whose amplitudes A can be expressed in the usual semiclassical approximation as\\n\\nA(0 → x 0 ) ∝ exp(-I e [x]),\\n\\nwhere I e [x] is the Euclidean and can be obtained as the integral of the Euclidean momentum evaluated on a solution of the Euclidean equation of motion (12). If, for simplicity, we define\\n\\nI e [x] = Ω N -1 16πG N +1 1 κ N -1 Īe [x],\\n\\nwhere Ω N -1 is the volume of S N -1 , then Īe [x] can be obtained as\\n\\nĪe [x] = x0 0 Pe (x)dx. ( 13\\n\\n)\\n\\nWe stress again that Pe (x) is the Euclidean momentum evaluated on a solution of the Euclidean equation of motion and can be obtained by substituting (11) . We thus have\\n\\nx ′ = V (x) in\\n\\nPe (x) = -x N -2 arctan 2 V (x) (β + ω)x , 7\\n\\nwhere, to make writing more compact 8 , we have introduced the quantities ω ± defined as ω ± = ±1. Note that in the Euclidean case we have some freedom in appropriately choosing one of the possible branches of the inverse tangent function. Different choices will affect, in general, the result that we obtain for the action. As in [16] we observe that non-careful choices will make the action a discontinuous function of the parameters; this can be seen without difficulties. Preliminarily, let us anticipate that with the symbol \\\"arctan\\\", we will indicate the branch of the inverse tangent function with range in [-π/2, π/2]. Let us then consider β = ω and let us integrate by parts the integral (13). We obtain\\n\\nĪe [x] = - x N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 + + x0 0 x N -1 N -1 d arctan 2 V (x) (β + ω)x .\\n\\nLet us now consider the first term above. Clearly, if we chose the branch of the arctan function to be the one with range in [-π/2, π/2], then we have lim β→-ω -\\n\\nx N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 = - ωπx N -1 0 2(N -1) but lim β→-ω + x N -1 N -1 arctan 2 V (x) (β + ω)x x0 0 = + ωπx N -1 0 2(N -1) ,\\n\\nso that the action develops a discontinuity at β = -ω.\\n\\nThis discontinuity can be eliminated if we choose different branches of the inverse tangent functions (please remember that in the expression that we are considering we are using a compact notation to indicate the difference of two inverse tangent functions); in particular the two discontinuities at β = ±1 can be eliminated by choosing: 1. the branch with range [-π, 0] for the arctan function containing quantities of the \\\"+\\\" spacetime; 2. the branch with range [-π, 0] for the arctan function containing quantities of the \\\"-\\\" spacetime.\\n\\nSince in our notation \\\"arctan\\\" has range [-π/2, π/2], this means that in the equations above must be rewritten with the substitutions (Θ is the step function) arctan(\\\" + \\\") -→ arctan(\\\" + \\\") -πΘ(β + 1) 8 In view of the definition of the ω's, and of the meaning of the square brackets, the equation above is a shorthand for Pe(x) = -x N-2\\n\\n\\\" ǫ + arctan \\\" 2 √ V (x) (β+1)x « -ǫ -arctan \\\" 2 √ V (x) (β-1)x «« .\\n\\nand arctan(\\\" -\\\") -→ arctan(\\\" -\\\") -πΘ(β -1);\\n\\nfor the jump of the quantity which appears inside the expression of the Euclidean momentum, this implies the following substitution:\\n\\n[arctan] -→ [arctan] -πΘ(1 -β 2 ).\\n\\nIn this way the action integral is continuous also at the points β = -ω and is given by the integral Īe [x] = -\\n\\nx0 0 dxx N -2 × × arctan 2 V (x) (β + ω)x -πΘ(1 -β 2 ) .( 14\\n\\n)\\n\\nAfter these preliminary considerations, we can proceed to evaluate the tunnelling amplitude in the WKB approximation; as we anticipated and as we will see, in arbitrary spacetime dimensions this result can be expressed analytically in terms of known functions.\"\n },\n {\n \"section_type\": \"RESULTS\",\n \"section_title\": \"B. General result for the tunnelling amplitude\",\n \"text\": \"To calculate the first contribution to the integral (14) we proceed as follows. As a preliminary step, we observe that it can be written as a difference of two integrals with the same general structure. Let us then consider the two integrals containing the inverse tangent functions: small differences, which do not substantially affect the calculation, can be taken into account by properly using the ω's introduced above, as we already did in the expressions for the momentum. This said, we can perform 9 an integration by parts in (14) . The terms evaluated at the limits of integration, which appear in this process, do vanish and by changing the integration variable from x to ζ = (x/x 0 ) 2 (which also transforms the integration domain into the unit interval (0, 1)) we obtain the following expression,\\n\\nĪe [x] = - x N 0 4(N -1) × × (β + ω) 1 0 ζ (N -2)/2 √ 1 -ζ 1 1 -z ω ζ dζ + ( 15\\n\\n) + πx N -1 0 (N -1) Θ(1 -β 2 ), 9\\n\\nStrictly speaking this can be done only when β = -ω. The cases in which β = -ω are trivial and can be dealt separately; or, more simply, since we already know from the discussion in the previous subsection that the action is continuous, we can just extend it to β = -ω by continuity.\\n\\nwhere\\n\\nz ω = 1 -x 0 β + ω 2 1 + x 0 β + ω 2 . ( 16\\n\\n)\\n\\nWhen z ω < 1 one of the above integrals diverges since there is a pole of the integrand on the domain of integration. On the other hand, it is easy to see that the condition z ω < 1 implies\\n\\nx 2 0 4 (β + ω) 2 < 0\\n\\nand cannot be realized for any choice of the parameters if a tunnelling trajectory has to exist. The value z ω = 1 can instead be obtained if β = -ω and we know that the action can be extended by continuity to these values, although the above procedure to calculate the integral is not valid. Thus, under the conditions β = -ω, we have that z ω > 1 is satisfied and equation (15) gives\\n\\nĪe [x] = - x N 0 4(N -1) Γ(N/2)Γ(1/2) Γ((N + 1)/2) × × (β + ω) 2 F 1 1, N 2 , N + 1 2 , z ω + (17) + πx N -1 0 (N -1) Θ(1 -β 2 ),\\n\\nwhere Γ is the Euler's gamma function, 2 F 1 the hypergeometric function. Note that x 0 depends on α, β and so the first factor of the formula, depending on the number of spacetime dimensions, cannot be ignored even for a qualitative description of the tunnelling amplitude.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Some cases of interest\",\n \"text\": \"It is useful to specialize the result (17) to particular situations. We are going to do this by considering the cases in which spacetime is three, four and five dimensional. Below we are going to explicitly discuss these three cases. A comparative presentation of the results can be found in the contour plots of figure 1 .\\n\\nWe start then with the three dimensional case, which is lower-dimensional gravity; in three spacetime dimensions it seems more clear how to build a quantum theory out of the classical junction conditions [46] . Moreover three dimensional gravity is an interesting system by itself, it allows an easier visualization of some results and can be used for interesting specific toy models. Anyway, in this case we have N = 2 and we can express 2 F 1 (1, 1; 3/2; x) in terms of elementary functions as\\n\\n2 F 1 1, 1; 3 2 ; y = arcsin( √ y) √ 1 -y √ y . ( 18\\n\\n)\\n\\nCorrespondingly the action becomes\\n\\nĪe = - x 2 0 4 Γ(1)Γ(1/2) Γ(3/2) × × (β + ω) arcsin( √ z ω ) √ 1 -z ω √ z ω + ( 19\\n\\n) + πx 0 Θ(1 -β 2 ).\\n\\nThe corresponding probability is plotted as a function of β in figure 2 for some non-negative values of α and in figure 3 for some negative values of α. Not many comments are necessary, of course, for the four dimensional case, the original arena on which this calculation was performed. Again we can take advantage of a simple expression for the corresponding hypergeometric function\\n\\n2 F 1 (1, 3/2; 2; y) = 2 y √ 1 -y - 2 y , ( 20\\n\\n)\\n\\nwhich brings the final result in the form\\n\\nĪe = - πx 2 0 4 × ( 21\\n\\n) × sgn(β + ω) 1 -(1 -z ω ) 1/2 z ω -Θ(1 -β 2 ) .\\n\\nThe plots of the probability as a function of β can be found in figure 4 for some non-negative values of α and in figure 5 for some negative values of α. This result is the same as the one obtained in [86] and it also reduces to the one calculated in [3]: it, thus, represents a useful consistency check. Finally, the five dimensional case is interesting in the context of the Randall-Sundrum scenario. In this case, using\\n\\n2 F 1 1, 2; 5 2 ; y = 3 2y arcsin( √ y) √ 1 -yy 1/2 -1 , ( 22\\n\\n)\\n\\nthe result for the action integral can be put in the following form:\\n\\nĪe = - x 4 0 12 Γ(2)Γ(1/2) Γ(5/2) × × (β + ω) arcsin( √ z ω ) √ 1 -z ω (z ω ) 3/2 - 1 z ω + ( 23\\n\\n) + πx 3 0 3 Θ(1 -β 2 ).\\n\\nAgain we present two plots of the corresponding probability as a function of β; figure 6 shows the behavior for some non-negative values of α, whereas plots for some negative values of α can be found in figure 7 .\"\n },\n {\n \"section_type\": \"DISCUSSION\",\n \"section_title\": \"IV. DISCUSSION\",\n \"text\": \"Up to this point we have discussed the physics of the system rather quickly, focusing mainly on the mathematics necessary to describe the classical and the semiclassical phases. There is still a point which deserves a 9 a b c +10 +5 0 -5 -10 +10 +5 0 -5 -10 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 -4 -2 0 +2 +4 α α β β β β β β no tunnel zone no tunnel zone no tunnel zone\\n\\nFIG. 1: Contour plots of the values of the tunnelling probability in 3 (case a), 4 (case b) and 5 (case c) spacetime dimensions in the β-α plane. The area inside the crosshatched parabola in the bottom of each picture is the region in parameter space where tunnelling cannot occur since no bounce solutions exists (no tunnel zone). A better understanding of what happens close to the limiting parabola can be understood from the following figures 3, 5 and 7. Next to next contours indicate a jump of the probability of 0.1, with the lightest gray tone indicating the range (0.9, 1.0) and the darkest the range (0.0, 0.1). The same color indicates the same range of values of the probability in all the three sub-diagrams.\\n\\ndetailed discussion, since it involves non-trivial aspects of the dynamics of the brane. Indeed, the physical process we have considered is the tunnelling of spacetime, from a classical situation representing a spacetime containing a small brane to another classical configuration describing a bouncing brane. In our picture, the pre-tunnelling state is represented by the x ≡ 0 solution of the junction condition. Although this case might appear rather simple at first sight and one could be tempted to just state that the initial state is, for example, the full M + spacetime (a point of view which has been taken for example in [67] ), much more care has to be taken. The main reason is that when x = 0, the brane's world-volume degenerates to a curve. Thus, the formalism that we used to describe the brane breaks down and the analysis that we have made cannot be considered as rigorous as it is in the case of the bouncing solutions. As a manifestation of this fundamental problem, we observe that the equations determining the ǫ ± signs do not hold for the particular solution x ≡ 0. In particular, for this solution the junction condition as written in (5) does not provide any mean to solve this problem.\\n\\nA way out of this situation appears when we reflect on the fact that this process, although described semiclassically, is quantum in its true nature. Thus an approximation of the system which considers it completely classical before the tunnelling, i.e. in the degenerate configuration, might be too rough and might require a more careful consideration. During the tunnelling trajectory, and even more for the x ≡ 0 classical solution, quantum effects are supposed to be, if not dominant, at least relevant enough to modify the classical picture of the junction.\\n\\nWe will propose here an attempt to address the problem. Our proposal should be considered a first step further, but far from a first principle solution of the complex quantum problem; it just aims to show that the mathematical and physical aspects of the problem stated above can be dealt with by giving an effective formulation for the phenomena that might arise at small scales. At the same time, although we will just build an effective model, we will not be very demanding about its main properties: in this way, hopefully, the model, although effective, could mimic well enough effects produced by quantum gravity whatever will be their, still undiscovered, true nature.\\n\\nIn this respect, we would also like to point out the following: i) the main effect of our proposal is to slightly perturb the effective potential (8) in the tunnelling region; then, we will ii) show that the perturbed problem is free from ambiguities and iii) use for the unperturbed case the results obtained in the perturbed one, when the perturbation becomes smaller and smaller. To have a definite model, we will regard the small scale behavior of the matter composing the brane/shell as the physical origin for the perturbations (see below). On the other hand, the consequence of these perturbations in the effective formulation is completely generic: it is, in fact, possible to show that other physical motivations, as for instance the quantum properties of spacetime at small scales, could be reflected in a similar way in the effective 10 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.005 0.01 0.015 0.02 0.025 0.03 0.03 0.02 0.01 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 2: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of β for fixed non-negative values of α as listed above. The detailed behavior for α = 0 around β = 0, is also shown (to better see that the probability is small but non-zero).\\n\\nformulation. For these reasons, we can regard our conclusions model independent to a high degree, at least within the context defined by the semiclassical approximation.\\n\\nTo introduce our model we can naturally think the brane as sourced by some matter fields whose nature is, ultimately, quantum. We thus argue that, although at large scales the approximation for the brane stress energy tensor that we made in subsection II B might be rough but still appropriate, at small scales it will instead break down due to quantum effects. We will model these additional quantum effects by adding a term to the brane stress-energy tensor as follows:\\n\\nS -→ S + S q , ( 24\\n\\n)\\n\\nwhere the \\\"quantum\\\" contribution S q will be parametrized as\\n\\nS q = ρ q u ⊗ u + σ q h. ( 25\\n\\n)\\n\\nThe conservation equation for S q under the assumption of spherical symmetry gives (we are going to use, from now on, the dimensionless versions of the parameters ρ q and σ q , which following the notation above, are called ρq and σq )\\n\\ndρ q dx + N ρq x = dσ q dx . ( 26\\n\\n)\\n\\nWe will choose preliminarily\\n\\nρq (x) = ax q ( 27\\n\\n) so that σq (x) = a q + N q x q . ( 28\\n\\n)\\n\\nIn this case the junction condition becomes 10\\n\\n-ǫ (q) ẋ2 + 1 -λx 2 x = x 2 µ(x), ( 29\\n\\n)\\n\\n10 We remember that we are following the convention in which square brackets indicate the jump of the enclosed quantity, i.e. the difference of it when evaluated in the '+' and '-' domains. We are now using the suffix \\\". . . q\\\" or \\\". . . (q) \\\" to indicate quantities when the energy momentum tensor is modified, as described in the text, to take into account quantum effects; thus the two signs will now be ǫ (q)± .\\n\\n11 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00\\n\\nFIG. 3: Plots of the values of the tunnelling probability in a spacetime of dimension 3, as a function of β for fixed negative values of α as listed above. For α < -1/2, not all values of β are allowed. We have put the mark on each curve in correspondence of the minimum value of β for which (at the given value of α) the tunnelling process can exist. For smaller values, the infinitely expanding solution, in fact, is not present.\\n\\nwith µ(x) = 1 + āx q (30) and, for short, ā = a(q + N )/q. Now we are going to slightly restrict the parameters ā and q to make these general settings appropriate for our model. In particular, the modification to the stress energy tensor, described by ā and q was introduced to model the quantum effects at small scales; thus it should be negligible at large scales and this can be achieved if q + 1 < 0, i.e. q < -1.\\n\\nWe can now compute the modified effective potential Vq . The potential turns out to be\\n\\nVq (x) = - x 2 β 2 + 2α(1 + āx q ) 2 + (1 + āx q ) 4 4(1 + āx q ) 2 ( 31\\n\\n) x 0 - ā2 4x -2q-2 ( 32\\n\\n) ā 0 V (x) + (1 -β 2 )ā 2 2x -q-2 . ( 33\\n\\n)\\n\\nExtracting the behavior of (31) for small x we get (32), whereas for small ā the leading contribution is given by (33). These are useful results. Indeed we see from (32) that, quite generally, and certainly under the above q < -1 condition, the potential satisfies\\n\\nlim x→0 + Vq (x) = -∞.\\n\\nThis implies that the Vq (x) allows not only the bounce brane junction, as V (x) does, but also bounded solutions. Thus the addition of the S q term to the stress energy tensor, which in our picture is supposed to take into account quantum effects at small scales, in fact does his job by trading the x ≡ 0 solution for a bounded one of finite (i.e. non-vanishing) size. Moreover this solution exists under very general assumptions about the form of S q , so that we do not have to commit ourselves too much about the underlying quantum gravity physics of which S q is roughly supposed to take into account some effective semiclassical description. Looking, now at equation (33) we also see that for small ā, apart from the evident qualitative difference at small scales, the potential resembles closely the non-perturbed one (and this happens, in particular, along the tunnelling trajectory). Thus it will be a sufficiently good approximation to evaluate the tunnelling probability in the, analytically much simpler, unperturbed case. At this point, our picture of the spacetime transition will be as follows. We will take as final configurations of the tunnelling process the junctions obtained without considering the correction, which in our setup is strongly suppressed at large distances; then, the initial state, which is the x ≡ 0 solutions, will be \\\"regularized\\\" by considering the junction as the limiting case of the bounded perturbed junction when ā → 0. In particular, the sign ambiguity, which affects the x ≡ 0 solution, will be solved by choosing the signs of the bounded trajectory 12 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.002 0.004 0.006 0.008 0.01 0.01 2 3 × 0.01 1 3 × 0.01 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 4: Plots of the values of the tunnelling probability in a spacetime of dimension 4, as a function of β for fixed non-negative values of α as listed above. Again the detailed behavior for α = 0 around β = 0 is shown (to better see that the probability, although very small, is non-vanishing).\\n\\nof the perturbed model. These signs can be obtained in closed form considering the x → 0 limit of the junction condition, which is dominated by the correction. Therefore\\n\\nǫ ± = ∓1 ( 34\\n\\n)\\n\\nUsing this prescription, then, one can build the spacetime diagrams representing the tunnelling process. We refer the reader to appendices for a complete discussion of the parameter space (appendix A) and of the global spacetime diagrams representing the various physical situations (appendix B).\\n\\nTo conclude the discussion, we would like to remark that, despite the fact that the regularization procedure we have described changes dramatically the situation in the region near x = 0 of the configuration space of the brane, the effect on the calculation we have performed is not significant. This is due to the fact that in the tunnelling region, the potential is substantially modified only in a narrow region close to the turning point corresponding to the maximum radius of the bounded solution of the modified classical junction condition. In the tunnelling region, the perturbation to the potential is finite, and by sending ā → 0 the tunnelling amplitude of the modified problem is approximated arbitrarily well by our calculation in section III. Of course, in the real physical problem we expect that quantum effects would prevent the brane from shrinking to zero radius: thus the mathematical limit ā → 0 should be read as a more physically sound limit in which ā tends to a small but finite value related to the ultimate physical nature of the brane itself (which is presently only partially understood). As a consequence, the analytical result that we have obtained (which relies on a WKB approximated description of the quantum process) can also be consider an approximation of the full quantum result at the lowest power in the ratio between the maximum radius of the bounded solution of the modified classical junction equations and x 0 .\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"V. CONCLUSIONS\",\n \"text\": \"In this paper we have calculated analytically the tunnelling amplitude for a domain of spacetime of de Sitter/anti-de Sitter type in a background which, again, is de Sitter/anti-de Sitter. The analytical result holds in arbitrary spacetime dimensions greater than three, and generalizes already existing four dimensional calcu-13 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00\\n\\nFIG. 5: Plots of the values of the tunnelling probability in a spacetime of dimension 4, as a function of β for fixed negative values of α as listed above. The same observations as in the case of 3 spacetime dimensions apply.\\n\\nlations. It is not difficult to see that this result reduces, for appropriate values of the parameters, to the result found by Coleman and de Luccia [3] for the false vacuum to true vacuum transition in four spacetime dimensions. Also the results by Parke [86] , again in the four dimensional case, are correctly reproduced.\\n\\nWe have, also, discussed extensively, in the text and in appendix B, the spacetime structures that can arise for all possible values of the parameters characterizing the model. Some of these spacetime structures, already discussed in the literature, are known as tunnelling from nothing configurations. We have also exposed a possible issue of the shell formalism in the description of the pre-tunnelling state and proposed a solution which relies on the above mentioned tunnelling from nothing configurations; in this way we have been able to make what we consider to be a consistent choice for the before tunnelling configurations. This proposal, which is implemented by considering a modification to the stress-energy tensor for the matter on the shell at small states, is motivated by the observation that if quantum effects are non-negligible on scales at which the tunnelling process occurs, they should also be non-negligible at smaller scales, where the before tunnelling configurations live. By modelling the influence of these quantum effects with a quite generic modification to the form of the stress-energy tensor at small scales, we have proposed an unambiguous rule to fix the initial configuration. This approach to the problem, seems to us consistent with the level at which we are modelling quantum effects for this gravitational system, which is the semiclassical approximation for an infinitesimally thin distribution of matter and energy. At the same time, it has already been shown that, if we add a more refined matter content on the shell (as for instance a collection of gauge fields), modifications similar to the one that we have considered in this paper naturally appear [87] . Moreover, we note that it gives a very consistent picture of the tunnelling process, since all the possible types of tunnelling result in a \\\"sudden expansion\\\" of a very small region of spacetime from a small size (where quantum effects are certainly non negligible) to a much bigger size, with radius of the order of x 0 . This shows that the regularized tunnelling always models what in the literature has been called tunnelling \\\"from nothing\\\". In our case the \\\"nothing\\\" is exactly the quantum state of the spacetime junction that we model, in an effective way, using (29). It thus seems that some tunnelling configurations present in the literature and of more difficult interpretation [67] might be ruled out by the quantum properties of matter and/or of spacetime at small scales. In fact, it is suggestive to reflect about the fact that, even in our very simplified and purely effective treatment of quantum effects before and during the tunnelling, all tunnelling processes start exactly \\\"from nothing\\\" (as interpreted above), i.e. from a configuration which is consistent with the quantum properties of the following tunnelling process.\\n\\nTo conclude, we would like to explicitly distinguish between the analytical result for the tunnelling probability and the proposal to interpret the pre-tunnelling 14 -4 -2 0 2 4 0 0.2 0.4 0.6 0.8 1 -4.0 -2.0 0.0 β +2.0 +4.0 0.0 0.2 0.4 P 0.6 0.8 1.0 × -0.4 -0.2 0 0.2 0.4 0.0002 0.0004 0.0006 0.0008 0.001 0.001 2 3 × 0.001 1 3 × 0.001 0 -0.04 -0.02 0 +0.02 +0.04 α = 10.0 α = 7.0 α = 4.0 α = 3.0 α = 2.5 α = 2.0 α = 1.5 α = 1.0 × α = 0.5 α = 0.0 FIG. 6: Plots of the values of the tunnelling probability in a spacetime of dimension 5, as a function of β for fixed non-negative values of α as listed above. Again the detailed behavior for α = 0 around β = 0 is shown (to better see that the probability, although much smaller than in the previous cases, is still non-vanishing).\\n\\nconfigurations. Indeed the analytical result is a natural generalization of already existing calculations and it incorporates them as special cases. Its validity is completely independent from our interpretation of the before tunnelling configurations and it can represent a useful limit case to check the results of more elaborated models: for instance, a de Sitter-Schwarzschild shell configuration should reproduce, in the limit of vanishing Schwarzschild mass, our result for a tunnelling between a de Sitter space of assigned cosmological constant and a de Sitter space with vanishing cosmological constant, i.e. Minkowski space.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"We would like to thank M. C. Johnson and S. Sonego for useful discussions and constructive criticism. This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative research agreement #DF-FC02-94ER40818. The work of S. Ansoldi is supported in part by a grant from the Fulbright Commission.\\n\\nAPPENDIX A: PARAMETER SPACE\\n\\nIn this appendix we will elaborate about the classification of the possible junctions between two (anti-)de Sitter spacetimes. After switching to the dimensionless formulation (see subsection II D), we remain with two parameters, i.e. λ ± , or, which is the same, α and β. We will mostly use the latter quantities in the following analysis, since many relevant features of the solutions to Israel's junction condition related to the causal structure of the full spacetime manifold M are easily deducible from the comparison between the two cosmological constants. In particular, in figure 8 we give a classification of the possible solutions. The diagram shows the parameter space of the variables (α, β) and (λ + , λ -). The classification of the solutions using α and β, looks nicely symmetric with respect to the axis β = 0, for which λ + = λ -; this is the primary reason why we will mostly use the (α, β) parametrization in our discussion. The (λ + , λ -) axes can, anyway, be conveniently used to single out the de Sitter from the anti-de Sitter spacetime. In particular, we have defined four main groups of solutions:\\n\\ntype A) these are solutions in which both spacetimes joined across the brane are anti-de Sitter space-15 0 1 2 3 4 5 0.2 0.4 0.6 0.8 1 0 1 2 β 3 4 5 0.2 0.4 P 0.6 0.8 1.0 × × α = -0.50 α = -1.00 × α = -1.50 α = -2.00 α = -2.50 α = -3.00 α = -4.75 α = -6.50 α = -8.25 α = -10.00 0.25 0.5 0.75 1 1.25 1.5 0.002 0.004 0.006 0.008 0.01 0.0 0.5 1.0 1.5 0.01 0.006 0.002 × FIG. 7: Plots of the values of the tunnelling probability in a spacetime of dimension 5, as a function of β for fixed negative values of α as listed above. For α < -1/2, not all values of β are allowed. We have put the mark on each curve in correspondence of the minimum value of β for which (at the given value of α) the tunnelling process can exist and, the zoomed diagram, helps resolving the superposition that arises between the curves obtained for α = -0.50, -1.00, -1.50.\\n\\ntimes, i.e we have an AdS (-) -AdS (+) junction; this part of the parameter space is bounded by the parabola α = -(β 2 + 1)/2 (equivalently x -1 0 = 0), whose inside is the black region where no solution exists; type B) these solutions describe a junction of one part of de Sitter spacetime in the M -manifold with a part of anti-de Sitter spacetime in the M + manifold, i.e. they are dS (-) -AdS (+) junctions, and play the role of counterparts of the below discussed type D solutions; type C) these are junctions in which both spacetimes have the de Sitter geometry, i.e. we have dS (-) -dS (+) junctions; in some cases, qualitatively different diagrams may arise depending on which cosmological constant is the bigger, i.e. depending on the sign of β; type D) as anticipated above, this last type of solutions is similar to the type B, with the role of \\\"-\\\" and \\\"+\\\" interchanged; we thus have AdS (-) -dS (+) junctions.\\n\\nThe above information is not enough to completely characterize the classical spacetime obtained from the junction. In addition we need to know the behavior of the normal to the brane travelling in the spacetimes that we have determined from the diagram in figure (8) . According to our convention the normal to the brane has its tail-tip direction going from the \\\"-\\\" to the \\\"+\\\" parts of the full spacetime M; on the other hand in each of the two spacetimes M ± the corresponding signs ǫ ± determine if the normal to the brane points in the direction of increasing (ǫ = +1) or decreasing (ǫ = -1) radius. Using the results in (7) for ǫ ± we can subdivide the parameter\\n\\n16 λ + λ - α β A B C A A D C A FIG. 8:\\n\\nThe space of parameters is subdivided according to the different kinds of junctions, which can be between anti-de Sitter and anti-de Sitter spacetimes (type A), anti-de Sitter and de Sitter spacetimes (type B), de Sitter and anti-de Sitter spacetimes (type D) or de Sitter and de Sitter spacetimes (type C). The black part of the parameter space, identified by the condition 1 + 2α + β 2 < 0, singles out the values of the parameters for which tunnelling is not possible since the infinitely expanding solution does not exist.\\n\\nλ + λ - α β 1 1 1 2 2 3 3 3\\n\\nFIG. 9: The parameter space can be subdivided in three other regions (strips) according to the values of the signs ǫ±. In region 1 we have ǫ± = -1, in region 2 we have ǫ± = ∓1 and, finally, in region 3 we have ǫ± = +1.\\n\\nλ + λ - α β A1 B1 C1 A2 B2 C2 D2 C2 D3 C3 A3 FIG. 10: In principle we have to study 10 different cases, corresponding to as many different regions in the parameter space. These regions are obtained combining the classifications in figures 8 and 9, so that in each region there is a well defined type of junction with an unique choice for the signs.\\n\\nAs we will see each of the pairs A1 and A3, B1 and D3, B2 and D2, C1 and C3 will correspond to a distinct processes in spacetime; this matches well with the analytical result, in which the action turns out to be an even function of β, and gives only a total of 6 different possible processes.\\n\\nspace in three main regions, as in figure 9. These regions correspond to the following situations:\\n\\nregion 1) ǫ -= ǫ + = -1, so that in both spacetimes M ± the normal n a | ± to\\n\\nthe brane trajectory points in the direction of decreasing r;\\n\\nregion 2) ǫ -= +1 but ǫ + = -1, and in M -normal to the brane trajectory n a | -points in the direction of increasing r -but in M + the normal n a | + to the\\n\\nbrane trajectory points in the direction of decreasing r + ;\\n\\nregion 3) ǫ -= ǫ + = +1; thus in both spacetimes M ± the normals n a | ± to the brane trajectory point in the direction of increasing r ± .\\n\\nWe thus have various combinations of the geometries of the two spacetimes M ± according to the classification in figure 8 ; moreover we have to combine them choosing the part of spacetime on the correct side of the brane trajectory following the classification in figure 9 . This gives a total of ten subcases, which are summarized in figure 10 . The naming convention is the most natural, so that, for instance, the junction named B2 is a junction of type B, i.e. a dS (-) -AdS (+) junction, with the signs as in region 2, i.e. ǫ -= +1 and ǫ + = -1. All other names 17 follow the same convention and we thus obtain solutions of the following types A1, A2, A3, B1, B2, C1, C2, C3, D2, D3; their causal structure is detailed in the following appendix.\\n\\nAPPENDIX B: PENROSE DIAGRAMS & TUNNELLING\\n\\nHere, following the classification given in the previous appendix, we show the corresponding Penrose diagrams for the classical solutions, and a pictorial representation of the corresponding tunnelling processes. For each case we give four diagrams: 1. the complete spacetime from which we have to \\\"cut\\\" the M -part of the bulk (with the trajectory of the bubble and the associated normal) in the top left diagram; 2. the complete spacetime from which we have to \\\"cut\\\" the M + part of the bulk (with the trajectory of the bubble and the associated normal) in the top right diagram; 3. the junction, i.e. the full manifold M (again with the shell trajectory and the corresponding normal) in the bottom left diagram; 4. a pictorial representation of the creation of the brane via a tunnelling process, in the bottom right diagram; in this case we have used time translation invariance to set the \\\"tunnelling time\\\" at the coordinate time t = 0 so that the top half of the diagram represents the final state (i.e. the top half of the junction in the bottom left diagram) whereas the bottom half of the diagram is a representation (detailed below) of the pre-tunnelling configuration.\\n\\nWe would like to discuss preliminarily in more detail the way in which the pre-tunnelling configuration is constructed. We remember that we are considering non negligible quantum effects in the regime of the dynamics before the tunnelling. We will thus use for the signs the results coming from the modified junction (29), which are those in (34) . This gives a spacetime structure consisting of two regions (which can be of the de Sitter or of the anti-de Sitter type depending on the values of α and β) both bounded by r = 0 and by the shell radius r = R(τ ): this is the junction corresponding to the bounded solution of the modified potential (31), which is a brane starting from zero radius and expanding to a maximum radius (much smaller than any other length scale present in the problem) before recollapsing to r = 0. In the limit in which ā → 0, as discussed in section IV, this maximum radius tends, in fact, to zero. Nevertheless, in our representation of the spacetime before the tunnelling we have kept an arbitrarily finite size for the maximum radius, to make the diagram more readable. At the same time, we have slightly \\\"blurred\\\" it to make pictorially explicit that we are not dealing with a purely classical configuration but with a somehow \\\"heuristic\\\" representation of a spacetime where quantum effects are highly non trivial and certainly non negligible. We would like, anyway, to stress again that these quantum configurations will be the initial state of tunnelling processes that correspond to what in the literature has also been called \\\"tunnelling from nothing\\\" for instance [67] and references therein).\\n\\nWe will now present in detail the various kinds of junctions.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Type A\",\n \"text\": \"As discussed in main text (and with reference to figure 10 ), this class of junctions consists of the matching of two anti-de Sitter spacetimes with different cosmological constants. There are three possibilities corresponding to type A1, A2 and A3 junctions, which are shown, respectively, in figures 11, 12 and 13. r = 0 r = 0 AdS -, ǫ-= -1 AdS+, ǫ+ = -1 r = 0 r = 0 AdS + /AdS-Tunnelling FIG. 11: Case A1: AdS/AdS, ǫ-= ǫ+ = -1\\n\\nThe junction of type A1 (figure 11 ) has ǫ ± = -1, so in both anti-de Sitter spacetimes the normal to the brane points in the direction of decreasing radii. For this junction (and for all the ones in the 1 sector), β > 1, so that λ -is always greater than λ + . The classical junction is seen (by an observer travelling toward increasing values of the radius) as a transition from a more negative to a less negative value of the cosmological constant, when he/she crosses the brane. The global picture of the tunnelling process is then obtained according to the 18 prescription discussed in section IV. The pre-tunnelling spacetime has been discussed above and now consists of two \\\"small\\\" parts of anti-de Sitter spacetime with different cosmological constant. The tunnelling process shows the transition between a compact spacetime composed by two anti-de Sitter regions to a spacetime similar to anti-de Sitter, except for the fact that at some radius the cosmological constant changes its value. r = 0 r = 0 AdS -, ǫ-= +1 AdS+, ǫ+ = -1 r = 0 r = 0 r = 0 r = 0 AdS + /AdS-Tunnelling FIG. 12: Case A2: AdS/AdS, ǫ-= +1; ǫ+ = -1\\n\\nThe junction of type A2, is instead a junction where effectively a tiny junction of two anti-de Sitter spacetimes is \\\"inflated\\\" (figure 12 ). In this case the signs are given by ǫ ± = ∓1. Thus the normal to the brane points in the direction of increasing radii in the anti-de Sitter spacetime with cosmological constant λ -, but it points in the direction of decreasing radii in the anti-de Sitter spacetime with cosmological constant λ + . The net effect of the tunnelling process, which starts from the already discussed initial state, is thus to inflate the pre-tunnelling compact spacetime into a similar, but much larger, one (note that for graphical reasons the scales in the pretunnelling and post-tunnelling parts of the diagram are different; we remember that the maximum radius of the pre-tunnelling state will be much smaller than any other length scale in the problem). The various diagrams are in figure 12 .\\n\\nThe final type A diagram is the A3 one. This is very much as the A1 type, only that now ǫ ± = +1; in both spacetime the normal to the shell trajectory points in the direction of increasing radii. Thus (apparently) the role of M ± is interchanged, as shown in figure 13 where the dark and light gray parts looks complementary to r = 0 r = 0 AdS-, ǫ-= +1 AdS+, ǫ+ = +1 r = 0 r = 0 AdS+/AdS-Tunnelling FIG. 13: Case A3: AdS/AdS, ǫ-= ǫ+ = +1\\n\\nthose in figure 11 ; on the other hand, now we have β < -1 so that λ + > λ -. The global spacetime structure in the bottom left corner of figure 13 implies that an observer crossing the brane in the direction of increasing values of the radius, perceives again a transition from a more negative to a less negative value of the cosmological constant. So this diagram describes exactly the same process described by type A1, as it is possible to see from the diagram in figure 13.\\n\\n2. Type B\\n\\nThe type B solutions, B1 and B2, correspond to the junction of a part of a de Sitter spacetime, M -, with a part of anti-de Sitter spacetime, M + . Although the pre-tunnelling picture does not change very much, the configurations before the tunnelling are now de Sitteranti-de Sitter junctions.\\n\\nLet us first discuss the case B1, shown in figure 14 . In both spacetimes, the normal to the shell points in the direction of decreasing radii, since we have ǫ ± = -1.\\n\\nThe M -part of spacetime corresponds to the region of de Sitter complementary to the region between r = 0 and the brane's world-volume, whereas the anti-de Sitter part M + is the bounded region in the upper right picture of figure 14 . Thus the spacetime M shows a transition from a negative to a positive value of the cosmological constant for an observer crossing the shell in the direction of increasing radius. The configuration before the tunnelling has been described above; the tunnelling pro-19 r = 0 r = 0\\n\\nI - I + r = H - r = H - r = 0 dS-, ǫ-= -1 AdS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H - r = H - I + r = H - r = H - r = 0 r = 0\\n\\nAdS+/dS-Tunnelling FIG. 14: Case B1: dS/AdS, ǫ-= ǫ+ = -1 cess again \\\"inflates\\\" an anti-de Sitter-de Sitter junction with small volume to a much larger one, as shown in the bottom right diagram of figure 14 .\\n\\nr = 0 r = 0 I - I + r = H - r = H - r = 0 dS-, ǫ-= +1 AdS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H - r = H - r = 0 r = 0 I + r = H -\\n\\nAdS+/dS-Tunnelling FIG. 15: Case B2: dS/AdS, ǫ-= +1; ǫ+ = -1\\n\\nWe then come to the case B2, shown in figure 15 . Since in this case the signs which fix the orientation of the normals are ǫ ± = ∓1, the situation for M + is as in the previous case, but the M -part of spacetime changes because of the change in the sign of ǫ + . This gives for the junction the diagram in the bottom left part of figure 15 . The configuration before and after the tunnelling is shown in the bottom right corner of figure 15 . The situation is very similar to the one in the previous case.\\n\\n3. Type C Type C solutions are the \\\"de Sitter counterpart\\\" of type A solution. The main difference (not related to the spacetime structure) is that whereas for anti-de Sitter junctions the values of the cosmological constant are restricted (i.e. given two arbitrary values a junction might not exists) de Sitter junctions exists for all values of the cosmological constants λ ± (this is clear from figure 10 ). Please, also remember that now the pre-tunnelling diagram will consist of a junction of two de Sitter spacetime with different cosmological constants and with all the other properties described above. r = 0 r = 0\\n\\nI - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS-, ǫ-= -1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H + r = H + r = H - r = H - r = 0 r = 0 I + r = H + r = H -\\n\\ndS+/dS-Tunnelling FIG. 16: Case C1: dS/dS, ǫ-= ǫ+ = -1\\n\\nLet us start with case C1, shown in figure 16 . For this junction, the signs are ǫ ± = -1, so in both de Sitter spacetimes the normals to the brane point in the direction of decreasing radii. Moreover, since in this case β > 1, the cosmological constant in M -will be bigger than the cosmological constant in M + , so that the cosmological horizon H -will be smaller than H + . The diagram for the junction is shown in the bottom left part of figure 16 . It is 20 interesting to see some effects of the brane on the spacetime structure. There are observers that crossing the brane can move behind the cosmological horizon of M - without going through H -. They can also come out, by crossing the brane in the opposite direction. At the same time observers, by moving accurately and with proper timing, might end up behind the cosmological horizon of M + without crossing any horizon but using the shell as a \\\"gate\\\" 11 . As usual no additional considerations are required for the before tunnelling configuration. We note, instead, that the semiclassical transition has again the effect of \\\"inflating\\\" a small quantum spacetime region, as clearly shown in figure 16 .\\n\\nr = 0 r = 0 I - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS -, ǫ-= +1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + I - I + r = H - r = H - r = H + r = H + r = 0 r = 0 I + I + r = H - r = H + r = H + dS + /dS- Tunnelling\\n\\nFIG. 17: Case C2: dS/dS, ǫ-= +1; ǫ+ = -1\\n\\nWe now discuss case C2, in which the signs are ǫ ± = ∓1. Then the normal to the brane in the de Sitter space with cosmological constant λ -points in the direction of increasing radii, whereas the normal to the brane in the de Sitter space with cosmological constant λ + points in the direction of decreasing radii. We have again two regions of spacetime of the de Sitter type but with a different cosmological constant joined across the brane (bottom right part of figure 17 ). The full picture of the tunnelling process requires the consideration of the same initial configuration used in case C1 above. We also point 11 These considerations might be modified if we consider the influence of the observer on spacetime. To neglect this influence, as a first approximation, is fairly common in the literature to obtain some hints about the global spacetime structure.\\n\\nout that in this case, the sign of β is not fixed, so either of the cosmological constants λ ± may be the bigger. In particular, the junction in the bottom left corner of figure 17 shows the situation in which λ + < λ -, so that H + > H -.\\n\\nIn the bottom right part of figure 17 a representation of the tunnelling process is shown. r = 0 r = 0\\n\\nI - I + r = H - r = H - r = 0 r = 0 I - I + r = H + r = H + dS-, ǫ-= +1 dS+, ǫ+ = +1 r = 0 r = 0 I - I + r = H + r = H + r = H - r = H - r = 0 r = 0 I + r = H + r = H -\\n\\ndS+/dS-Tunnelling FIG. 18: Case C3: dS/dS, ǫ-= ǫ+ = +1\\n\\nThe last junction of the C type is C3, for which the usual set of diagrams is shown in figure 18 . As all C type ones, this is a junction between two de Sitter spacetimes and since β < -1 the relation between the cosmological constants is λ + > λ -, so that H + < H -. At this stage we might wonder if this junction is the same as C1 (mirroring what happens between the junctions of the A1 and A3 type). We will see that this is indeed the case. The normals now point in both spacetimes toward the direction of increasing radii (since ǫ ± = +1). Thus the junction is as in the bottom left part of figure 18 . When we consider also the configuration before tunnelling, we see that the diagram in the bottom right part of figure 16 is the \\\"switched color and reflected\\\" version of the one in the bottom right part of figure 18 : we remember now that the in the C1 case λ -in the light gray part was bigger than λ + in the dark side, but now exactly the opposite happens. Thus, despite the different coloring, case C3 represents the same physical situation of case C1. This observation makes interesting to complete the analysis for the remaining two cases, which is done in the following subsection. 21 4. Type D\\n\\nTo conclude the analysis of the global spacetime structure we have to analyze what happens for the last kind of junctions, those of type D. These are junctions between de Sitter and anti-de Sitter spacetimes, so that the pretunnelling configuration will also change accordingly, as we have discussed for the previous cases.\\n\\nr = 0 r = 0 r = 0 I - I + r = H + r = H + AdS-, ǫ-= +1 dS+, ǫ+ = -1 r = 0 r = 0 I - I + r = H + r = H + r = 0 r = 0 I + r = H + r = H +\\n\\ndS + /AdS-Tunnelling FIG. 19: Case D2: AdS/dS, ǫ-= +1;ǫ+ = -1\\n\\nThe case of the junction of type D2 is characterized by the following signs: ǫ ± = ∓1. This means that in the anti-de Sitter side the normal points in the direction of increasing radii, whereas in the de Sitter part, it points in the direction of decreasing radii. Thus the junction, shown in the bottom left corner of figure 19 , is the mirror image of the B2 case. Again, the considerations in section IV determine the geometry before the tunnelling. This brings, in complete analogy with the B2 case, to the picture of the tunnelling process given in the bottom right part of figure 19 ; again, apart from the different colors, this case is the same as B2 also in connection with the semiclassical tunnelling process.\\n\\nWe have thus only one case left, which is quickly dealt with, namely D3. It is now not difficult to anticipate that this process will turn out to be identical to B1, that appears in figure 14 . Indeed the signs are now ǫ ± = +1, so that in both spacetime the normals point in the direction of increasing radii. By the usual procedure we are thus led to the spacetime diagram shown in the bottom left part of figure 20 . Part of this diagram will constitute the final state of the tunnelling process. The initial state is obtained as usual and the picture of the tunnelling r = 0 r = 0 r = 0\\n\\nI - I + r = H + r = H + AdS-, ǫ-= +1 dS+, ǫ+ = +1 r = 0 r = 0 I - I + r = H + r = H + I + r = H + r = H + r = 0 r = 0\\n\\ndS+/AdS-Tunnelling FIG. 20: Case D3: AdS/dS, ǫ-= ǫ+ = +1\\n\\nprocess is as in the bottom right part of figure 20.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Comments\",\n \"text\": \"From the analysis developed so far we have seen that, although in principle we have ten different tunnelling processes, in fact process A1 is the same as A3, process B1 is the same as D3, process B2 is the same as A2 and process C1 is the same as C3. We are thus left with only six distinct processes. From \\\"symmetry consideration\\\", it is natural to notice that |β|, not β itself, is the relevant quantity (together with α) to classify the possible tunnelling configurations. We will thus not be surprised if the tunnelling amplitude will be a function of |β|, or, which is the same, an even function of β. We would also like to note already here that all the tunnelling processes can be interpreted as a very large expansion of a mall quantum region of spacetime.\\n\\n[1] S. Coleman, Phys. Rev. D 15, 2929 (1977) [Erratum-ibid. D 16, 1248 (1977)]. [2] J. C. G. Callan and S. Coleman, Phys. Rev. D 16, 1762 (1977).\\n\\n22 [3] S. Coleman and F. D. Luccia, Phys. Rev. D 21, 3305 (1980). [4] K. Lee and E. J. Weinberg, Phys. Rev. D 36, 1088 (1987). [5] S. W. Hawking, I. G. Moss, and J. M. Stewart, Phys. Rev. D 26, 2681 (1982). [6] W. Z. Chao, Phys. Rev. D 28, 1898 (1983). [7] K. Sato, M. Sasaki, H. Kodama, and K.-I. Maeda, Progr.\\n\\nTheor. Phys. 65, 1443 (1981). [8] H. Kodama, M. Sasaki, K. Sato, and K.-I. Maeda, Progr.\\n\\nTheor. Phys. 66, 2052 (1981). [9] K. Sato, Progr. Theor. Phys. 66, 2287 (1981). [10] K.-I. Maeda, K. Sato, M. Sasaki, and H. Kodama, Phys.\\n\\nLett. B 108, 98 (1982). [11] K. Sato, H. Kodama, M. Sasaki, and K.-I. Maeda, Phys.\\n\\nLett. B 108, 103 (1982). [12] H. Kodama, M. Sasaki, and K. Sato, Progr. Theor. Phys.\\n\\n68, 1979 (1982). [13] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys.\\n\\nRev. D 36, 2919 (1987). [14] S. K. Blau, E. I. Guendelman, and A. H. Guth, Phys.\\n\\nRev. D 35, 1747 (1987). [15] A. Aurilia, R. S. Kissack, R. Mann, and E. Spallucci, Phys. Rev. D 35, 2961 (1987). [16] E. Farhi, A. H. Guth, and J. Guven, Nucl. Phys. B 339, 417 (1990). [17] V. A. Berezin, V. A. Kuzmin, and I. I. Tkachev, Phys.\\n\\nRev. D 43, R3112 (1991). [18] M. Sasaki, T. Tanaka, K. Yamamoto, and J. Yokoyama, Phys. Lett. B 317, 510 (1993). [19] T. Tanaka, Nucl. Phys. B 556, 373 (1999). [20] A. Khvedelidze, G. V. Lavrelashvili, and T. Tanaka, Phys. Rev. D 62, 083501 (2000). [21] A. Vilenkin, Phys. Rev. D 27, 2848 (1983). [22] A. D. Linde, Lett. Nuovo Cim. 39, 401 (1984). [23] A. D. Linde, Sov. Phys. JETP 60, 211 (1984). [24] D. Stojkovic, G. D. Starkman, and R. Matsuo, arXiv:hepph/0703246 (2007). [25] K. V. Kuchar and M. P. J. Ryan, Phys. Rev. D 40, 3982 (1989). [26] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). [27] R. Bousso and S. W. Hawking, Phys. Rev. D 52, 5659 (1995). [28] R. Bousso and S. W. Hawking, Phys. Rev. D 54, 6312 (1996). [29] R. R. Caldwell, H. A. Chamblin, and G. W. Gibbons, Phys. Rev. D 53, 7103 (1996). [30] T. Vachaspati and M. Trodden, Phys. Rev. D 61, 023502 (2000). [31] J. Gariel and G. L. Denmat, Class. Quantum Grav. 16, 149 (1999). [32] P. Wang and X.-H. Meng, Class. Quantum Grav. 22, 283 (2005). [33] A. Aguirre and M. C. Johnson, Phys. Rev. D 72, 103525 (2005). [34] S. W. Hawking, Phys. Rev. D 18, 1747 (1978). [35] J. B. Hartle and S. W. Hawking, Phys. Rev. D 28, 2960 (1983). [36] A. Vilenkin, Phys. Rev. D 30, 509 (1984). [37] S. Carlip, Phys. Rev. D 46, 4387 (1992). [38] A. Vilenkin, Phys. Rev. D 58, 067301 (1998). [39] R. Bousso and A. Chamblin, Phys. Rev. D 59, 084004 (1999). [40] G. W. Gibbons, Class. Quantum Grav. 15, 2605 (1998).\\n\\n[41] A. Vilenkin, Phys. Lett. B 117, 25 (1982). [42] W. Fischler, D. Morgan, and J. Polchinski, Phys. Rev. D 41, 2638 (1990). [43] W. Fischler, D. Morgan, and J. Polchinski, Phys. Rev. D 42, 4042 (1990). [44] J. Garriga, Phys. Rev. D 49, 6327 (1994). [45] B. Freivogel, V. E. Hubeny, A. Maloney, R. C. Myers, M. Rangamani, and S. Shenker, JHEP 03, 007 (2006). [46] L. Anchordoqui, C. Nunez, and K. Olsen, JHEP 10, 50 (2000). [47] S. Kar, JHEP 10, 052 (2006). [48] S. Kar, Phys. Rev. D74, 126002 (2006). [49] S. J. Kolitch and D. M. Eardley, Phys. Rev. D 56, 4651 (1997). [50] S. J. Kolitch and D. M. Eardley, Phys. Rev. D 56, 4663 (1997). [51] Y. Nutku, M. B. Sheftel, and A. A. Malykh, Class. Quantum Grav. 14, L59 (1997). [52] A. Aurilia, H. Nicolai, and P. K. Townsend, Nucl. Phys. B 176, 509 (1980). [53] M. Henneaux and C. Teitelboim, Phys. Lett. B 143, 415 (1984). [54] A. Aurilia, G. Denardo, F. Legovini, and E. Spallucci, Phys. Lett. B 147, 258 (1984). [55] A. Aurilia, G. Denardo, F. Legovini, and E. Spallucci, Nucl. Phys. B 252, 523 (1985). [56] J. D. Brown and C. Teitelboim, Phys. Lett. B 195, 177 (1987). [57] J. D. Brown and C. Teitelboim, Nucl. Phys. B 297, 787 (1988). [58] F. Mellor and I. Moss, Class. Quantum Grav. 6, 1379 (1989). [59] J. L. Feng, J. March-Russel, S. Sethi, and F. Wilczek, Nucl. Phys. B 602, 307 (2001). [60] S. Ansoldi, A. Aurilia, and E. Spallucci, Phys. Rev. D 64, 025008 (2001). [61] A. Gomberoff, M. Henneaux, C. Teitelboim, and F. Wilczek, Phys. Rev. D 69, 083520 (2004). [62] J. Garriga and A. Megevand, Int. J. Th. Phys. 43, 883 (2004). [63] C. Barrabes and W. Israel, Phys. Rev. D 43, 1129 (1991). [64] W. Israel, Nuovo Cimento B44, 1 (1966) [Erratum-ibid.\\n\\nB48, 463 (1967)]. [65] R. Geroch and J. Traschen, Phys. Rev. D 36, 1017 (1987). [66] R. Maartens, Living. Rev. Relativity 7, 7 (2004); http://www.livingreviews.org/lrr-2004-7 . [67] S. Ansoldi, A. Aurilia, R. Balbinot, and E. Spallucci, Class. Quantum. Grav. 14, 2727 (1997). [68] P. Hajicek and J. Bicak, Phys. Rev. D 56, 4706 (1997). [69] J. L. Friedman, J. Louko, and S. N. Winters-Hilt, Phys.\\n\\nRev. D 56, 7674 (1997). [70] P. Hajicek and J. Kijowski, Phys. Rev. D 57, 914 (1998). [71] P. Hajicek, Phys. Rev. D 57, 936 (1998). [72] J. L. Friedman, J. Louko, and B. F. Whiting, Phys. Rev. D 57, 2279 (1998). [73] P. Hajicek, Phys. Rev. D 58, 084005 (1998). [74] P. Hajicek and J. Kijowski, Phys. Rev. D 61, 129901(E) (2000). [75] S. Mukohyama, Phys. Rev. D 65, 024028 (2002). [76] R. Capovilla, J. Guven, and E. Rojas, Class. Quantum Grav. 21, 5563 (2004). [77] C. Barrabes and W. Israel, Phys. Rev. D71, 064008 (2005), gr-qc/0502108.\\n\\n[78] G. L. Alberghi, R. Casadio, and G. Venturi, Phys. Rev. D 60, 124018 (1999). [79] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2752 (1977). [80] Misner, Thorne, and Wheeler, Gravitation (Freeman, 1973). [81] V. I. Arnold, Mathematical methods of classical mechanics (Springer Verlag, New York, 1978). [82] A. Aurilia, M. Palmer, and E. Spallucci, Phys. Rev. D 40, 2511 (1989). [83] S. Ansoldi and L. Sindoni (2004), gr-qc/0411042 (In the proceedings of the 6th International Symposium on Frontiers of Fundamental Physics (FFP6), Udine, Italy, 26-29 Sep 2004, published in Udine 2004, Frontiers of funda-mental physics 69-73). [84] S. Ansoldi (2007), gr-qc/0701117 (In the proceedings of the 16th Workshop on General Relativity and Gravitation (JGRG16), Niigata, Japan, 27 Nov -1 Dec 2006, Ken-ichi Oohara, Testuya Shiromizu, Kei-ichi Maeda and Misao Sasaki editors, Kyoto, (2007), page 114). [85] S. Ansoldi (2007), in preparation (to appear in the proceedings of \\\"From Quantum to Emergent Gravity: Theory and Phenomenology\\\", SISSA, Trieste, Italy, 11th-15th June 2007; http://pos.sissa.it ). [86] S. Parke, Phys. Lett. B 121, 313 (1983). [87] E. I. Guendelman and J. Portnoy, Class. Quantum Grav.\\n\\n16, 3315 (1999).\"\n }\n]","isConversational":false}}},{"rowIdx":189,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1079"},"arxiv_id":{"kind":"string","value":"0704.1079"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1126-6708/2007/06/011"},"content_hash":{"kind":"string","value":"07092b48df99e03a14bc7cb9c6dea5dc9b655146e584c77ad1809f8431641638"},"title":{"kind":"string","value":"Instanton Induced Neutrino Majorana Masses in CFT Orientifolds with\n MSSM-like spectra"},"abstract":{"kind":"string","value":"Recently it has been shown that string instanton effects may give rise to\nneutrino Majorana masses in certain classes of semi-realistic string\ncompactifications. In this paper we make a systematic search for supersymmetric\nMSSM-like Type II Gepner orientifold constructions admitting boundary states\nassociated with instantons giving rise to neutrino Majorana masses and other L-\nand/or B-violating operators. We analyze the zero mode structure of D-brane\ninstantons on general type II orientifold compactifications, and show that only\ninstantons with O(1) symmetry can have just the two zero modes required to\ncontribute to the 4d superpotential. We however discuss how the addition of\nfluxes and/or possible non-perturbative extensions of the orientifold\ncompactifications would allow also instantons with $Sp(2)$ and U(1) symmetries\nto generate such superpotentials. In the context of Gepner orientifolds with\nMSSM-like spectra, we find no models with O(1) instantons with just the\nrequired zero modes to generate a neutrino mass superpotential. On the other\nhand we find a number of models in one particular orientifold of the Gepner\nmodel $(2,4,22,22)$ with $Sp(2)$ instantons with a few extra uncharged\nnon-chiral zero modes which could be easily lifted by the mentioned effects. A\nfew more orientifold examples are also found under less stringent constraints\non the zero modes. This class of $Sp(2)$ instantons have the interesting\nproperty that R-parity conservation is automatic and the flavour structure of\nthe neutrino Majorana mass matrices has a simple factorized form."},"authors":{"kind":"list like","value":["L.E. Ibanez","A.N. Schellekens","A. M. Uranga"],"string":"[\n \"L.E. Ibanez\",\n \"A.N. Schellekens\",\n \"A. M. Uranga\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th","hep-ph"],"string":"[\n \"hep-th\",\n \"hep-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"String Theory, as the leading candidate for a unified theory of Particle Physics and Gravity, should be able to describe all observed particle phenomena. One the most valuable experimental pieces of information obtained in the last decade concerns neutrino masses. Indeed the evidence from solar, atmospheric, reactor and accelerator experiments indicates that neutrinos are massive. The simplest explanation of the smallness of neutrino masses is the see-saw mechanism [1] . The SM gauge symmetry allows for two types of operators bilinear on the neutrinos (with dimension ≤ 4) :\n\nL ν = M ab ν a R ν b R + h ab ν a R HL b (1.1)\n\nwhere ν R is the right-handed neutrino, L is the left-handed lepton doublet and H is the Higgs field. In supersymmetric theories, this term arise from a superpotential with the above structure, upon replacing fields by chiral superfields. If M ab is large, the lightest neutrino eigenvalues have masses\n\nM ν = < H > 2 h T M -1 h (1.2)\n\nFor M ∼ 10 10 -10 13 GeV and Dirac neutrino masses of order charged lepton masses, the eigenvalues are consistent with experimental results.\n\nWhat is the structure of neutrinos and their masses in string theory? In specific compactifications giving rise to the MSSM spectra singlet fields corresponding to righthanded neutrinos ν R generically appear. Dirac neutrino masses are also generically present but the required Majorana ν R masses are absent. This is because most MSSMlike models constructed to date have extra U(1) symmetries, under which the righthanded neutrinos are charged, which hence forbid such masses. In many models, such symmetries are associated to a U(1) B-L gauge boson beyond the SM. In order to argue for the existence of ν R masses, string model builders have searched for nonrenormalizable couplings of the type (ν R ν R NR NR ) with extra singlets N R . Once the latter fields get a vev, U(1) B-L is broken and a Majorana mass appears for the ν R .\n\nAlthough indeed such couplings (or similar ones with higher dimensions) exist in some semi-realistic compactifications, such a solution to the neutrino mass problem in string theory has two problems: 1) The typical ν R masses so generated tend to be too small due to the higher dimension of the involved operators and 2) The vevs for the N R fields breaks spontaneously R-parity so that dimension 4 operators potentially giving rise to fast proton decay are generated. This is in a nutshell the neutrino problem in string compactifications (see [2] for a recent discussion in heterotic setups). 1 In [3] (see also [4] ) two of the present authors pointed out that there is a builtin mechanism in string theory which may naturally give rise to Majorana masses for right-handed neutrinos. It was pointed out that string theory instantons may generate such masses through operators of the general form M string e -U ν R ν R . (1.3) Here U is a linear combination of closed string moduli whose imaginary part gets shifted under a U(1) B-L gauge transformation in such a way that the operator is fully gauge invariant. The exponential factor comes from the semi-classical contribution of a certain class of string instantons. This a pure stringy effect distinct from the familiar gauge instanton effects which give rise to couplings violating anomalous global symmetries like (B+L) in the SM. Here also (B-L) (which is anomaly-free) is violated.\n\nThis operator is generated due to existence of instanton fermionic zero modes which are charged under (B -L) and couple to the ν R chiral superfield. Although the effect can take place in different constructions, the most intuitive description may be obtained for the case of Type IIA CY orientifold compactifications with background D6-branes wrapping 3-cycles in the CY. In the simplest configurations one has four SM stacks of D6-branes labeled a, b, c, d which correspond to U(3), SU(2) (or U(2)), U(1) R and U(1) L gauge interactions respectively, which contain the SM group. One can construct compactifications with the MSSM particle spectrum in which quarks and leptons lie at the intersections of those SM D6-branes. Then the relevant instantons correspond to euclidean D2-branes wrapping 3-cycles in the CY (satisfying specific properties so as to lead to the appropriate superspace interaction). The D2-D6 intersections lie the additional fermionic zero modes which are charged under (B -L). For instantons with the appropriate number of intersections with the appropriate D6-branes, and with open string disk couplings among the zero modes and the ν R chiral multiplet (see fig.(3.1)), the operator in (1.3) is generated.\n\nThe fact that the complex modulus U transforms under U(1) B-L gauge transformations indicates that the U(1) B-L gauge boson gets a mass from a Stückelberg term.\n\nSo a crucial ingredient in the mechanism to generate non-perturbative masses for the ν R 's is that there should be massless U(1) B-L gauge boson which become massive by a Stückelberg term. It turns out that not many semi-realistic models with U(1) B-L mass from Stückelberg couplings have been constructed up to date. In the literature there are two main classes of RR tadpole free models with massive B-L. The first class are non-susy type IIa toroidal orientifold models first constructed in [5] . The second class are the type II Gepner orientifold models constructed by one of the present authors and 2 collaborators [6, 7] . The former were already considered in [3] . In the present paper we will concentrate on the RCFT Gepner model constructions, which lead to a large class of MSSM like models, more representative of the general Calabi-Yau compactifications (for a recent discussion of instanton-induced neutrino masses in a model with no RR tadpole cancellation, see [8] ).\n\nThe class of constructions in [6, 7] start with any of the 168 Type II compactifications obtained by tensoring N = 2 SCFT minimal models. In addition one can choose a number of modular invariant partition functions (MIPF), leading to a total of 5403. Then different consistent orientifold projections are performed on the different models. This yields a total of 49304 Type II orientifolds. The open string sector of the theory is defined in terms of the boundary states of the theory. Intuitively, they play the same role as D-branes wrapping cycles in the geometrical settings. Thus one associates boundary states a, b, c, d to the gauge groups giving rise to the SM. Different choices for the SM boundary states lead to different spectra. In the present paper we will make use of the data in [6] which contains 211634 different MSSM-like spectra (including also different hidden sectors). Although this number is huge, most of these models are really extensions of the MSSM, since they have either an extra U(1) B-L or SU(2) R × U(1) B-L group factor beyond the SM group. As we said, we are actually only interested in models in which the U(1) B-L gets a Stückelberg mass. Then we find that the number of MSSM-like models with these characteristics is dramatically reduced: only 0.18 percent of the models (391) have a massive U(1) B-L .\n\nAs we said, in the geometrical setting of IIA orientifolds with intersecting D6-branes [9, 10] (see [11] for reviews and [12, 13] for the IIB counterparts), instantons are associated to D2-branes wrapping 3-cycles, like the background D6-branes do. Analogously, in the RCFT setting the same class of boundary states appearing in the SM constructions are the ones corresponding to instantons. The zero modes on the instanton is computable from the overlaps of instanton brane boundary states (zero modes uncharged under the 4d gauge group) or of instanton and 4d spacefilling brane boundary states (zero modes charged under the corresponding gauge factor). We find that the criteria for a non-perturbative superpotential to be generated [14] are only fulfilled if the Chan-Paton (CP) symmetry of the instantons is O(1). For instantons with CP symmetry 1 Sp(2) or U(1) we find that there are a few extra uncharged fermionic zero modes which would preclude the formation of the searched superpotentials. On the other hand we argue that the addition of fluxes and/or possible non-perturbative 1 We adopt the convention that the fundamental representation of Sp(m) is m-dimensional.\n\nextensions of the orientifold compactifications would allow also instantons with Sp(2) and U(1) symmetries to generate such superpotentials. We thus include all O(1), Sp(2) and U(1) instantons 2 in our systematic search. The computation of charged and uncharged fermion zero modes may be easily implemented as a routine in a systematic computer search for instanton zero modes in Gepner MSSM-like orientifolds. Results of such a systematic computer search are presented in this article.\n\nWe find that out of the 391 models with massive U(1) B-L , there are very few admitting instantons with the required minimal O(1) CP symmetry, and in fact none of them without additional vector-like zero modes. On the other hand we do find 32 models admitting Sp(2) symmetric instantons with just the required charged zero mode content (and the minimal set of non-chiral fermion zero modes). In fact they are all variations of the same orientifold Gepner model based on the tensor product 2, 4, 22, 22) . These models all in fact correspond to the same MIPF and orientifold projection, they only differ on which particular boundary states corresponding to the four a, b, c, d SM 'stacks'. All models have the same chiral content, exactly that of the MSSM , with extra vectorlike chiral fields which may in principle become massive in different points of the CY moduli space. They have no hidden sector, i.e., the gauge group is just that of the SM. For each of the models there are 8 instantons with Sp(2) CP symmetry with just the correct charged zero mode structure allowing for the superpotential coupling (1.3) giving rise to ν R Majorana masses. As we said, they have extra uncharged fermion zero modes beyond the two required to generate a superpotentials. However one would expect that these unwanted zero modes might be lifted in more generic situations in which e.g. NS/RR fluxes are added.\n\n(k 1 , k 2 , k 3 , k 4 ) = (\n\nWe thus see that, starting with a 'large' landscape of 211634 MSSM-like models, and searching for instantons inducing neutrino masses, we find there are none admitting the O(1) instantons with exactly the required zero mode structure, and only few (32) examples with Sp(2) instantons with next-to-minimal uncharged zero mode structure (and exactly the correct charged zero modes). Let us emphasize though that it is the existence of massive U(1) B-L models which is rare. Starting with the subset of models with a massive U(1) B-L , finding models with correct instanton charged zero modes within that class is relatively frequent, 10 percent of the cases. Furthermore, we will see that there are further models beyond those 32 which contain extra non-chiral instanton zero modes and which may also be viable if these modes get massive by some 2 We refer to the different kinds of instanton by their Chan-Paton symmetry on their volume. Since we are not interested in gauge theory instantons, this notation should not be confusing.\n\n4 effect (like e.g. the presence of RR/NS fluxes).\n\nInstantons may generate some other interesting superpotential couplings in addition to ν R masses, some possibly beneficial and others potentially dangerous. In particular we find that in the models which contain Sp(2) instantons which might induce ν R masses, there are also other instantons which would give rise directly to the Weinberg operator [15] L W = λ M (LHLH) (1.4) Once the Higgs field gets a vev, this gives rise directly to left-handed neutrino masses.\n\nThus we find that in that class of models both the see-saw mechanism (which also gives rise to a contribution to the Weinberg operator) and an explicit Weinberg operator might contribute to the physical masses of neutrinos. Which effect dominates will depend on the relative size of the corresponding instanton actions as well as on the size of the string scale. Among potentially dangerous operators which might be generated stand the R-parity violating operators of dimension < 5, which might give rise e.g. to fast proton decay. We make an study of the possible generation of those, and find that in all models in which ν R masses might be generated R-parity is exactly conserved. This is a very encouraging result.\n\nA natural question to ask is whether one can say something about the structure of masses and mixings for neutrinos. As argued in [3] generically large mixing angles are expected, however to be more quantitative we also need to know the structure of Yukawa couplings for leptons. In principle those may be computed in CFT but in practice this type of computation has not yet been developed for CFT orientifolds.\n\nNevertheless we show that, in the case of instantons with Sp(2) CP symmetry, a certain factorization of the flavor structure takes place, which could naturally give rise to a hierarchical structure of eigenvalues for neutrino masses.\n\nThe structure of this article is as follows. In the next section we present a discussion of instanton induced superpotentials in Type II orientifolds. This discussion will apply both to Type IIA and Type IIB CY orientifolds as well as to more abstract CFT orientifolds. We discuss the structure of both uncharged and charged instanton zero modes. In particular we show that only instantons with O(1) CP symmetry have the appropriate uncharged zero mode content to induce a superpotential contribution. We also discuss how Sp(2) and U(1) might still generate superpotential contributions if extra ingredients are added to the general setting. In section 3 we apply that discussion to the specific case of the generation of ν R Majorana masses, showing what is the required zero mode structure in this case. We show how the flavor structure of the 5 Majorana mass term factorizes in the case of instantons with Sp(2) CP symmetry, leading potentially to a hierarchical structure of eigenvalues. We further discuss the generation of other B/L-violating operators including the generation of the Weinberg operator as well as R-parity violating couplings. In section 4 we review the RCFT Type II orientifold constructions in [6, 7] . A general discussion of zero fermion modes for instantons in RCFT orientifolds is presented in section 5.\n\nThe results of our general search for instantons generating ν R masses are presented in section 6. We provide a list of all Gepner orientifolds which admit instanton configurations potentially giving rise to ν R Majorana masses. We describe the structure of the models with Sp(2) instantons having the required charged zero modes for that superpotential to be generated. We also describe the boundary states of the corresponding instantons and provide the massless spectrum of the relevant MSSM-like models. Other orientifolds with zero mode structure close to the minimal one are also briefly discussed. A brief discussion about the possible generation of R-parity violating superpotentials is included. We leave section 7 for some final comments. Some notation on the CFT orientifold constructions, and a discussion of the CFT symmetries in the Sp(2) examples are provided in two appendices.\n\nAs this paper was ready for submission, we noticed [16, 17] , whose discussion of instanton zero modes partially overlaps with our analysis in Section 2.2.\n\nIn this Section we review the generation of superpotentials involving 4d charged fields via D-brane instantons in type II compactifications. The discussion applies both to type IIA and IIB models, and to geometrical compactification as well as to more abstract internal CFT's. For recent discussions on D-brane instantons, see [4, 3, 25, 8] .\n\nBefore starting, a notational remark in in order. Our notation is adapted to working in the covering theory, namely the type II compactification, and orientifolding in a further step. Thus we describe the brane configurations as a system of branes (described by boundary states for abstract CFT's), labeled k, and their orientifold images labeled k ′ . Similarly, we denote M the brane / boundary state corresponding to the instanton brane, and M ′ its orientifold image. If a brane is mapped to itself under the orientifold action, we call it a 'real' brane / boundary, and 'complex' otherwise. 6 2.1 D-brane instantons, gauge invariance and effective operators A basic feature of type II orientifold compactifications with D-branes is the generic presence of Stückelberg couplings between the U(1) gauge fields on the D-branes, and certain 4d RR closed string 2-forms. These couplings are required by the Green-Schwarz mechanism when the U(1)'s have non-zero triangle contributions to mixed anomalies [18, 19] , but can also exist for certain non-anomalous U(1)'s [5, 20]. These couplings make the U(1) gauge bosons massive, but the symmetry remains as a global symmetry exact in perturbation theory. Since the closed string moduli involved are scalars (0-forms) in the RR sector, the natural candidate non-perturbative effects to violate these U(1) symmetries are instantons arising from euclidean D-branes coupling to these fields.\n\nIn computing the spacetime effective interaction mediated by the instanton, one needs to integrate over the instanton zero modes. In the generic case (and in particular for the case of our interest) there are no bosonic zero modes beyond the universal ones (namely, the four translational bosonic zero modes associated to the position of the instanton). On the other hand, the structure of fermion zero modes will be crucial.\n\nSince we are interested in models with non-trivial 4d gauge group, arising from a set of background 4d spacetime filling branes, we consider separately fermion zero modes which are uncharged under the 4d gauge group and those which are charged. In this paper we restrict our discussion to 4d N = 1 supersymmetric models, and this will simplify the analysis of zero modes.\n\nFermion zero modes which are uncharged under the 4d gauge group determine the kind of 4d superspace interaction which is generated by the instanton. We are interested in generating superpotential interactions, which, as is familiar, requires the instanton to have two fermion zero modes to saturate the d 2 θ superspace integration.\n\nFor this, a necessary (but not sufficient!) condition is that the D-branes are half-BPS, so these fermion zero modes are the Goldstinos of the two broken supersymmetries.\n\nIn the string description, uncharged zero modes arise from open strings in the MM sector (in our notation, the one leading to adjoint representations), which in particular contain these Goldstinos, and the MM ′ sector (in our notation, the one leading to two-index symmetric or antisymmetric tensors). Note that both are the same for 'real' branes. Hence we are primarily interested in D-branes whose MM sector contains just two fermion zero modes, and whose MM ′ sector (for 'complex' branes) does not contain additional fermion zero modes.\n\nIn analogy with the familiar case of gauge theory instantons [21], charged fermion zero modes determine the violation of perturbative global symmetries by the instantoninduced interaction. Namely, in order to saturate the integration over the charged fermions zero modes, the spacetime interaction must contain insertions of fields charged under the 4d gauge symmetry, and in particular under the global U(1) factors, which are thus violated by the D-brane instanton. Notice that this holds irrespectively of the number of uncharged fermion zero modes, namely of the kind of superspace interaction induced by the instanton. Restricting to superpotential interactions, the resulting operator in the 4d effective action has roughly the form W inst = e -U Φ 1 . . . Φ n (2.1) Here the fields Φ 1 , . . . , Φ n are 4d N = 1 chiral multiplets charged under the 4d gauge group, and in particular also under the U(1) symmetries. Note also that the instanton amplitude contains a prefactor (which in general depends on closed and open string moduli) arising from the Gaussian path integral over (massive) fluctuations of the instanton (hence described by an open string annulus partition function, see [22, 23] for related work), which we can ignore for our purposes in this paper.\n\nFor D-brane instantons, U is the closed string modulus to which the euclidean Dbrane couples. In the D-brane picture, instanton fermion zero modes charged under the gauge factor carried by the k th stack of 4d space-filling branes (and its image k ′ ) arise from open strings in the Mk and Mk ′ sectors, transforming as usual in the ( M , k ) and ( M , k ) representations, respectively (with both related in the case of 'real' branes). The (net) number of instanton fermion zero modes with such charges is given by certain multiplicities 3 I M k , I M k ′ .\n\nA D-brane instanton, irrespectively of the superspace structure of the 4d interactions it may generate, violates U(1) k charge conservation by an amount I M k -I M k ′ for 'complex' branes and I M k for 'real' branes. In particular, this is the total charge of the field theory operator Φ 1 . . . Φ n in (2.1). From the Stückelberg couplings, it is possible to derive [3] (see [23, 24, 25] for related work, also [26]) 3 In geometric type IIA compactifications with 4d spacetime-filling branes and instanton branes given by D6-and D2-branes wrapped on Special Lagrangian 3-cycles, I Mk corresponds to the intersection number between the 3-cycles corresponding to the k th D6-and the D2-brane M (and similarly for I Mk ′ ). In geometric type IIB compactifications, it corresponds to the index of a suitable Dirac operator. In general (even for abstract CFT's) it can be defined as the Witten index for the 2d theory on the open string with the boundary conditions corresponding to the two relevant branes. We will often abuse language and refer to this quantity as intersection number, even in Section 6 where we work in the non-geometric regime of type IIB compactifications.\n\nthat for 'complex' instantons, gauge transformations of the U(1) k vector multiplets V k → V k + Λ k , transform U as U → U + k N k (I M k -I M k ′ )Λ k (2.2) For 'real' brane instantons, which were not considered in [3], the shift is given by 4 U → U + k N k I M k Λ k (2.3) (this new possibility will be an important point in our instanton scan in Section 6).\n\nThe complete interaction (2.1) is invariant under the U(1) gauge symmetries. However, from the viewpoint of the 4d low-energy effective field theory viewpoint, it leads to non-perturbative violations of the perturbative U(1) global symmetries, by the amount mentioned above.\n\nIn the string theory construction there is a simple microscopic explanation for the appearance of the insertions of the 4d charged fields (related to the mechanism in [27] ). The instanton brane action in general contains cubic terms α Φ γ, involving two instanton fermions zero modes α in the ( M , k ) and γ in the ( p , M ) coupling to the 4d spacetime field Φ in the ( k , p ) of the 4d gauge group 5 . Performing the Gaussian path integral over the instanton fermion zero modes leads to an insertion of Φ in the effective spacetime interaction. In general, and for a 'complex' instanton, there are several fermion zero modes α i , γ i in the fundamental (resp. antifundamental) of the instanton gauge group, coupling to a 4d spacetime chiral operators O ij (which could possibly be elementary charged fields, or composite chiral operators). Gaussian integration over the fermion zero modes leads to an insertion of the form det O (for 'real' brane instantons, det O should be interpreted as a Pfaffian). It is straightforward to derive our above statement on the net charge violation from this microscopic mechanism.\n\nNote that the above discussions show that instantons in different topological sectors (namely with different RR charges, and different intersection numbers with the 4d spacefilling branes) contribute to different 4d spacetime operators. In particular, multiwrapped instantons, if they exist as BPS objects, contribute to operators different 4 Equivalently, one may use (2.2), but must include an additional factor of 1/2 from the reduction of the volume for a real brane (which is invariant under the orientifold action). 5 Although there is no chirality in 0+0 dimension, the fermion zero modes α and γ are distinguished by their chirality with respect to the SO(4) global symmetry of the system (which corresponds to rotations in the 4d spacetime dimensions, longitudinal to the space-filling branes, and transverse to the instanton brane). Supersymmetry of the instantons constrains the couplings on the instanton action (such as the cubic couplings above) to have a holomorphic structure.\n\n9 from the singly wrapped instanton. This implies that the instanton expansion for a fixed operator is very convergent, and could even be finite.\n\nAnother important implication of the above discussion is that, in order to generate a specific operator via an instanton process, a necessary condition is that the instanton has an appropriate number and structure of charged zero modes. However, this is not sufficient. Insertions of 4d fields appear only if the fields couple to the instanton fermion zero modes via terms at most quadratic in the zero modes. In equivalent terms, only zero modes appearing in the Gaussian part of the instanton action can be saturated by insertions of 4d fields (those to which they couple). The requirement that the zero modes have appropriate couplings to the 4d fields may be non-trivial to verify in certain constructions. This is the case for the Gepner model orientifolds in coming sections, whose couplings are computable in principle, but unknown in practice. In such cases we will assume that any coupling which is not obviously forbidden by symmetries will be non-vanishing. Unfortunately there are no arguments to estimate the actual values of such non-vanishing couplings, hence we can argue about the existence of certain instanton induced operators, but not about the coefficients of such terms.\n\nIn this section we describe more concretely different kinds of instantons and the structure of interesting and unwanted zero modes. Our discussion will be valid for general D-brane models in perturbative type II orientifolds without closed string fluxes, although we also make some comments on more general F-theory vacua and the effects of fluxes. A more specific discussion is presented in Section 5.\n\nWe start discussing zero modes uncharged under the 4d gauge group. These are crucial in determining the kind of superspace interaction induced by the instanton on the 4d theory. In particular, we are interested in instantons contributing to the 4d superpotential, namely those which contain just two fermion zero modes in this sector. We are also interested in instantons which may contain additional fermion zero modes, and the possible mechanisms that can be used to lift them. Let us discuss 'real' and 'complex' brane instantons in turn.\n\n• Real brane instantons Real brane instantons correspond to branes which are mapped to themselves by the 10 orientifold action, hence M = M ′ . Uncharged zero modes arise from the MM open string sector. As discussed in Section 5, there is a universal sector of zero modes, in the sense that it is present in any BPS D-brane instanton, which we now describe. Before the orientifold projection, we have a gauge group U(n) on the volume of n coincident instantons. Notice that, although there are no gauge bosons in 0 + 0 dimensions, the gauge group is still well-defined, since it acts on charged states (open string ending on the instanton brane). There are four real bosonic zero modes and four fermion zero modes in the adjoint representation. For the minimal U(1) case, the four bosons are the translational Goldstones. The four fermions arise as follows. This sector is insensitive to the extra 4d spacefilling branes, and feels an accidental 4d N = 2 supersymmetry.\n\nThe BPS D-brane instanton breaks half of this, and leads to four Goldstinos, which are the described fermions 6 .\n\nThe orientifold projection acts on this universal sector as follows (see Section 5 for further discussion). The gauge group is projected down to orthogonal or symplectic.\n\nHence instanton branes with symplectic gauge group must have even multiplicity (a related argument, in terms of the orientifold action on Chan-Paton indices, is given in Section 5). For instantons with O(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes (related by the two supercharges of 4d N = 1 supersymmetry broken by the instanton) to the two-index symmetric representation, and the other two fermion zero modes (related by the other two supercharges of the accidental 4d N = 2 in this sector) to the antisymmetric representation. Hence for O(1) instantons (namely instantons with O(1) gauge group on their volume), we have just two fermion zero modes, which are the Goldstinos of 4d N = 1 supersymmetry, and the instanton can in principle contribute to the superpotential (if no additional zero modes arise from other non-universal sectors). For instantons with Sp(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes to the two-index antisymmetric representation, and the other two fermion zero modes to the symmetric representation. Hence, even for the minimal case of Sp(2)\n\ninstantons, we have just two fermion zero modes in the triplet representation, in addition to the two 4d N = 1 Goldstinos. Hence Sp(2) instantons cannot contribute to the superpotential in the absence of additional effects which lift these zero modes (see later) 7 .\n\n6 We thank F. Marchesano for discussions on this point. 7 For D-brane instantons corresponding to gauge instantons, the additional fermion zero modes in the universal sector couple to the boson and fermion zero modes from open strings stretched between the instanton and the 4d spacefilling brane. They act as Lagrange multipliers which impose 11 In addition to this universal sector, there exist in general additional modes, whose presence and number depends on the detailed structure of the branes. Namely, on the geometry of the brane in the 6d compact space in geometric compactifications, or on the boundary state of the internal CFT in more abstract setups. They lead to a number of boson and fermion zero modes in the symmetric or antisymmetric representation. The computation of these multiplicities in terms of the precise properties of the instanton branes is postponed to Section 5. In order to generate a superpotential, one must require these modes to be absent, except for the case of antisymmetrics of O(1) instantons, which are actually trivial.\n\nAn important point is that extra fermion zero modes (including the extra triplet fermion zero modes in the universal sector of Sp(2) instantons, and any two-index tensor fermion zero mode in the non-universal sectors) are in principle not protected against acquiring non-zero masses once the model is slightly modified. In other words, such fermions are non-chiral, in terms of the SO(4) chirality in footnote 5. One such modification is motion in the closed string moduli space, which can lift the nonuniversal modes if there are non-trivial couplings between them and closed string moduli (unfortunately, such correlators are difficult to compute, even in cases where the CFT is exactly solvable, like the Gepner models). Note that extra zero modes in the universal sector of Sp(2) instantons cannot be lifted by this effect, since it does not break the accidental 4d N = 2 in this sector. A second possible modification which in general can lift extra zero modes is the addition of fluxes, generalizing the results for D3-brane instantons in geometric compactifications [29] (for non-geometric CFT compactifications, we also expect a similar effect, once fluxes are introduced following [30] ). Note that fluxes can lift extra zero modes in the universal sector as well, since they can break the accidental 4d N = 2 susy in this sector. A last mechanism arising in more general F-theory compactifications and discussed below for complex instantons, is valid for real instanton branes as well.\n\nThe bottom line is that in the absence of such extra effects, only O(1) instantons can contribute to superpotential terms. However, in modifications of the model such extra effects can easily lift the extra fermion zero modes. Hence, this kind of extra vector-like zero modes will not be considered catastrophic, and real instantons (including the O(1) and Sp(2) cases) with such zero modes are considered in our scan in Section 6.\n\nthe fermionic constraints in the ADHM construction [28] , and may not spoil the generation of a superpotential.\n\n• Complex brane instantons Zero modes uncharged under the 4d gauge group can arise from the MM and MM ′ open string sectors. Notice that the orientifold action maps the MM sector to the M ′ M ′ , hence we simply discuss the former and impose no projection. The discussion of the MM sector is as for real brane instantons before the orientifold projection. The universal sector leads to a U(n) gauge symmetry, and four bosonic and four fermionic zero modes in the adjoint representation. The bosons are translational Goldstones, while the fermions are Goldstinos of the accidental 4d N = 2 present in this sector. Hence, even in the minimal case of U(1) brane instantons there are two extra fermion zero modes, beyond the two fermion zero modes corresponding to the 4d N = 1 Goldstinos. Hence U(1) instanton (except for those corresponding to gauge instantons, see footnote 7) cannot contribute to superpotential terms in the absence of additional effects, like closed string fluxes . However, keeping in mind the possibility of additional effects lifting them in modifications of the model, we include them in the discussion. Also, in what follows we will use the U(n) notation for the different fields to keep track of the Chan-Paton index structure.\n\nThe above statement would seem in contradiction 8 with computations of nonperturbative superpotentials [14] induced by M5-branes instantons in M-theory compactifications on Calabi-Yau fourfolds, which are dual to D3-brane instantons (with world-volume U(1) gauge group) on type IIB compactifications. The resolution is that the M5-branes that contribute are intersected by different (p, q) degenerations of the elliptic fiber. This implies that U(1) D3-brane instanton only contribute if they are intersected by mutually non-local (p, q) 7-branes. The two extra fermion zero modes exist locally on the D3-brane volume, but cannot be defined globally due to the 7-brane monodromies. Hence such effect can take place only on non-perturbative type IIB compactifications including (p, q) 7-branes. Note that in perturbative compactifications, namely IIB orientifolds, the (p, q) 7-branes are hidden inside orientifold planes [31] with their monodromy encoding the orientifold projection; hence the only branes that can contribute to the superpotential are real branes, for which the projection/monodromy removes the extra fermion zero modes, as discussed above.\n\nIn addition to this universal sector, the MM sector may contain a non-universal set of fermions and bosons, in the adjoint representation (hence uncharged under U(1)).\n\nThey depend on the specific properties of the brane instanton, and will be discussed in Section 5. These additional zero modes should be absent in order for the instanton 8 We thank S. Kachru for discussions on the ideas in this paragraph.\n\nto induce a non-trivial superpotential. Notice however that these zero modes are uncharged under any gauge symmetry, and hence vector-like. Thus, there could be lifted in modifications of the model, as discussed for real instantons.\n\nThe MM ′ sector is mapped to itself under the orientifold action. Hence it leads to a number of bosons and fermions in the two index symmetric or antisymmetric representations. Notice that the two-index antisymmetric representation is trivial for U(1), so these modes are actually not present. On the other hand, fermion zero modes in the two-index symmetric representation are chiral and charged under the brane instanton gauge symmetry. Hence they cannot be lifted by any of the familiar mechanisms, and thus spoil the appearance of a non-perturbative superpotential, even if the model is modified. Such fermion zero modes are considered catastrophic and we will look for models avoiding them in our scan in Section 6.\n\n• Real brane instantons Instanton zero modes charged under the 4d gauge group arise from Mk open string sectors (and their image Mk ′ ). In the generic case, there are no scalar zero modes in these sectors. This is because in mixed Mk open string sectors the 4d spacetime part leads to DN boundary conditions, which already saturate the vacuum energy in the NS sector. Only in the special case where the internal structure of the spacetime filling brane k and the instanton brane are the same, there may be NS ground states of the internal CFT which do not contribute extra vacuum energy, hence leading to massless scalars. However, this corresponds to brane instantons which can be interpreted as instantons of the 4d gauge theory on the 4d space-filling branes. The instantons we are interested in for the generation of neutrino Majorana mass terms are not of this kind [3] (see e.g. [32, 33, 28, 25, 34] for discussions on gauge theory instantons from D-brane instantons).\n\nHence we focus on charged fermion zero modes, which are generically present in any mixed Mk sector. Let us define L M k , L M k ′ the (positive by definition) number of left-handed chiral fermion zero modes in the representations\n\n( M , k ), ( M , k ),\n\nrespectively. The net number of chiral fermion zero modes in the ( M , k ) is given\n\nby I M k = L M k ′ -L M k .\n\nThis controls the violation of the U(1) a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φ kp and/or Φ kp ′ .\n\nIn addition, there are P M k = min(L M k ′ , L M k ) vector-like pairs of fermion zero 14 modes. Since they are vector-like, they may be lifted by slight modifications of the model, like moving in the closed string moduli space, or by introducing additional ingredients, like fluxes. In addition, they may be lifted by moving in the open string moduli space of the 4d spacefilling branes, as follows. The zero modes may lead to insertions of 4d fields Φ kk , if the kk sector contains such 4d chiral multiplets (or to insertions of composite 4d operators in the adjoint of the k th 4d gauge factor), and if they couple to the zero modes. Although this may not be generically not the case, many of our models in coming section contain such adjoint fields. Hence, a nontrivial vev for the latter can lift these extra vector-like zero modes, hence leading to instanton generating the superpotential of interest. Given these diverse mechanisms to lift these zero modes, their presence of such zero modes is thus unwanted, but again not necessarily catastrophic.\n\nOne last comment, related to the concrete kind of instanton search we will be interested in. Namely, we will be searching for instantons leading to a specific operator, carrying non-trivial charges under a specific set of 4d gauge factors. Postponing the detailed discussion to Sections 3.1, 4.2 , let us denote a, b, c, d the set of branes leading to a field theory sector, denoted 'observable' (and which reproduces the SM in our examples). We will require the instanton to have a prescribed number of chiral fermion zero modes charged under these branes, namely we require the intersection numbers of the instanton with these branes I M a , . . . , I M d to have specific values (as mentioned above, in the most restrictive scan we forbid vector-like pairs of zero modes under these branes). In addition, the model in general contains an additional sector of branes, denoted 'hidden' (since there is zero net number of chiral multiplets charged under both sectors) and labeled h i , required to fulfill the RR tadpole cancellation conditions. In general there may be instanton fermion zero modes from e.g. the Mh 1 , h 2 M sectors, which would contribute to insertions of the 4d fields in the h 1 h 2 sector if there are suitable cubic couplings. These extra insertions could be avoided if such 4d fields in the hidden sector acquire vevs (note that vevs for the (vector-like) fields charged under the visible and hidden sectors would typically break hypercharge, and should be avoided), and hence lift the zero modes. Equivalently, from the 4d perspective, the unwanted extra h 1 h 2 field insertion is replaced by its vev. However, this renders the discussion very model dependent. Moreover, the possibility of hidden brane recombination was not included in the search for SM-like models in [6, 7] (namely, the possibility of allowing for chiral fields charged under the observable and hidden gauge groups, which may become non-chiral and possibly massive upon hidden brane 15 recombination). Hence we will consider these chiral fermion zero modes as unwanted (as usual, non-chiral modes are unwanted but not catastrophic, hence they are allowed for in a more relaxed search).\n\nThe discussion of 'complex' brane instantons is somewhat analogous to the previous one, with the only complication that the brane M and its image M ′ lead to independent modes, leading to a more involved pattern of fermion zero modes. Instanton zero modes charged under the 4d gauge group arise from the Mk,Mk ′ and related sectors. As for 'real' brane instantons, there are generically no scalars in these sectors (and certainly not in our case of interest). Hence we focus on charged fermion zero modes, which are generically present in any mixed sector.\n\nIn contrast with 'real' brane instantons, a net combination of fermion zero modes in the ( M , k ) + ( M , k ) is not vector-like, but chiral under the instanton gauge symmetry. Such a pair cannot therefore be lifted even by modifications of the theory.\n\nIn general, if the instanton has a mismatch in the total numbers n α , n γ of fermion zero modes α i in the M and γ j in the M , the instanton amplitude automatically vanishes.\n\nNamely, the matrix of operators O ij coupling to the zero modes necessarily has rank at most min(n α , n γ ). That is , if n α > n γ there are linear combinations of the α i which do not couple, and cannot lead to insertions. Moreover, they are not liftable by the familiar mechanisms 9 , thus in our instanton search in Section 6 such excess zero modes are forbidden even in relaxed scans.\n\nLet us thus discuss a sector of fermion zero modes with equal number n α = n γ .\n\nConsidering a given 4d space-filling brane k, let us denote\n\nL M k , L M ′ k ′ , L M k ′ , L M ′ k the\n\n(positive by definition) number of left-handed chiral fermion zero modes in the repre-\n\nsentations ( M , k ), ( M , k ), ( M , k ), and ( M , k ) respectively. The net number of chiral fermion zero modes in the ( M , k ) and ( M , k ) is given by I M k = L M k ′ -L M ′ k ′ and I M k ′ = L M k -L M ′ k , respectively\n\n. This net number of fermions zero modes controls the violation of the U(1) a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp, Mk ′ , Mp ′ sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φ kp and/or Φ kp ′ .\n\n9 Note that such a mismatch is always correlated with the existence of extra chiral zero modes in the M M ′ sectors, discussed above. Denoting Q a , Q orient the vector of RR charges of the 4d spacefilling branes and orientifold planes, they satisfy the RR tadpole conditions a\n\nN a Q a + a ′ N a Q a ′ + Q orient. = 0.\n\nBy taking the 'intersection' bilinear with the RR charges Q M of the brane instanton, we have I Ma +I Ma ′ +I M,orient = 0. This implies that the number of fundamentals minus anti-fundamentals of the instanton gauge group is related to the number of two-index tensors.\n\nThe remaining fields in this sector are vector-like pairs, in the ( M , k ) + ( M , k ) or the ( M , k ) + ( M , k ), which in principle lead to the vanishing of the instanton amplitude, but which can be lifted by additional effects (motion in closed or open string moduli space, or addition of fluxes), in a way consistent with the gauge symmetries in 4d spacetime and on the instanton. Just like for 'real' brane instantons, we conclude by commenting on our concrete instanton search in models with a set of visible branes a, b, c, d and a set of hidden branes h i . The requirement that the instanton leads to an operator with specific charges under the visible branes fixes the values of the quantities I M a -I M a ′ , etc. As we described for real branes, one may still have fermion zero modes charged under the hidden sector branes, but they lead to additional insertions, hence we rather focus on\n\ninstantons with I M h i -I M h ′ j = 0.\n\nThe two kinds of conditions, on intersection numbers with visible and hidden branes, still leave the possibility of combinations of fermion zero modes of the kind\n\n( M , k ) + ( M , k ), which do not contribute to I M k , or of the kind ( M , k ) + ( M , k ), which does not contribute to I M k ′ .\n\nautomatically vector-like, and thus may be lifted in modifications of the theory. But the condition also allow combinations like\n\n( M , k ) + ( M , k ), which exploit a cancellation between I M k and I M k ′ (as also does ( M , k ) + ( M , k )\n\n). Such combinations are chiral by themselves, and in general imply a mismatch of modes in the M and the M . The total mismatch can be arranged to vanish using combinations of the kind\n\n( M , k ) + ( M , k ) and ( M , p ) + ( M , p )\n\nfor different branes. However, the only way to lift these pairs is by breaking the gauge symmetry on the 4d space-filling branes k and p. This can be done without damage to the visible sector if these are hidden branes, but this corresponds to the recombination of hidden branes that, as mentioned already, we are not going to consider. Hence only vector-like pairs with respect to each brane are considered to be liftable in simple modifications of the theory. In our instanton search, these are the only additional fermion zero modes which we allow in relaxed scans (but they are clearly not allowed for in restricted scans)\n\nIn this Section we discuss the possible physical effects of D-brane instantons in string models with SM-like spectrum. In particular we describe the conditions to generate right-handed neutrino Majorana masses. We also comment on other possible B and/or L violating operators that can be generated by instantons. In this section we will again\n\n17 ν Sp(2),U(2) R L L L R E L Q U , D R R W gluon U(1) U(1) U( 3\n\n)\n\nd-Leptonic a-Baryonic b-Left c-Right R Figure 1: Quarks and leptons at intersecting branes use the geometrical language of IIA intersecting D-branes but it should be clear that our discussion equally applies to general CFT orientifolds like the ones presented in the next section.\n\nLet us now specify the discussion in the previous section to the case of the generation of a right-handed neutrino mass term. In order to do that we need some realistic orientifold construction with the gauge group and fermion spectrum of the Standard Model (SM). In the context of Type II orientifolds perhaps the most economical brane configuration leading to a SM spectrum is the one first considered in [5] . This consists of four stacks, labelled a, b, c, d. The gauge factor on branes a is U(3), and contains QCD and baryon number. The d factor is U(1) d , and corresponds to lepton number.\n\nStack b contains SU(2) Weak either embedded in U(2) or Sp(2). Finally brane c can either provide a U(1) or an O(2) factor. In the brane intersection language, the chiral fermions of the SM live at the intersections of these branes, as depicted in Fig. 3.1.\n\nThe U(1) Y factor of the standard model is embedded in the Chan-Paton factors of branes a,c and d as\n\nY = 1 6 Q a - 1 2 Q c - 1 2 Q d = 1 2 (Q B-L -Q R ) (3.1)\n\nwhere Q x denotes the generator of the U(1) of brane stack x (in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator).\n\nNote that in this convention the Q d generator appears with sign opposite to other conventions in the literature, e.g. in [3] . In addition to Y these models have two 18 additional U(1) gauge symmetries:\n\nQ anom = 3Q a + Q d = 9Q B + Q L Y ′ = 1 3 Q a + Q c -Q d = Q B-L + Q R (3.2)\n\nThe first is anomalous whereas the second, which we will call B -L (with a slight abuse of language, since it is in fact a linear combination of B -L and hypercharge), is anomaly free. In models in which the electroweak gauge group is embedded in U(2), rather than in Sp(2), there is a second anomalous U(1) b . The charges of the SM particles under these U(1) symmetries are given in table 1.\n\nIntersection D = 4 fields/ zero modes\n\nQ a Q c Q d Y Q M (ab),(ab') Q L 3(3, 2) 1 0 0 1/6 0 (ca) U R 3( 3, 1) -1 1 0 -2/3 0 (c'a) D R 3( 3, 1) -1 -1 0 1/3 0 (db),(db') L 3(1, 2) 0 0 1 -1/2 0 (c'd) E R 3(1, 1) 0 -1 -1 1 0 (cd) ν R 3(1, 1) 0 1 -1 0 0 (Mc) α i 2(0, 0) 0 -1 0 1/2 1 (dM) γ i 2(0, 0) 0 0 1 -1/2 -1\n\nTable 1: Standard model spectrum and U (1) charges of particles and zero modes. Q M stands for the world-volume gauge symmetry in the case of U (1) complex instantons.\n\nThe U(1) k gauge symmetries have couplings with the RR 2-forms B r of the model, as follows\n\nS BF = k,r N k (p kr -p k ′ r ) 4d B r ∧ F k (3.3) where p kr , p k ′ ,r\n\nare given by the RR charges of the D-branes. These imply that under\n\na U(1) k gauge transformation A k → A k + dΛ k the scalar a r dual to B r transforms as a r → a r + k N k (p kr -p k ′ r ) Λ k (3.4)\n\nThis has two effects: 1) The linear combination of axion fields r (p kr -p k ′ ,r )a r is eaten up by the U(1) k massless gauge boson, making it massive. 2) For anomalous U(1) k , the anomalies cancel through a 4d version of the Green-Schwarz mechanism.\n\nThis works due to the existence of appropriate a r F ∧ F couplings, involving the gauge fields in the theory.\n\nIt is obvious that all anomalous U(1)'s become massive by this mechanism. However it is important to realize [5] that gauge bosons of anomaly-free symmetries like U(1) B-L may also become massive by combining with a linear combination of axions. This is 19 interesting since it provides a mechanism to reduce the gauge symmetry of the model without needing explicit extra Higgsing. In the models in which U(1) B-L becomes massive in this way, the gauge group left over is purely that of the SM. Moreover, we will see that having (B-L) massive by this Stückelberg mechanism is crucial to allow the generation of instanton-induced Majorana neutrino masses.\n\nNote that the B ∧ F couplings may also be potentially dangerous, since in principle they could also exist for hypercharge, removing U(1) Y from the low-energy spectrum.\n\nAs we will see in our RCFT examples later on, having massless U(1) Y but massive U(1) B-L turns out to be a strong constraint in model building.\n\nAs discussed in the previous section, string instantons can give rise to non-perturbative superpotentials breaking explicitly the perturbative global U(1) symmetries left-over from U(1) gauge bosons made massive through the Stückelberg mechanism. The kind of operator we are interested in has the form\n\nW ≃ e -S ins ν R ν R (3.5)\n\nwhere ν R is the right-handed neutrino superfield 10 . Here S ins transforms under both U(1) B-L and U(1) R in such a way that the overall operator is gauge invariant. This operator may be created if the mixed open string sectors lead to fermionic zero modes α i , γ i , i = 1, 2, appropriately charged under the 4d gauge factors. As we discussed in the previous section, to generate a superpotential one needs instanton with O(1) Chan-Paton symmetry, in order to lead to two uncharged fermion zero modes to saturate the d 2 θ 4d superspace integration. On the other hand, as we argued, instantons with Sp(2) or U(1) CP symmetries may also induce the required superpotentials if there is some additional dynamics getting rid of the extra uncharged zero modes which in principle appear in instantons with these symmetries. We thus consider all O(1), Sp(2) and U(1) instantons in our discussion.\n\nIn order to to get a ν R bilinear, the intersection numbers of instanton M and d, c branes are as follows Sp(2) case :\n\nI M c = 1 ; I M d = -1 (3.6)\n\n(since there is an extra multiplicity from the two branes required to produce Sp(2)) O(1) case :\n\nI M c = 2 ; I M d = -2 (3.7)\n\n10 Actually we denote by ν R the left-handed ν c L field following the usual (a bit confusing) convention.\n\n20 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 D2 D6 D6 ν R γ α Figure 2: Disk amplitude coupling two charged zero modes to ν R in the geometrical Type IIA intersecting brane approach.\n\nU(1) case :\n\nI M c = 2 ; I M d = -2 or I M d ′ = 2 ; I M c ′ = -2 (3.8)\n\nFurthermore there must be cubic couplings involving the right-handed neutrino superfield ν a in the a th family and the fermionic zero modes α i , γ j\n\nL cubic ∝ d ij a (α i ν a γ j ) , a = 1, 2, 3 (3.9)\n\nIn type IIA geometric compactifications, this coupling arises from open string disk instantons, see Fig. 3.1. In general type IIA models (resp. IIB models), the coefficients d ij a depend on the Kähler (resp. complex structure) moduli, and possibly on open string moduli. In simple CFT models (like e.g. in toroidal cases) these quantities may be in principle explicitly computed.\n\nThese trilinear couplings appear in the instanton action and after integration of the fermionic zero modes α i , γ i one gets a superpotential coupling proportional to\n\nd 2 θ d 2 α d 2 γ e -d ij a (α i ν a γ j ) = d 2 θ ν a ν b ( ǫ ij ǫ kl d ik a d jl b ) (3. 10\n\n) yielding a right-handed neutrino mass term. This term is multiplied by the exponential of the instanton euclidean action so that the final result for the right-handed neutrino mass (up to a 1-loop prefactor) has the form\n\nM R ab = M s ( ǫ ij ǫ kl d ik a d jl b ) exp( - V Π M g s + i r q M,r a r ) (3.11)\n\nFor geometric compactifications V Π M is roughly related to the wrapped volume. We keep the same notation to emphasize that the effect is non-perturbative in g s . In supersymmetric models the term in the exponential is the linear combination U of complex structure moduli to which the instanton D-brane couples, as described in the previous section. As explained, the gauge U(1) c , U(1) d transformation of the bilinear 21 piece and the e -S D2 factor nicely cancel. Note that from the viewpoint of the 4d SM effective field theory, the instanton has generated a Majorana neutrino mass violating B -L. Notice also that since this symmetry is non-anomalous, its violation cannot be associated to a gauge instanton, hence this is a pure string theory instanton effect.\n\nIn order to extract more specific results for the flavor structure of the obtained Majorana mass operator, one needs to know more details about the quantities d ij a coming from the disk correlators. However in the particular case of Sp(2) instantons, the labels i, j are Sp(2) doublet indices, and the symmetry requires d ij a = d a ǫ ij . The mass matrix for the three neutrinos is given by M R ab =2M s d a d b exp (-U), so that the flavour dependence on a, b = 1, 2, 3 factorizes. More generally, as we will see in our RCFT search in Section 6, there are typically several different instantons contributing to the amplitude, so that we actually have a result for the mass\n\nM R ab = 2M s r d (r) a d (r) b e -Ur (3.12)\n\nwhere the sum goes over the different contributing instantons. One thus has a structure of the form\n\nM R ∼ r e -Ur diag (d (r) 1 , d (r) 2 , d (r) 3 ) • 1 1 1 1 1 1 1 1 1 • diag (d (r) 1 , d (r) 2 , d\n\n(r) 3 ) . (3.13) This structure is very interesting. Indeed, each instanton makes one particular (instantondependent) linear combination of the neutrinos massive, leaving two linear combinations massless. Hence, for three or more instantons, one generically has a matrix with three non-zero eigenvalues. It is easy to imagine a hierarchical structure among the three eigenvalues if e.g. the exponential suppression factors exp(-Re U r ) are different for each instanton.\n\nOur main focus in this paper is on the generation of right-handed neutrino Majorana masses. However instantons may induce other L-and B-violating operators which we briefly summarize in this subsection. 22\n\nA right-handed neutrino Majorana mass term is not the only possible operator violating lepton number. Instanton effects may also give rise to dimension 5 operators not involving ν R . Specifically, the Weinberg operator L W = λ M (LHLH) . (3.14) might be generated. Once Higgs fields get a vev v this operator gives rise directly to Majorana masses for the left-handed neutrinos of order ≃ v 2 /M. Indeed, it is easy to check that in this case the required instanton M must verify Sp(2) case :\n\nI M c = -1 ; I M d = 1 (3.15) O(1) case : I M c = -2 ; I M d = 2 (3.16) U(1) case : I M c = -2 ; I M d = 2 or I M c ′ = 2 ; I M d ′ = -2 ( 3\n\n.17) (here we are assuming SU(2) weak to be embedded in an Sp(2)). Note that these intersection numbers are different to those giving rise to ν R mass terms. In particular they lead to a transformation under B -L opposite to that of ν R mass operators 11 .\n\nIn the present case there are altogether four fermionic zero modes α i ,γ i corresponding to the intersections of the instanton M with the branes c, d. These zero modes can have couplings involving the left-handed leptons L and the u-type Higgs multiplet H\n\nL disk ∝ c ij a (α i (L a H)γ j ) . (3.18)\n\nUpon integration over the fermionic zero modes one recovers the Weinberg operator.\n\nIn the present case the scale M of the Weinberg operator will be the string scale M s and the coupling λ ≃ exp(-S ins ). Again, in the particular case of Sp(2) instantons the situation simplifies (c ij a = c a ǫ ij ) and one gets left-handed neutrino Majorana masses\n\nM L ab = < H > 2 M s r 2c (r) a c (r) b e -Sr (3.19)\n\nwhere r runs over the different contributing instantons and S r is their corresponding action. The flavour structure of this left-handed neutrino mass matrix is the same as in eq.(3.13) and again may potentially lead to a hierarchical structure of left-handed neutrino masses, as is experimentally observed.\n\n11 Instantons with these intersection numbers will be denoted with a plus sign in the instanton search later on 23 In a given model both this kind of instanton and the one giving rise to righthanded neutrino masses (which is different) may be present. This contribution to the left-handed neutrino Majorana mass is in principle sub-leading compared to the see-saw contribution\n\nM L ab (see-saw) = < H > 2 2M s h T D ( r d (r) a d (r) b e -Sr ) -1 h D (3.20)\n\nwhere is the ordinary Yukawa coupling constant h ab D (ν a R HL b ). In principle the former is doubly suppressed both by 1/M s and the exponential factor. On the other hand if the exponential suppression is not too large this mechanism involving directly the Weinberg operator may be the most relevant source of neutrino masses. This is because the see-saw contribution coming from ν R exchange is proportional to the square of the ordinary Yukawa couplings h ab D which could be small. One could even think of having just the Weinberg operator as the unique source of the observed left-handed neutrino masses. Note however that in string vacua like this, in which the ν R 's are present and massless at the perturbative level, having just the Weinberg operator would not be phenomenologically correct, and instantons of the first class are still needed so that the ν R 's get a sufficiently large mass.\n\nIn the case of N = 1 SUSY models like the MSSM there might be operators of dimension 3 and 4 violating lepton and/or baryon number. These are the superpotential couplings\n\nW Rp = µ L a L a H + λ abc Q a D b L c + λ ′ abc U a D b D c + λ ′′ abc L a L b E c (3.21)\n\nin standard notation. Unlike the neutrino operators mentioned above, these operators violate B -L in one unit (rather than 2). It is well known that the standard R-parity of the MSSM may be identified with a Z 2 subgroup of U(1) B-L , so these terms are odd under R-parity. The simultaneous presence of all these couplings is phenomenologically unacceptable. Indeed, the third coupling violates baryon number, and the other three violate lepton number. Together they lead to proton decay at an unacceptably large rate. On the other hand couplings violating either B or L are phenomenologically allowed.\n\nIt is an interesting question whether any of these operators may be induced by string instanton effects. A first point to note is that instantons with Sp(2) Chan-Paton symmetry can never generate operators of this type. The reason is that all charged zero modes will necessarily come in Sp(2) doublets and hence the charged operators induced will always involve an even number of charged D = 4 fields and R-parity is 24 automatically preserved. On the other hand O(1) and U(1) instantons may generate R-parity violating operators. In particular, the LH bilinear is essentially the square root of the Weinberg operator, and may be induced if a U(1) or O(1) instanton M exists with\n\nI M c = -1 ; I M d = 1 or I M c ′ = 1 ; I M d ′ = -1 . (3.22)\n\n(in the O(1) case the second option is not independent from the first). Again, if the appropriate disk couplings are non-vanishing a term with µ a L ∼ M s exp(-S ins ) is generated. The rest of the operators in W Rp may also be generated. Possible instanton zero modes which may induce them are shown in table 2. For example, the QDL operator may be induced if a U(1) instanton M with intersection numbers\n\nI M b = -1 ; I M c ′ = 1 ; I M d = 1 (3.23)\n\nis present and in addition couplings\n\nL disk ∝ c ab (α(U a Q b j )γ j ) + c ′ a (βL a j γ j ) (3.24)\n\nexist. Here α, β, γ are zero modes corresponding to (Mc ′ ), (Md) and (bM) intersections and a, b(j) are flavor(SU(2) L ) indices. Analogous trilinear or quartic disk amplitudes involving two charged zero modes should exist to generate the rest of the R-parity violating amplitudes in table 2.\n\nD = 4 Operator I Ma I Ma ′ I Mb I Mc I Mc ′ I Md I Md ′ ν R ν R 0 0 0 2 0 -2 0 L HL H 0 0 0 -2 0 2 0 L H 0 0 0 -1 0 1 0 QDL 0 0 -1 0 1 1 0 U DD -1 0 0 1 2 0 0 LLE 0 0 -1 0 1 1 0 QQQL 1 0 -2 0 0 1 0 U U DE -1 0 0 2 2 -1 0\n\nTable 2: Zero modes required to generate Lepton/Baryon-number violating superpotential operators. Sp(2) instantons cannot give rise to R-parity violating operators whereas 1) instantons may in principle contribute to all of them. In the case of U (1) instantons there are additional zero mode possibilities which are obtained by exchanging\n\nO(1),U (\n\nI M x ↔ -I M x ′ . 25\n\nThere are also superpotential dimension-5 operators violating B and L which may be constructed from the MSSM matter superfields. Indeed the dimension 5 operators ( 1 M ) QQQL ; ( 1 M ) UUDE (3.25) are in fact the leading source of proton decay in SUSY GUT models with R-parity.\n\nUnlike the other operators considered here these ones preserve B -L (hence R-parity) but not B + L. These operators do not contribute directly to a proton decay but need to be 'dressed' by a one loop exchange of some fermionic SUSY particle. This makes that, even although they are suppressed only by one power of the relevant fundamental scale, the loop factor and the corresponding couplings make the overall rate in SUSY-GUTS (barely) consistent with present experimental bounds for M of order the GUT scale or larger.\n\nThese dimension 5 operators may also be induced in D-brane models of the class here considered by the presence of instantons with appropriate intersection numbers.\n\nFor instance, the first operator may be induced through O(1) or U(1) instantons M with\n\nI M b = I M b ′ = -2 ; I M a = 1 ; I M d = 1 (3.26)\n\nAgain Sp(2) instantons cannot induced this operator, since the the Ma intersection would yield 6 (rather than 3) colored fermionic zero modes. The proton decay rate obtained from these operators depend on the ratio exp(-S ins ) × 1/M s . For M s of order 10 16 GeV, the rate is consistent with present bounds if exp(-S ins ) provides a suppression of a few orders of magnitude. On the other hand, models with a low string scale may be in danger unless the exponential suppression is sufficiently large (or such particular instantons are absent).\n\nAs a general conclusion, these phenomenological aspects of instanton induced operators very much depend on the action of the instanton, e.g. the volume of the wrapped D2-instanton in the intersecting D-brane constructions. In any event it is clear that the instantons here considered may indeed induce proton decay at a model-dependent rate. However in certain models R-parity will be preserved and prevent too rapid proton decay. Indeed, this is what we find in our instanton search in Gepner orientifolds.\n\nAs we said Sp(2) instantons automatically preserve R-parity. More generally, models that violate R-parity are rare, and the corresponding instantons actually generate very high dimensional operators, so R-parity breaking effects seems quite suppressed. In 26 fact in our search within MSSM-like models in Gepner model orientifolds we do not find instantons with just the correct charged zero modes to generate the low dimensional couplings discussed above. So, at least within our class of RCFT constructions, R-parity preservation is quite a common feature.\n\nIn this section we describe the 4d string models we consider, namely orientifolds of type IIB Gepner model compactifications. This is a very large class, on which one can carry out large scans for certain desired properties. And moreover at present the only known class of (SUSY) models with massive B -L.\n\nIn general, RCFT orientifolds are orientifold projections of closed string theories constructed using rational conformal field theory. Although this includes in principle rational tori and orbifolds, the real interest lies in cases where the two-dimensional CFT is interacting, because such theories are hard to access by other methods. A disadvantage of the use of RCFT is that this method is algebraic, and not geometric in nature, so that one cannot easily explore small deformations of a certain string theory.\n\nIt is best thought of as a rational scan of moduli spaces.\n\nThe most easily accessible examples are the orientifolds of tensor products of minimal N = 2 conformal field theories (\"Gepner models\") forming a type IIB closed string theory. During the last decade, examples in this class have been studied by many authors (see [35] 42] ), and searched systematically in [6] and [7] .\n\n[36][37][38][39][40][41][\n\nAlthough the Gepner models form only a small subset of RCFT's, they already offer a large number of possibilities. The total number of tensor products with the required central charge c = 9 is 168. On top of this, one can choose a large number of distinct modular invariant partition functions on the torus. The orientifold formalism is not available for all of them, but it has been completely worked out [43] for all simple current invariants (based on the charge conjugation invariant). This yields a total of 5403 distinct MIPFs. On top of this, we may choose various orientifold projections. Here the only known possibilities are a class of simple-current based choices [44][45] [46] [47] .\n\nThis then yields a total of 49304 orientifolds.\n\nFor each orientifold choice, the full open string partition function is 1 2 a,b,i\n\nN a N b A i ab χ i ( τ 2 ) + a,i N a M i a χi ( τ 2 + 1 2 ) (4.1)\n\nHere A i ab are the annulus coefficients, M i a the Moebius coefficients, N a the Chan-Paton multiplicities and χ(τ ) are the closed string characters, and χi (τ ) = T -1/2 χ i (τ ). The set of integers i is simply the set of primary fields of the closed string CFT, and depends only on the tensor product. The integers a, b are the boundary labels; this set depends on the MIPF. Our notation and labelling conventions for these CFT quantities are explained in Appendix A. The integers A i ab and M i a depend in addition also on the orientifold choice; in the case of A i ab the latter dependence is very simple: all distinct annuli can be written as\n\nA Ω,i ab = c A i c a C Ω cb ,\n\nwhere Ω is the orientifold choice (which we usually do not specify explicitly) and C Ω cb is the boundary conjugation matrix, which acts as an involution on the set of boundaries. Suppressing some details (which can be found in [43] ) we may write these integers as\n\nA Ω,i ab = m,J,K S i m R a,(m,J) g Ω,m JK R b,(m,K) S 0m (4.2) M Ω,i a = m,J,K P i m R a,(m,J) g Ω,m JK U Ω (m,K) S 0m (4.3)\n\nHere m is the label of an Ishibashi-state (the set of states that propagates in the transverse (or closed string) channel of the the annulus or Moebius diagrams). It is a subset of the set of closed string labels i, but in general there are degeneracies, so that more than one distinct Ishibashi state belongs to a given closed string label.\n\nThese degeneracies are distinguished by the labels J, K (see Appendix A). The complex numbers R and U are respectively the boundary and crosscap coefficients. Note that the latter depend on the orientifold choice, but the former do not. The only dependence of the annulus coefficients on the orientifold choice is through the Ishibashi metric g Ω JK , which is a matrix on each Ishibashi degeneracy space, and which can be a sign if there are no degeneracies. Finally, the matrix P is given by P = √ T ST 2 S √ T , where S and T are the generators of the modular group of the torus. Similar expressions exist for the Klein bottle multiplicities defining the unoriented closed sector, but they will not be needed in this paper.\n\nThe boundary labels a, b, . . . refer to all boundaries that respect the bulk symmetries of the CFT. This includes the individual N = 2 chiral algebras of the factors in the tensor product, the alignment currents 12 that ensure the proper definition of world-sheet supersymmetry and the space-time supersymmetry generator that imposes a generalized GSO-projection on the spectrum. The latter implies that all characters χ i respect (at least) N = 1 space-time supersymmetry. By construction, the boundary states are then supersymmetric as well. Both conditions (boundary and bulk space-time supersymmetry) can in principle be relaxed within the formalism, but this leads to a much larger set of bulk and boundary states. The precise labelling of the boundaries is explained in Appendix A and involves a subset of the closed string labels i and a degeneracy label, distinct from the one used for the Ishibashi states. The set of boundary labels is complete in the sense of [45] . This means that no additional boundary states exist that respect all the aforementioned symmetries. It also means that the matrices R are square matrices (although their rows and columns are defined in terms of different index sets). It is in principle possible to write down additional boundary states that break some of the world-sheet symmetries. This is an important possibility to keep in mind, but we will not consider it here.\n\nThe massless spectrum is obtained by restricting the characters χ i to massless states. Since the characters are supersymmetric those massless states are either vector multiplets or chiral multiplets. The latter can be restricted to one chirality (e.g lefthanded); the other choice merely produces the CPT conjugates. Boundaries are called real if a = a ′ , where the conjugate boundary a ′ is defined by C Ω a,a ′ = 1, and complex otherwise. The Chan-Paton multiplicities N a give rise to gauge groups U(N a ) for complex boundaries and SO(N a ) or Sp(N a ) for real ones. In the latter case N a must be even. To count bi-fundamentals we define\n\nL ab ≡ i A i ab χ i ( τ 2 ) massless,L . (4.4)\n\nNote that because of the factor 1 2 in (4.1) and the fact that L ab is symmetric, the value of L ab is indeed precisely the number of bi-fundamentals in the representation (N a , N b ).\n\nThese are spin-3 currents consisting of products of the world-sheet supercurrents of the factors in the tensor product, including the NSR space-time factor.\n\nIt is convenient to introduce the intersection matrix 13\n\nI ab ≡ L ab ′ -L a ′ b , (4.5)\n\nwhich is manifestly antisymmetric in a and b. Note that for a pair of complex boundaries a, b with conjugates a ′ , b ′ one can define four quantities that are relevant for the massless spectrum, two of which are chiral, namely I ab and I ab ′ .\n\nIt is often convenient to associate a geometric picture to these integers. Thus we will often refer to the boundary labels and their multiplicities as \"stacks of branes\", and view the integers I ab as brane intersection numbers. This is only done for convenience and does not imply a concrete brane realization; indeed, it does not make sense to say that a given boundary label corresponds to a Dp-brane for some give p. Such an interpretation might be valid in a large radius limit, assuming such a limit exists.\n\nIn general, for a choice of Chan-Paton multiplicities N a there will be tadpoles in the one-point closed string amplitudes on the disk and the crosscap. These have to be cancelled in order to make the theory consistent (since we work with supersymmetric strings we do not have the option of cancelling RR and NS-NS tadpoles separately).\n\nThis leads to a condition on the Chan-Paton multiplicities:\n\na N a R a,(m,J) = 4η m U m,J (4.6)\n\nwhere η 0 = 1 and all other η's are -1; there is such a condition for any Ishibashi label (m, J) that leads to a massless scalar in the transverse channel. The one for m = 0 (which is non-degenerate) is the dilaton tadpole condition. It has the special feature that all coefficients R a0 are real and positive. The crosscap coefficient U 0 is also real and can be chosen positive (in the CFT both signs are acceptable). If U 0 = 0 (4.6) limits the Chan-Paton multiplicities; if U 0 = 0 the only solution is N a = 0 for all a, which rules out any realization of the Standard Model. This reduces the number of usable orientifolds to 33012.\n\nTadpole cancellation condition implies cancellation of RR-charges coupling to longrange fields, and absence of local anomalies. There is a second condition that has 13 Note that L ab is a symmetric matrix giving the number of chiral multiplets in the ( a , b ) bifundamental. This is a natural quantity in unoriented CFT's, where a symmetric definition for the annulus amplitude exists. In oriented CFT the annulus is, in general, not symmetric, but on the other hand it is possible to choose the branes in such a way that only ( , ) bi-fundamentals appear.\n\nThis has become the customary way of counting states in the intersecting brane literature, even for orientifold models. The quantity I ab is defined in such a way that it is anti-symmetric in a and b. This is why boundary conjugations appear in the right hand side. This has the additional advantage of making I a more familiar quantity for readers used to the standard intersection brane conventions.\n\nto be taken into account, which has to do with Z 2 charges that do not couple to long-range fields, usually referred to as \"K-theory charges\" in geometric constructions.\n\nUncancelled K-theory charges may lead to global anomalies in symplectic factors of the gauge group. But even if this symptom is absent, the disease may still exist. A much more general way to probe for uncancelled K-theory charges is to require the absence of global anomalies not only in the Chan-Paton gauge group but also on all symplectic brane-anti-brane pairs that can be added to it as \"probe-branes\" [48] . Presently this is the most general constraint that be imposed in these models, but it is not known if additional ones are required. This probe brane constraint leads to a large number of mod-2 constraint and is potentially very restrictive, but almost harmless in practice [49] . It is satisfied by all models we consider in the present paper.\n\nThe complete set of solutions to these conditions is finite but huge, but the vast majority is of no phenomenological interest. In the last few years systematic searches have been carried out for models that contain the Standard Model. The models that were considered have the property that the set of Chan-Paton labels can be split into two subsets, the observable and the hidden sector. The former has been limited, for practical reasons, to at most four complex brane stacks, required to contain the Standard Model gauge group and the right intersections to yield three families of quarks and leptons, plus (in general) some non-chiral (vector-like) additional matter. The hidden sector is only constrained by the requirement that there be no net number of chiral multiplets charged under both the observable and hidden sector, and by practical computational limitations. The main purpose of the hidden sector in these models is to provide variables that can be used to satisfy the tadpole and global anomaly conditions, since the multiplicities in the observable sector are already fixed. In some cases the observable sector already satisfies the constraints by itself, and there is no hidden sector.\n\nThe observable sector can be realized in many different ways if one only imposes the constraint that the standard model should be contained in it. These possibilities were recently explored in [7] . We will focus on the realization described in Section 3.1, first considered in [5] . There are four stacks, namely a (containing QCD and baryon number as U(3)), b (containing electroweak SU(2) embedded as U(2) or Sp(2)), c (providing a U(1) or an O(2) factor 14 , and d (providing another U(1) factor). 14 In [6] also Sp(2) was considered, but this requires an additional Higgs mechanism.\n\nThe standard model hypercharge generator is , defined in (3.1):\n\nY = 1 6 Q a - 1 2 Q c - 1 2 Q d (4.7)\n\nwhere Q x denotes the generator of the U(1) of brane stack x; in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator. In addition to Y these models have two or three additional U(1) gauge symmetries (the latter case if electroweak SU(2) arises from U(2)). These (except the combination B -L) are anomalous, with anomaly cancelled by the Green-Schwarz mechanism, implying the existence of a B ∧ F coupling making them massive. In fact, as already mentioned, such Stückelberg couplings may be present for non-anomalous U(1)'s as well. We are interested in models where the hypercharge gauge boson does not have such couplings (otherwise the model would be phenomenologically unacceptable), but\n\nwhere the B -L gauge boson is massive by such couplings (both in order that the gauge group reduces to the SM one, and that neutrino Majorana masses may be induced by string instantons, as discussed in previous sections).\n\nThe combined requirements of having a massive B -L and a massless Y turn out to be difficult to satisfy. In fact, if the group on brane c is O(2) they are impossible to satisfy simultaneously, because the O(2) component of the vector boson does not couple to any axions, and hence the B -L and Y bosons have the same mass. But even in models with a U(1) group on brane c it happens rather rarely that both constraints are satisfied simultaneously, at least in the searches that have been done so far.\n\nWe will make use here of the data presented in [6, 7] , which are available in slightly improved form on the website www.nikhef.nl/ ∼t58/filtersols.php. This database consist of 211634 distinct spectra. Here \"distinct\" means that they are physically different for a given MIPF 15 if the hidden sector is ignored. Hence the differences can be the number of vector-like states of various kinds or the dilaton couplings of branes a, b, c, d. Geometrically, these spectra may originate from the same moduli space, but then in any case from different points on this moduli space. The improvements in comparison with the data presented in [6] consist of taking into account the full global anomaly conditions from probe branes. In some cases this required nothing more than checking these conditions for an existing solution of the tadpole conditions, but in other cases a new solution had to be found. As a result, a few models disappeared from the original database, but due to improved algorithms a few new ones could be added. The net re-15 Rare cases of identical spectra and couplings originating from different MIPFs are treated as distinct.\n\n32 sult is some small but inconsequential changes in the total number of models of various kinds. The numbers we will mention below are based on the improved database.\n\nThe total number of models in that database with a Chan-Paton group U(3) × Sp(2) × U(1) × U(1) is 10587. Of these, 391 (about 4%) have a massive B -L vector boson. For U(3) × U(2) × U(1) × U(1) these numbers are, respectively, 51 and 0. Hence no examples of the latter type were found, although they were found with 1,2 and 4 families (in a limited search), in a few percent of the total number of models. It seems therefore reasonable to expect that U(3) × U(2) × U(1) × U(1) with massive B -L do exist, and that their absence is just a matter of statistics. Just for comparison, the total number of U(3) × Sp(2) × O(2) × U(1) models is 56627.\n\nIn this section we discuss D-brane instantons for general compactifications, including abstract CFT ones. We also provide the spectrum of zero modes on an instanton brane, using the information about their internal structure i.e. in the compactified dimension in geometric models, or of the internal CFT in more abstract setups like in previous section. We will be interested in the latter case.\n\nA first question that should be addressed is what this internal structure is. For instance, in type IIA geometric compactifications, it corresponds to a supersymmetric (i.e. special lagrangian) 3-cycle. Notice that these are the same kind of 3-cycles already used to wrap the D6-branes that give rise to the 4d gauge symmetry of such models.\n\nFor general CFT's, D-branes are described as boundary states. To describe instantons, one can simply use the same boundary state of the internal CFT to describe the 4d space-filling branes present in the model and the instanton branes. The only difference is that boundaries satisfy Neumann conditions in the 4d space-filling case, and Dirichlet in the instanton case. This exploits the fact that whenever a boundary state of the internal CFT, and with Neumann boundary conditions in the 4d space is an acceptable state of the full CFT, the same boundary state of the internal CFT, combined with Dirichlet boundary conditions in the 4d space also gives an acceptable state of the full CFT. For geometric compactifications this is related to Bott periodicity of the K-theory classes associated to the D-brane charges, but it is possible to show it in general.\n\nSince instanton D-branes can thus be naturally associated to the boundary states of 4d space-filling branes, it is convenient to express the spectrum of zero modes of the former in terms of the massless states of the latter. This is particularly useful, since 33 the computation of the spectra on 4d space-filling branes for Gepner model orientifolds has already been described (although the arguments below are valid also for geometric compactifications). Hence, let us denote by M a 4d space-filling brane associated with the same boundary state of the internal CFT as the instanton brane M of interest.\n\nNote that the 4d space-filling brane M is an auxiliary tool, and need not be (and, for our instantons of interest, will not be) one of the 4d space-filling branes present in the model.\n\nLet us first consider the case of 'real' brane instantons. Consider a set of m 4d spacefilling branes M, and focus first on the massless spectrum in the MM sector. Before the orientifold projection, it leads to a universal 4d N = 1 U(m) vector multiplet, and a number L MM of adjoint chiral multiplets. The orientifold operation maps this sector to itself, acting on the Chan-Paton with a matrix γ Ω,M . This matrix satisfies\n\nγ T Ω,M γ -1 Ω,M = ±1 m (5.1)\n\nThe two possibilities can be chosen to correspond to\n\nγ Ω,M = 1 m or γ Ω,M = ǫ m , with ǫ m = 0 1 r -1 r 0\n\n, and m = 2r hence necessarily even in the latter case. They correspond to the SO and Sp projections, respectively.\n\nThe orientifold projection on the N = 1 vector multiplet Chan-Paton matrices is given by\n\nλ = -γ Ω,M λ T γ -1 Ω,M (5.2)\n\nand leads to SO(m) or Sp(m) vector multiplets for the SO or Sp projection (hence the name). Concerning the N = 1 chiral multiplets, they fall in two classes of p -, p +\n\n(with p -+ p + = L MM ) which suffer the projections λ = ±γ Ω,M λ T γ -1 Ω,M (5.3)\n\nFor the SO projection, this leads to p + , p -chiral multiplets in the , representation.\n\nFor the Sp projection, there are p + , p -chiral multiplets in the , representation.\n\nThe sectors Ma (where a is a 4d space-filling branes present in the model) are mapped to sectors Ma ′ , so it is enough to focus on the former. After the orientifold projection one gets L Ma , L Ma ′ chiral multiplets in the ( M , a ), ( M , a ).\n\nLet us now obtain the zero modes for a set of m instanton branes M in terms of the above spectrum. The MM sector is closely related to the MM sector, by changing the NN boundary conditions in 4d spacetime to DD boundary conditions 34 (which can be done in a covariant formalism, but not in the light-cone gauge). Before the orientifold projection, one obtains the same set of states (since moddings for NN and DD boundary conditions are identical, both in the NS and R sector), but with different world-volume interpretation. Also, the change in boundary conditions implies that some polarization states which are unphysical for the 4d spacefilling brane are physical in the instanton brane. Hence, the U(m) gauge bosons on the 4d space-filling brane M correspond to four adjoint real scalars in the instanton brane M. Similarly, the 4d spinors in M, correspond to four fermion zero modes on M, transforming as two spinors of opposite chiralities θ α , θ α of the SO(4) rotation group in transverse space. The orientifold projection maps the MM sector to itself, acting on Chan-Paton indices with a matrix γ Ω,M . In close analogy with the argument in [50] for the familiar D5-D9-brane system in type I (see [51, 52] for related derivations), one can show that the condition (5.1) flips sign upon changing four NN boundary conditions to DD, hence\n\nγ T Ω,M γ -1 Ω,M = ∓1 m (5.4)\n\nNamely, the instanton brane has Sp(m) gauge group when the 4d space-filling brane (with same internal boundary state) has gauge group O(m), and vice-versa. We still refer to these projections as SO and Sp, hoping no confusion arises. Note that, as mentioned in Section 2.2, although there are no gauge bosons in 0 + 0 dimensions, the gauge group is present on the instantons in that it acts on open string endpoints.\n\nLet us consider the effect of the orientifold projection on the MM states, as compared with the effect on MM states. Again, following arguments familiar in the D5-D9 brane system in type I, one can show that the signs in conditions like (5.2), (5.3) remain unchanged upon changing four NN dimensions to DD, except for bosonic modes polarized along the directions longitudinal to these four dimensions (and for fermions related to them by the unbroken susy of the total system). To be concrete, considering the four MM adjoint bosons, and two MM adjoint fermions θ α associated to the universal MM vector multiplets, they suffer the projection\n\nλ = +γ Ω,M λ T γ -1 M (5.5)\n\nHence they transform in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection. On the other hand, for the two fermion zero modes θ α, the projection is\n\nλ = -γ Ω,M λ T γ -1 M (5.6) 35\n\nand leads to two fermion zero modes in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection.\n\nThis implies that in order to obtain two fermion zero modes from this universal multiplet, in order to generate a superpotential, one should consider instantons with orthogonal gauge group and multiplicity one (O(1) instantons). For instantons with symplectic gauge group and multiplicity two (Sp(2) instantons), there are two additional fermion zero modes in the triplet representation. As mentioned, we will continue to consider such instantons in our relaxed scan. Multiple instantons, i.e. boundary states with higher multiplicity, lead to a larger amount of additional fermion zero modes (due to the larger gauge representations for the fermions), and do not contribute to superpotentials; we will not consider such cases even in relaxed scans, since they also very often lead to too many charged fermion zero modes and cannot contribute to the operators of interest (except possibly for O(2) and U(2) instantons with low intersections, which are kept in our scan as a curiosity).\n\nSimilarly, for the p ± sets of MM scalars and fermions associated to the MM 4d chiral multiplets, the projection is\n\nλ = ±γ Ω,M λ T γ -1 Ω,M (5.7)\n\nwith the same sign choice as in (5.3). The different structure of γ Ω implies that, for the SO projection we get p + , p -sets of scalars and fermions in the , , while for the Sp projection there are p + , p -sets of scalars and fermions in the , .\n\nThis concludes the discussion of the MM sector. Let us not consider the Ma sectors, from the information from the Ma sectors. Notice that this implies changing four NN boundary conditions to DN, which have different moddings. Hence the states are different in both situations, but the information on the multiplicities is preserved.\n\nSpecifically, in the NS sector the DN boundary condition introduce an additional vacuum energy which generically makes all states massive. Hence there are no massless scalar zero modes in generic Ma sectors. In the R sector, the change in the moddings reduces the dimension of the massless ground state, leading to a single (chiral) fermionic degree of freedom. Since the orientifold action maps the Ma sector to Ma ′ sectors, there are no subtleties in the orientifold projection. The end result is L Ma , L Ma ′ fermion zero modes in the ( M , a ), ( M , a ). The net number of chiral fermion zero modes in the ( M , a ) is given by I Ma = L Ma ′ -L Ma , i.e. the net number of chiral multiplets in the related Ma sector.\n\nThe results for orientifold projections for real branes are shown in table 3. 36 Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) SO N = 1 vect. mult. U(m) O(m) Sp(m)\n\n2 f + 2 f + 4 b N = 1 ch. mult. (p + + p -) Ad p + + p - 2p + ( f + b ) + 2p -( f + b ) Sp N = 1 vect. mult. U(m) Sp(m) O(m) 2 f + 2 f + 4 b N = 1 ch. mult. (p + + p -) Ad p + + p - 2p + ( f + b ) + 2p -( f + b ) Any N = 1 ch. mult. L Ma ′ ( M , a )+ L Ma ′ ( M , a )+ L M a ′ ( M , a ) f L Ma ( M , a ) L Ma ( M , a ) L M a ( M , a ) f net I Ma ( M , a ) net I M a ( M , a ) f\n\nTable 3: Orientifold projection for real branes: Massless modes of the 4d space-filling branes M (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes)\n\nWe now consider the case of complex brane instantons. The arguments are very similar, hence the discussion is more sketchy. Consider m 4d spacefilling branes M, associated to the internal boundary state of the instanton brane M of interest. The MM leads to a 4d N = 1 U(m) vector multiplet and a number L MM ′ of adjoint chiral multiplets. The orientifold action maps it to the M ′ M ′ sector, hence we may keep just the former and impose no projection. The MM ′ sector is mapped to itself under the orientifold projection. Denoting by γ Ω,M the action on Chan-Paton indices, the MM ′\n\nmodes split into sets L ± M M , L ± M ′ M ′ , which suffer a projection λ = ±γ Ω,M λ T γ -1 Ω,M (5.8) leading, for γ Ω,M = 1 m , to L + MM , L - MM chiral multiplets in the , , and L + M ′ M ′ , L - M ′ M ′ chiral\n\nmultiplets in the , . The net number of chiral multiplets in the , is\n\nI + MM ′ = L + M,M -L + M ′ M ′ , I - MM ′ = L - M,M -L - M ′ M ′ .\n\nAnd oppositely for γ Ω,M = ǫ m . Finally, the Ma, Ma ′ and related sectors lead, after the orientifold projection, to\n\nL Ma ′ , L Ma , L M ′ a ′ , L M ′ a chiral multiplets in the ( M , a ), ( M , a ), ( M , a ), ( M , a ).\n\nIn order to simplify notation, we replace M → M in these expressions in our discussions of instanton zero modes.\n\nLet us now consider m brane instantons M and compute their zero mode spectrum in terms of the above. In the MM (and its image M ′ M ′ ) sector there are four scalar modes and four fermions in the adjoint of the U(m) gauge symmetry group; these are related to the 4d vector multiplet in the MM sector. In addition, there are L M M ′ sets of scalars and fermions in the adjoint, related to the L MM ′ non-universal chiral multiplets in the MM sector. The MM ′ sector is mapped to itself, and one has to impose the orientifold projection (recalling that the matrix γ Ω,M differs from γ Ω,M ).\n\nFor γ Ω,M = 1, hence γ Ω,M = ǫ, we obtain L + M M , L - M M chiral multiplets in the , , and L + M ′ M ′ , L - M ′ M ′ chiral multiplets in the , . The net number of chiral multiplets in the ,\n\nis I + M M ′ = L + M M -L + M ′ M ′ , I - M M ′ = L - M M -L - M ′ M ′ .\n\nAnd oppositely for γ Ω,M = ǫ hence γ Ω,M = 1.\n\nIn the Ma, Ma ′ and related sectors, there are generically no bosonic zero modes, and there are and ( M , a ) respectively. The net number of chiral fermion zero modes in the ( M , a ) and\n\nL Ma , L M ′ a ′ , L Ma ′ , L M ′ a chiral fermion zero modes in the ( M , a ), ( M , a ), ( M , a ),\n\n( M , a ) is given by I Ma = L Ma ′ -L M ′ a ′ and I Ma ′ = L Ma -L M ′ a . In\n\norder to simplify notation, we replace M → M in these expressions in our discussions of instanton zero modes.\n\nThe results for orientifold projections for real branes are shown in table 4.\n\nIn this section we perform a search of models which admit an instanton inducing a right-handed neutrino Majorana mass operator. Namely, for each model with the chiral content of the SM in the classification described in Section 4.2, we first scan over boundary states, searching for all instantons with the required uncharged and charged fermion zero mode structure to yield neutrino masses. We then relax our criteria a bit and allow for instantons with correct charged zero mode structure but having extra non-chiral zero modes (both charged and uncharged). The idea is that these non-chiral zero modes could be lifted by diverse effects, as discussed.\n\nIt is important to recall that the cubic couplings between instanton zero modes and 4d chiral multiplets are difficult to compute in Gepner model orientifolds. Hence, we will simply assume that such couplings are non-zero if there is no symmetry forbidding them. 38 Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) Any N = 1 vect. mult. U(m) × U(m) ′ U(m) U(m) 4 Ad f + 4 Ad b N = 1 ch. mult. p adj Ad + p adj Ad ′ p adj Ad 2p adj ( Ad f + Ad b ) SO N = 1 ch.mult.\n\nL MM ( M , M ′ ) L + MM M + L - MM M 2L + M M b,f + 2L - M M b,f L M ′ M ′ ( M , M ′ ) L + M ′ M ′ M + L - M ′ M ′ M 2L + M ′ M ′ b,f + 2L - M ′ M ′ b,f Sp N = 1 ch.mult. L MM ( M , M ′ ) L + MM M + L - MM M 2L + M M b,f + 2L - M M b,f L M ′ M ′ ( M , M ′ ) L + M ′ M ′ M + L - M ′ M ′ M L + M ′ M ′ b,f + L - M ′ M ′ b,f Any N = 1 ch. mult. L Ma ′ ( M , a )+ L Ma ′ ( M , a ) L M a ′ ( M , a ) f . . . L Ma ( M , a ) L M a ( M , a ) f . . . L M ′ a ′ ( M , a ) L M ′ a ′ ( M , a ) f . . . L M ′ a ( M , a ) L M ′ a ( M , a ) f net I Ma ( M , a ) net I M a ( M , a ) f net I Ma ′ ( M , a ) net I M a ′ ( M , a ) f\n\nTable 4: Orientifold projection for complex branes: Massless modes of the 4d space-filling branes M (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes)\n\nOur detailed strategy will become clear along the description of the results. Given a set of a,b,c,d standard model branes, we must look for additional boundary states M that satisfy the requirements of a (B -L)-violating instanton. From the internal CFT point of view this is just another boundary state, differing from 4d spacefilling branes only in the fully localized 4d spacetime structure. The minimal requirement for such a boundary state is B -L violation, which means explicitly\n\nI M a -I M a ′ -I M d + I M d ′ = 0 (6.1)\n\nIt is easy to see that the existence of such an instanton implies (and hence requires) the existence of a Stückelberg coupling making B -L massive. To see this, consider adding to the Standard Model configuration a 4d spacefilling brane M (in fact used in Section 5) associated to the boundary state M (RR tadpoles can be avoided by simultaneously including M antibranes, which will not change the argument). The new sector in the chiral spectrum charged under the branes M can be obtained by reversing the argument in Section 5, and is controlled by the intersection numbers of M. From 39 the above condition it follows that the complete system has mixed U(1) B-L × (G M ) 2 anomalies, where G M is the Chan-Paton-factor of brane M. These anomalies are cancelled by a Green-Schwarz mechanism involving a (B -L)-axion bilinear coupling, which ends up giving a mass to B -L via the Stückelberg mechanism. This coupling is in fact not sensitive to the presence of the brane M, hence it must have been present already in the initial model (without M).\n\nHence the existence of a boundary label M that satisfies (6.1) implies that B -L is massive. Unfortunately the converse is not true: even if B -L has a Stückelberg mass, this still does not imply the existence of suitable instantons satisfying (6.1) 16 Indeed, in several models we found not a single boundary state satisfying (6.1).\n\nNote that, since hypercharge must be massless, one can use the reverse argument and obtain that\n\nI M a -I M a ′ -I M c + I M c ′ -I M d + I M d ′ = 0 (6.2)\n\nin all models. We verified this for all models we considered as a check on the computations.\n\nAs already discussed in Section 4.2, in the search for SM constructions in Gepner orientifold, there are 391 models with massless hypercharge and massive B-L. In these models we found a total of 29680 instantons with B -L violation, i.e. with intersection numbers satisfying (6.1). Of course, in order to serve our purpose of generating a Majorana mass superpotential, the instantons have to satisfy some more conditions.\n\nLet us consider them in order of importance, and start with the conditions on the net number of chiral fermion zero modes charged under the 4d observable sector. Clearly we need I M a = I M a ′ and I M b = I M b ′ . The latter condition is automatically satisfied in this case, because the b-brane is real in all 391 models. The chiral conditions on the zero modes charged under the branes c and d are as in [3] 17 and are given in equations (3.6), (3.7) (3.8) of the present paper. These are the instantons of most interest, and on which we mainly focus. However, as discussed in Section 3.4, other important Band/or L-violating operators (such as the Weinberg operator or the LH operator) can 16 From intuition in geometric compactifications, one expects that there may always exist a D-brane with the appropriate topological pairings, but there is no guarantee that there is a supersymmetric representative in that topological sector, and even less that it would have no additional fermion zero modes. Note also that even if such D-brane instantons exists, there is no guarantee that it will fall in the scan over RCFT boundary states. 17 Note that there is a sign change in the contribution of the U (1) d generator to Y in comparison to [3] 40 be generated by instantons with similar intersection numbers, up to a factor of 2 and a sign, see table 2 . For this reason we also allow at this stage any instanton which has the correct number of charged zero modes to generate them. Imposing these conditions reduces the number of candidate instantons potentially contributing to neutrino masses in any of the models to 1315.\n\nAll instantons satisfying these requirements are summarized in the table 5. In columns 1,2 and 3 we list the tensor combination, MIPF and orientifold choice for which the model occurred. The latter two numbers codify simple current data that describe respectively a MIPF and an orientifold. MIPFs are in general defined by means of a subgroup H of the simple current group G, plus a certain matrix X of rational numbers [55] . Orientifolds are defined by a simple current and a set of signs [43] . In previous work [6] we have enumerated these quantities (up to permutation symmetries) and assigned integer labels to them for future reference. We only refer to these numbers here, but further details are available upon request. Usually for each MIPF and orientifold which contains the standard model there are several choices a,b,c,d for which it is obtained. For a given choice of tensor combination, MIPF and orientifold and SM branes there may be several instantons. For clarity we put all such instantons together in the information in table 5 . In column 4 we indicate which type of instanton branes were found. Five types are distinguished: O1, O2, S2, U1 and U2, corresponding to O(1), O(2), Sp(2), U(1) and U(2) Chan-Paton symmetry on the instanton volume. The number indicates the instanton brane multiplicity that gives the correct number of instanton charged zero modes from the a, b, c, d branes, to lead to right-handed neutrino Majorana masses. The number of zero modes is in general the product of the instanton brane multiplicity and 'intersection number' with the corresponding 4d spacefilling brane. As discussed in Section 5, for symplectic branes the smallest possible brane multiplicity is 2. As we discussed there, only O1 instantons may have the required universal minimal set of two zero modes in the uncharged sector.\n\nStill we look for all O(1), Sp(2) and U(1) instantons which may yield a superpotential if the extra uncharged fermion zero modes. In this vein we also include a search for O2 and U2 instantons. Note also that such O2 or U2 instantons imply the existence of other instantons involving the same boundary state, but with multiplicity 1, which may lead to the R-parity violating operator LH. We will discuss the generation of R-parity violating operators at the end of this section. The third character (+ or -)\n\nin the instanton in table 5 is the sign of I M c ′ -I M c . For the instantons giving rise to right-handed neutrino Majorana masses this sign should be negative, whereas it should 41 be positive for instantons giving rise to the Weinberg operator (or the LH operator), see table 2 .\n\nThe 1315 instantons are divided in the following way over the different types: 3 of types O1+ and O1-, 46 of type U1+, 24 of type U1-, 550 S2+, 627 S2-, 27 of types U2+ and U2-and four of types O2+ and O2-. Notice that the vast majority (97.5%) of the instanton solutions are of type S2+ and S2-. This is encouraging given the nice properties of such instantons, concerning e.g. R-parity conservation. Note also that in almost all cases both S-and S+ are simultaneously present, 18 so both sources of physical neutrino Majorana masses (from the see-saw mechanism or the Weinberg operator) are present. The other instanton classes possibly generating right-handed neutrino masses are O1-and U1-, which are much less abundant. There is just one orientifold with O1-instantons, for which one can obtain cancellation of RR tadpoles, see below. On the other hand we have found no orientifold with U1-instantons and cancellation of tadpoles, see below.\n\nTable 5: Summary of instanton branes. (1, 16, 16, 16) (2, 4, 12, 82) 2, 4, 12, 82) (2, 4, 12, 82) (2,4,14,46) 10 0 (2,4,14,46) 16 0 (2,4,16,34) 15 0 (2,4,16,34) 15 1 (2, 4, 16, 34) (2,4,16,34) 16 1 (2,4,16,34) (2, 4, 16, 34) (2, 4, 16, 34)\n\n12 0 S2 + , S2 - Yes\n\n19 0 S2 -! ? (\n\n19 0 U2 + !, U2 -! No\n\n19 0 U1 + , U1 - No\n\n16 0 S2 + , S2 - Yes\n\n18 0 S2 - Yes\n\n18 0 U1 + , U1 -, U2 + , U2 -No\n\n49 0 U2 + , S2 -!, U1 + Yes\n\nContinued on next page 18 In some models contributing many instantons there is an exact symmetry between S-and S+.\n\nThis explains the approximate symmetry in the full set. In some cases this symmetry can be understood in terms of flipping the degeneracy labels of boundary states. We regard it as accidental, since it is not found in all models.\n\nTable 5 -continued from previous page Tensor MIPF Orientifold Instanton Solution (2,4,16,34) 49 0 U1 - No (2,4,18,28) 17 0 2, 4, 22, 22) 2, 4, 22, 22) (2, 4, 22, 22) (2, 4, 22, 22) (2, 4, 22, 22) (2,4,22,22) 20 0 (2,4,22,22) 46 0 (2,4,22,22) (2, 6, 14, 14) 1 1 U1 + No (2,6,14,14) 22 2 (2,6,14,14) 60 2 (2,6,14,14) 64 0 (2,6,14,14) 65 0 (2,6,10,22) 22 2 2,6,8,38) 16 0 (2,8,8,18) (2, 8, 8, 18) (2, 10, 10, 10) (4,6,6,10) 41 0 (4,4,6,22) 43 0 (6,6,6,6) 18 0 Most models have a hidden sector containing extra boundary states beyond the SM ones. In the same spirit of imposing chiral conditions first, we should require\n\n(\n\n13 3 S2 + !, S2 -! Yes! (\n\n13 2 S2 + !, S2 -! Yes\n\n13 1 S2 + , S2 - No\n\n13 0 S2 + , S2 - Yes\n\n31 1 U1 + , U1 - No\n\n49 1 O2 + , O2 -, O1 + , O1 -Yes\n\n(\n\n14 2 S2 + !, S2 -! Yes\n\n14 0 S2 + !, S2 -! No\n\n52 0 U1 + , U1 - No\n\nthat I M h = I M h ′ ,\n\nwhere h is a hidden sector brane. This is to guarantee that the generated superpotential does not violate some hidden sector gauge symmetry which would require the presence of hidden sector fields along with the ν R bilinear. The latter condition is not imposed on the previously known hidden sector (i.e. the one in [6, 7] ), but instead a new search for tadpole solutions was performed, for each M, restricting the candidate hidden sector branes to those satisfying I M h = I M h ′ (as discussed in 43 Section 5). This is because in general the known hidden sector in [6, 7] is just a sample out of a huge number of possibilities.\n\nIn column 5 we indicate for which instantons it was possible to satisfy the tadpole conditions with this additional constraint. With regard to observable-hidden matter we use the same condition as in [6] , namely that it is allowed only if it is vector-like. Such a solution could be found for 879 of the 1315 instantons, with ten cases inconclusive (i.e it was computationally too difficult to decide if a solution does or does not exist).\n\nThe latter are indicated with a question mark in column 5 (for most of the undecidable cases there is a tadpole solution for a different instanton with the same characteristics;\n\nfor that reason just one question mark appears).\n\nIndependently of the RR tadpole condition (since there may be alternative sources for its cancellation, or hidden sectors which fall beyond the reach of RCFT), we can also consider the further constraint that the number of charged fermion zero modes is exactly right, not just in the chiral sense. This means\n\nI M a = I M a ′ = I M b = I M b ′ = 0, I M c = 2, I M c ′ = 0 and I M d = -2, I M d ′ = 0 or vice-versa.\n\nFurthermore we require that there are no adjoint or rank-2 tensor zero-modes (note that the latter could be chiral if the instanton brane is complex, and indeed they are in some of the 1315 cases).\n\nThis reduces the 1315 instantons to 263. In column 4 we indicate those cases with an exclamation mark. It is noteworthy that the success rate for solving the tadpole conditions is highest for these instantons: 254 of the 263 allow a solution (with 3 undecided). If an exclamation mark appears in column 4, this only indicates that some of the instantons are free of the aforementioned zero modes, not that all of them are. But in all cases, if there are tadpole solutions, they exist in particular for the configurations with an exclamation mark. Finally we may impose the condition that I M h and I M h ′ are separately zero. This is indicated with and exclamation mark in column 5. This turns out to be very restrictive. The only cases where this happens have no hidden sector at all.\n\nIt is worth remarking that the only instantons having exactly the correct set of charged zero modes and cancelling tadpoles are of S2 ± type. Also those instantons are the only cases marked with an exclamation mark in column 4 and 5. These examples, which will be discussed below in some detail, also have just the minimal set of fermion zero modes, except for the universal sector (which for Sp(2) instantons contains two extra triplets).\n\nThe main conclusion about this scan is that we did not find any instantons with exactly the zero mode fermions to generate the neutrino mass superpotential. However 44 we have found a number of examples which come very close to that, with exactly the required charged zero modes and a very reduced set of extra uncharged zero modes from the universal sector. These extra zero modes are non-chiral and hence one expects that e.g. RR/NS fluxes or other effects may easily lift them, as we discussed in section 2. Concerning O(1) instantons, which have just the two required fermion zero modes in the universal sector, we have found one example, with the appropriate net structure of charged zero modes. However, it has plenty of other extra zero modes. We discuss examples of O(1) and Sp(2) instantons in the following subsections.\n\nLet us first discuss the case of O(1) instantons. In principle they would be the more attractive since they have no undesirable universal zero modes at all. Unfortunately this type of instanton is rare within the set we scanned, and we found just one example with a solution to the tadpole equations without any unwanted chiral zero-modes. The instanton however has a very large number of uncharged and charged vector-like zero modes.\n\nThe standard model brane configuration occurs for tensor product (2, 4, 22, 22), MIPF 49, orientifold 1, boundaries (a,b,c,d) = (487, 1365, 576, 486) . As usual we only provide this information in order to locate this model in the database. Further details are available on request.\n\nThe bi-fundamental fermion spectrum of this model in the (a,b,c,d) sector is fairly close to the MSSM: there is an extra up-quark mirror pair, two mirror pairs of leptoquarks with down quark charges and one with up-quark charges, plus two extra righthanded neutrinos (i.e. a total of five right-handed neutrinos). There are three MSSM Higgs pairs. The tensor spectrum is far less appealing, in particular for brane c: this has 25 adjoints and 7 vector-like pairs of anti-symmetric tensors.\n\nAs we said, there is just one instanton brane of type O1-. It has exactly the right number of zero-modes with brane d, but five superfluous pairs of vector-like zeromodes with brane c, plus one vector-like pair with brane a. In addition there are four symmetric tensor zero-modes on the instanton brane (which of course are vector-like, since it is a real brane): the parameter p + in table 3 is equal to 2.\n\nThe tadpole solution that is (chirally speaking) compatible with this instanton has a large hidden sector: O(1)×O(2) 4 ×O(3)×U(1) 2 ×Sp(2) 2 ×U(3) (there are other possibilities, but no simple ones). This hidden sector introduces more undesirable features: vector-like observable/hidden matter, vector-like instanton/hidden sector modes, plus 45 chiral and non-chiral matter within the hidden sector. Finally the coupling ratios are as follows: α 3 /α 2 = .54, sin 2 θ w = .094, and the instanton coupling is 3.4 times weaker than the QCD coupling (α 3 /α Instanton = 3.4).\n\nDespite these unappealing features this model does demonstrate the existence of this kind of solution.\n\nAs we have mentioned, these are the examples which come closer to the minimal set of fermion zero modes. As we see in Table 5 , all such instantons satisfying the criteria on the zero mode structure (except for the extra universal zero modes) appear for models based on the same CFT orientifold. It is the one obtained from the (2, 4, 22, 22) Gepner model with MIPF 13 and orientifold 3 in the table. The model is obtained as follows.\n\nWe start with the tensor product (2, 4, 22, 22) . This yields a CFT with 12060 primary fields, 48 of which are simple currents, forming a discrete group G = Z 12 ×Z 2 ×Z 2 . After taking into account the permutation symmetry of the last two factors, we find that this tensor product has 54 symmetric MIPFs, and we choose one of them to build the model of interest. For convenience we specify all quantities in terms of a standard minimal model notation, but also in terms of the labelling of the computer program \"kac\" that generates the spectrum. This particular MIPF is nr. 13. To build it we choose a subgroup of G, which is isomorphic to H = Z 12 × Z 2 . The generator of the Z 12 factor is primary field nr. 1, (0, 0, 0, {24, -24, 0}, {24, 20, 0}), and the Z 2 factor is generated by primary field nr. 24, (0, 0, 0, 0, {24, 20, 2}). The representations are specified on a basis (NSR, k = 2, k = 4, k = 22, k = 22), i.e. the boundary conditions of the NSRfermions and the four minimal models in the tensor product. Here 0 indicates the CFT vacuum, and for all other states we use the familiar (l, q, s) notation for the N = 2 minimal models. The first generator has conformal weight h = 11 12 and has ground state dimension 1. The second has weight h = 11 2 and has ground state dimension 2: the ground state contains both (0, 0, 0, 0, {24, 20, 2}) and (0, 0, 0, {24, 20, 2}, 0) . The matrix X defining the MIPF according to the prescription given in [53] [54][55]\n\nis 1 12 0 0 1 2 (6.3) 46\n\nThis simple current modification is applied to the charge conjugation invariant of the tensor product. This defines a MIPF that corresponds to an automorphism of the fusion rules, and that pairs all the primaries in the CFT off-diagonally. The number of Ishibashi states, and hence the number of boundary states is 1080. The MIPF is invariant under exchange of the two k = 22 factors: this maps current 24 to itself, and current 1 to current 11, which is also in H. Hence this symmetry of the tensor product maps H into itself, and it also preserves the matrix X.\n\nTo define an orientifold, we must specify a \"Klein bottle current\" plus two signs defined on the basis of the simple current group. For the current K we use the generator of the second Z 2 in G, primary field nr. 12. This is the representation (0, 0, {4, -4, 0}, {(24, 16, 2)}, {(24, -12, 2)}) which is degenerate with nine other states, all of dimension 1 and conformal weight 7. The crosscap signs are chosen, on the aforementioned basis of H as (+, -). This results in a crosscap coefficient of 0.0464731, and it is orientifold nr. 3 of a total of 8. The orientifold is also invariant under permutation of the identical factors.\n\nThe closed string spectrum contains 14 vector multiplets and 60 chiral multiplets.\n\nTo build a standard model configuration we have to specify the boundary state labels.\n\nIt turns out that we have four choices for label a and b, one for c and two for d.\n\nThis leads to a total of 32 possibilities. Among these 32 there are 22 have distinct spectra (distinguished by the number of vector-like states), but for all 32 choices one obtains the same set of dilaton couplings. It seems plausible that these choices simply correspond to putting the a, b and d branes in slightly different positions, so that we move the configuration in brane moduli space. The choices are as follows (these are boundary labels assigned by the computer program, and can be decomposed in terms of minimal model representations; this will be explained in table 6 below) a : 10, 22, 130, 142 b : 210, 282, 290, 291 c : 629 d : 712, 797 There are additional possibilities, but they do not give rise to additional distinct spectra. 47 Table 6: Branes appearing in standard model configurations Label Orbit/Deg. Reps Weight Dimension 10 240 (0, 0, 0, 0, {10, 0, 0}) 5/4 1 130 2760 (0, 0, 0, {10, 0, 0}, 0) 5/4 1 22 [528,0] (0, 0, 0, {1, -1, 0}, {11, 1, 0}) 3/2 1 (0, 0, 0, {1, 1, 0}, {11, -1, 0}) 3/2 1 142 [3048,0] (0, 0, 0, {11, -1, 0}, {1, 1, 0}) 3/2 1 (0, 0, 0, {11, 1, 0}, {1, -1, 0}) 3/2 1 210 4248 (0, 0, {3, 3, 0}, {3, -3, 0}, {9, -9, 0}) 1/2 1 282 5760 (0, 0, {3, 3, 0}{9, -9, 0}{3, -3, 0}) 1/2 1 290 [5952,0] (0, 0, {1, 1, 0}{9, 7, 0}{11, -11, 0}) 5/6 1 291 [5952,24] (0, 0, {1,\n\n1, 0}{9, 7, 0}{11, -11, 0}) 5/6 1 629 [9348,30] (0, (1, -1, 0), 0, {9, 9, 0}{5, -3, 0} 7/12 1 712 [9852,0] (0, {1, 1, 0}{3, -3, 0}{1, 1, 0}{5, 5, 0}) 1/2 2 (0, {1, 1, 0}{1, -1, 0}{1, 1, 0}{5, -3, 0}) 1/2 2 797 [10356,30] (0, {1, 1, 0}{3, -3, 0}{5, 5, 0}{1, 1, 0}) 1/2 2 (0, {1, 1, 0}{1, -1, 0}{5, -3, 0}{1, 1, 0}) 1/2 2\n\nThe second column gives the boundary labels in terms of a primary field label and a degeneracy label (boundaries not indicated by square brackets are not degenerate).\n\nThe labels appearing in columns 1 and 2 are assigned by the computer program, and are listed here only for the purpose of reproducing the results using that program.\n\nIn column 2, the boundary labels are expressed in terms of primary field labels, as in formula (A.4). If a single number appears, this is a representative of an H-orbit corresponding to the boundary. If square brackets are used, this means that the Horbit has fixed points, and that it corresponds to more than one boundary label. The second entry in the square brackets is the degeneracy label, and refers to a character of the \"Central Stabilizer\" defined in [43] ; the details of the definition and the labelling will not be important here. In this case the first entry within the square brackets refers to an orbit representative.\n\nThese orbit representatives can also be expressed in a standard form for minimal model tensor products. This is done in column 3. This is basically the same expansion shown in (A.4), except that the degeneracy label Ψ I turns out to be trivial in all cases, 48 both for the standard model and for the instanton branes shown below (although the theory does contain primaries with non-trivial Ψ's). In columns 4 and 5 we specify the weight and ground state dimension of the corresponding highest weight representation.\n\nThese data are not directly relevant for the boundary state, but helps in identifying it.\n\nSince boundaries are specified by orbit representatives, it is not straightforward to compare them, since the standard choice (the one listed in column 2) is arbitrary. For this reason we have used another representative in columns 3, 4 and 5, selected by an objective criterion: we choose the one of minimal dimension and minimal conformal weight (in that order). If there is more than one representative satisfying these criteria we list all.\n\nIn Table 7 we summarize the spectra of the 32 models. The first four columns list the a,b,c,d brane labels. The last eight columns specify the total number of multiplets of types Q (quark doublet), U (up quark singlet), D (down quark singlet), L (lepton doublet), E (charged lepton singlet), N (neutrino singlet), Y (lepto-quark) and H (Higgs).\n\nThe numbers given are for the total number of lefthanded fermions in the representation, plus their complex conjugates. So for example a 7 in column \"Q\" means that there are 5 quark doublets in the usual representation (3, 2, 1 6 ), plus two in the complex conjugate representation (3 * , 2, -1 6 ). This yields the required three families of quark doublets, plus two mirror pairs.\n\nHence the smallest number that can occur in the six columns QUDLEN is three, if there are no mirrors (note that cubic anomaly cancellation requires three right-handed neutrinos in this class of models). The lepto-quarks Y are all in the same representation as the down-quarks (D), or the conjugate thereof, and they occur only as vector-like mirror pairs. They differ from D-type mirror quarks because they carry lepton number, because they come from open strings ending on the d-brane instead of the c-brane.\n\nIn general, there can also exist U-type lepto-quarks, but in these models they do not occur. Finally the numbers 10, 18 and 26 in column 'H' mean that there are 5, 9 or 13 MSSM Higgs pairs H + H. It is worth noticing that right-handed quarks U, D and neutrinos N = ν R do not have vectorlike copies. On the other hand right-handed\n\nleptons E always have one and the left-handed fields Q, L may have up to 3 vector-like copies.\n\nTable 7: Spectrum all 32 configurations. U(3) Sp(2) U(1) U(1) Q U D L E N Y H 10 210 629 712 7 3 3 9 5 3 6 10 22 210 629 712 7 3 3 9 5 3 6 10 130 210 629 712 3 3 3 9 5 3 2 10 142 210 629 712 3 3 3 9 5 3 2 10 10 282 629 712 3 3 3 5 5 3 6 26 22 282 629 712 3 3 3 5 5 3 6 26 130 282 629 712 7 3 3 5 5 3 2 26 142 282 629 712 7 3 3 5 5 3 2 26 10 290 629 712 3 3 3 3 5 3 6 18 22 290 629 712 3 3 3 3 5 3 6 18 130 290 629 712 3 3 3 3 5 3 2 18 142 290 629 712 3 3 3 3 5 3 2 18 10 291 629 712 3 3 3 3 5 3 6 18 22 291 629 712 5 3 3 3 5 3 6 18 130 291 629 712 3 3 3 3 5 3 2 18 142 291 629 712 3 3 3 3 5 3 2 18 10 210 629 797 7 3 3 5 5 3 2 10 22 210 629 797 7 3 3 5 5 3 2 10 130 210 629 797 3 3 3 5 5 3 6 10 142 210 629 797 3 3 3 5 5 3 6 10 10 282 629 797 3 3 3 9 5 3 2 26 22 282 629 797 3 3 3 9 5 3 2 26 130 282 629 797 7 3 3 9 5 3 6 26 142 282 629 797 7 3 3 9 5 3 6 26 10 290 629 797 3 3 3 3 5 3 2 18 22 290 629 797 3 3 3 3 5 3 2 18 130 290 629 797 3 3 3 3 5 3 6 18 142 290 629 797 3 3 3 3 5 3 6 18 10 291 629 797 3 3 3 3 5 3 2 18 22 291 629 797 5 3 3 3 5 3 2 18 130 291 629 797 3 3 3 3 5 3 6 18 Continued on next page 50 Table 7 -continued from previous page U(3) Sp(2) U(1) U(1) Q U D L E N Y H 142 291 629 797 3 3 3 3 5 3 6 18 In the following table we list the multiplicities L aa and L aa ′ of the branes that occur in these models, leading to vector-like sets of adjoints and rank-2 tensors. Since brane b is symplectic, the number of adjoints is equal to the number of symmetric tensors.\n\nTable 8: 4d matter from the aa and aa ′ sectors.\n\nBoundary Adjoints Anti-symm. Symm. a(10) 2 2 6 a(22) 2 2 2 a(130) 2 2 6 a(142) 2 2 2 b(210) -14 10 b(282) -14 10 b(290) -14 6 b(291) -14 6 c(629) 9 -14 d(712) 3 -6 d(797) 3 -6 It should be emphasized that CFT constructions generically correspond to particular points in moduli space of CY orientifolds. Due to this, they usually have an 'enhanced' massless particle content with extra vector-like matter and closed string gauge interactions. Thus one would expect that many of the massless vector-like chiral fields present in this class of models could gain masses while moving to a nearby point in moduli space. 51 6.3.4 The instantons Each of these 32 Standard Model compactifications admits 8 instantons. The instanton labels are identical for all the 32 models. They are listed in Table 9 . The first five columns use the same notation as for the standard model boundary labels. In column 6 we list the numerical value of the dilaton coupling to the instanton brane. This quantity is proportional to 1 g 2 . It is instructive to compare these couplings to the gauge couplings, in order to gain intuition on the suppression factor for our instantons. In these models the U(3) dilaton couplings are 0.00622, so that the instantons are more strongly coupled than QCD 19 On the other hand in this particular model the ratio α 3 /α 2 at the string scale is 3.23 (the value of sin 2 θ w at the string scale is 0.527). All of these couplings are subject to renormalization group running, and there are plenty of vector-like states to contribute to this, if one assumes that they acquire masses at a sufficiently low scale. One should perform a detailed renormalization group analysis to check whether one may obtain consistency with the gauge couplings measured at low-energies. Let us emphasize however that one expects that moving in moduli space many of these vector-like states will gain masses and also the values of the different gauge couplings will also generically vary.\n\nSince the value of the Type II dilaton is a free parameter at this level, one can get the appropriate (intermediate) mass scale for the right-handed neutrino Majorana masses by choosing an appropriate value for the dilaton. In this context, it is satisfactory to verify that the instanton couplings are unrelated to the gauge couplings, as expected since they do not correspond to gauge instantons [3], and are in fact less suppressed than the latter.\n\nNote that the 8 instantons fall into two distinct classes (evidently not related by any discrete symmetry, since the conformal weight on the boundary orbit is distinct, and the coupling is different as well). Within each class, the orbits of the four instanton boundaries appear to be related by the Z 2 symmetries of interchange of the last two tensor factors, and simultaneous inversion of the charge q of the minimal model. However, one has to be very careful in reading off symmetries directly from the labels in columns 3 of Tables ( 6) and (9) for a number of reasons. First of all the entries in column 3 are representatives of boundary orbits, and these representatives themselves are merely representatives of extension orbits. Secondly the action of any discrete 19 Note that the Type II dilaton in this compactifications is an arbitrary parameter which can always be chosen so that we consistently work at weak coupling. It is the relative value of gauge couplings which we are comparing here.\n\nTable 9: Instantons for all 32 configurations Lbl. Orbit/Deg. Reps Weight Dim. coupling 414 [8064,0] (0, {1, 1, 0}, 0, {22, -22, 0}, {20, 16, 0}) 5/2 1 0.0016993 417 [8076,30] (0, {1, -1, 0}, 0, {22, 22, 0}, {20, -16, 0}) 5/2 1 0.0016993 456 [8316,0] (0, {1, 1, 0}, 0, {20, 16, 0}, {22, -22, 0}) 5/2 1 0.0016993 459 [8328,30] (0, {1, -1, 0}, 0, {20, -16, 0}, {22, 22, 0}) 5/2 1 0.0016993 418 [8088,0] (0, {1, 1, 0}, 0, {22, -22, 0}, {18, 16, 0}) 5/3 1 0.0027033 420 [8100,0] (0, {1, -1, 0}, 0, {22, 22, 0}, {18, -16, 0}) 5/3 1 0.0027033 502 [8592,0] (0, {1, 1, 0}, 0, {18, 16, 0}, {22, -22, 0}) 5/3 1 0.0027033 505 [8604,30] (0, {1, -1, 0}, 0, {18, -16, 0}, {22, 22, 0}) 5/3 1 0.0027033 symmetry on the degeneracy labels can be non-trivial. In appendix B we discuss these symmetries in more detail.\n\nThe Sp(2) instanton examples just discussed are the ones which get closer to the required minimal set of fermion zero modes. Under slightly weaker conditions, we find many more solutions. In all these cases some additional mechanism beyond exact RCFT will be needed to lift some undesirable zero modes.\n\nThe simplest such case is the following. The tensor product is (2, 8, 8, 18) , MIPF nr. 14, orientifold 2 (the precise spectra may be found using this information in the database www.nikhef.nl/ ∼t58/filtersols.php). There are three distinct brane configurations for which almost perfect instantons exist, namely (a, b, c, d) = (64, 562, 389, 67) and (64, 577, 389, 67) and (65, 560, 189, 66) . Each has six instantons, three of type S2+ and three of type S2-. As in the foregoing example, the six instantons are identical for the three standard model configurations. In this example, they have three different dilaton coupling strengths: .00254, .00665 and .0108 (each value occurs once for S2+ and once for S2-). By comparison, the U(3)-brane dilaton coupling strength is 0.0119338, so that the instanton brane coupling is quite a bit stronger than the QCD coupling. This is again an interesting point if we want that ν R masses are not too much 53 suppressed. Furthermore in this example there are three distinct instanton couplings, so that one may expect three non-zero eigenvalues (with a hierarchy) in the mass matrix. As in the previous examples there is not gauge coupling unification, one rather has α 3 /α 2 = .4813 and sin 2 (θ w ) = .183 at the string scale. Again a full renormalization group analysis should be performed in order to check consistency with the measured low-energy gauge coupling values.\n\nThese models all have a hidden sector consisting of a single Sp(2) factor. They have respectively 3, 1 and 3 susy Higgs pairs, and a spectrum of bi-fundamentals that is closer to that of the standard model than the previously discussed Sp(2) examples: quarks and leptons do not have vector-like copies (there are only some vector-like leptoquarks), and even one of the three models have the minimal set of Higgs fields of the MSSM. The rest of the spectrum is purely vector-like, and contains a number of rank-2 tensors, including eight or six adjoints of U(3). Furthermore there is vectorlike observable-hidden matter. The only undesirable instanton zero-mode is a single bi-fundamental between the hidden sector Sp(2) brane and the instanton brane. Still, these SM brane configurations without the hidden sector, provide interesting and very simple local models of D-brane sectors admitting instantons generating neutrino masses (with the additional ingredients required to eliminate the extra universal triplets of fermion zero modes).\n\nWe now turn to the generation of other possible superpotentials violating B -L. An instanton violates R-parity if the amount of B -L violation,\n\nI M a -I M a ′ -I M d + I M d ′ (6.4)\n\nis odd. Examples of instantons with that property were found in the following tensor product/MIPF/orientifold combinations: [(1, 16, 16, 16) , 12, 0], [(2, 4, 16, 34) , 49, 0], [(2, 4, 12, 82) , 19, 0] [(2, 4, 22, 22), 49, 0] and [(2, 4, 16, 34), 18, 0] . Note that all cases for which O2 or U2 instantons were found necessarily have R-parity violating instantons as well: the corresponding O1 and U1 instantons have I M d or I M d ′ equal to ±1, whereas the intersection with the a is non-chiral. In principle, there are many more ways to obtain R-parity violating instantons (either due to non-vanishing contributions to\n\nI M a -I M a ′ or higher values of I M d -I M d ′ )\n\n, and indeed, many such instantons turn out to exist. But the number of tensor product/MIPF/orientifold combinations where they occur hardly increases: only in the case [(1, 16, 16, 16), 12, 0] we found R-parity 54 violating instantons, but no U1 or O1 instantons. This suggests that in the other cases R-parity is a true symmetry of the model. Unfortunately we have no way of rigorously ruling out any other non-perturbative effects, but at least the set we can examine respects R-parity. This includes in particular the models without hidden sector (found for [(2, 4, 22, 22), 13, 3] ) discussed above.\n\nThe following table list the total number of instantons with the chiral intersections listed in table 2. The total number of instantons (boundaries violating the sum rule, as defined in (6.1)) is 29680, for all standard model configurations combined. The last four columns indicate how many unitary instantons satisfy the sum rule exactly as listed in table (6.1), how many satisfy it with I M x ↔ -I M x ′ (the column U'), and how many O-type and S-type instantons there are. Here 'S' refers to boundaries with a symplectic Chan-Paton group if the boundary is used as an instanton brane. All intersection numbers for type S have been multiplied by 2 before comparing with table 2. For real branes, the relevant quantities used in the comparison are\n\nI M a -I M a ′ , I M c -I M c ′ and I M d -I M d ′ , while I M b = 0.\n\nThere are fewer unitary instantons possibly generating Majorana masses then the numbers mentioned above because the conditions we use here are stricter: we require here that I M x and I M x ′ match exactly, not just their difference. Note however that this still allows additional vector-like zero-modes. If we only wish to consider cases without any spurious zero-modes, we may limit ourselves to the O-type instantons in the last column. There are very few to inspect, and all of them turn out to have a few non-universal zero modes.\n\nD = 4 Operator U U' S O ν R ν R 1 2 627 3 L HL H 0 5 550 3 L H 3 3 0 4 QDL 8 4 0 4 UDD 0 0 0 4 LLE 8 4 0 4 QQQL 0 4 0 3892 UUDE 4 0 0 3880\n\nTable 10: Number of instantons in our search which may induce neutrino masses (first 2 rows), R-parity violation (next 4 rows) or proton decay operators (last 2 rows).\n\nThe last two cases are B -L preserving dimension five operators, and obviously do not come from the set of 29680 B -L violating instantons. They were searched for 55 separately, but the search was limited to the same 391 models we used in the rest of the paper. Obviously, one could equally well look for such instantons in the full database, since their existence does not require a massive B -L.\n\nIt is interesting to note that in the classes of MSSM-like models discussed earlier in this section with the closest to minimal zero mode structure, there are no instantons al all generating either R-parity violating or the B -L dim=5 operators in the table.\n\nThis makes them particularly attractive.\n\nNote that all numbers in table 10 refer to the occurrence of instantons in the set of 391 tadpole-free models with massive B-L, but without checking the presence of zero-modes between the hidden sector and the instanton. It makes little sense to use the hidden sector in the database for such a check, since this is just one sample from a (usually) large number of possibilities. A meaningful question would be: can one find a hidden sector that has no zero-modes with the instanton. We have done such a search for the B -L violating instantons (see the exclamation marks in the last column of table (5)), but not for the B -L preserving instantons.\n\nIn this paper we have presented a systematic search for MSSM-like Type II Gepner orientifold models allowing for boundary states associated to instantons giving rise to neutrino Majorana masses. This search is very well motivated since neutrino masses are not easily accommodated in the semi-realistic compactifications constructed up to now. String instanton induced Majorana masses provides a novel and promising way to understand the origin of neutrino masses in the string theory context.\n\nThe string instantons under discussion are not gauge instantons. Thus, for example, they not only break B + L symmetry (like 't Hooft instantons do) but also B -L, allowing for Majorana neutrino mass generation. The obtained mass terms are of order M s exp(-V /g 2 ) but this suppression is unrelated to the exponential suppression of e.g. electroweak instantons and may be mild. In fact we find in our most interesting examples that the instanton action is typically substantially smaller than that of QCD or electroweak instantons, and hence these effects are much less suppressed than those coming from gauge theory instantons.\n\nTo perform our instanton search we have analyzed the structure of the zero modes that these instantons must have in order to induce the required superpotential. This analysis goes beyond the particular context of Gepner orientifolds and has general 56 validity for Type II CY orientifolds. We have found that instantons with O(1) CP symmetry have the required universal sector of just two fermionic zero modes for the superpotential to be generated. Instantons with Sp(2) and U(1) CP symmetries have extra unwanted universal fermionic zero modes, which however may be lifted in a variety of ways in more general setups, as we discuss in the text. In fact we find in our search that around 98 % of the instantons with the correct structure of charged zero modes have Sp(2) CP symmetry. Indeed, from a number of viewpoints the Sp(2)\n\ninstantons are specially interesting. The instantons we find with the simplest structure of fermionic zero modes are Sp(2) instantons which are also the ones which are present more frequently in the MSSM-like class of Gepner constructions considered. They have also some interesting features from the phenomenological point of view. Indeed, due to the non-Abelian structure of the CP symmetry, the structure in flavor space of the neutrino Majorana masses factorizes. This makes that, irrespective of what particular compactification is considered, Sp(2) instantons may easily lead to a hierarchical structure of neutrino masses. It would be important to further study the possible phenomenological applications of the present neutrino mass generating mechanism.\n\nString instanton effects can also give rise to other B-or L-violating operators. Of particular interest is the dimension 5 Weinberg operator giving direct Majorana masses to the left-handed neutrinos. We find that in the most interesting cases, different instantons giving rise to the Weinberg operator and to ν R Majorana masses are both simultaneously present. Which effect is the dominant one in the generation of the physical light neutrino masses depends on the values of the instanton actions and amplitudes as well as on the value of the string scale. Instantons may also generate dim< 5 operators violating R-parity. We find however that instantons inducing such operators are extremely rare, and in fact are completely absent in the Gepner models with the simplest Sp(2) instantons inducing neutrino masses.\n\nThere are many avenues yet to be explored. It would be important to understand better the possible sources (moving in moduli space, addition of RR/NS backgrounds etc.) of uplifting for the extra uncharged fermionic zero modes in the most favoured Sp(2) instantons. A second important question is that we have concentrated on checking the existence of instanton zero modes appropriate to generate neutrino masses; one should further check that the required couplings among the fermionic zero modes and the relevant 4d superfields (i.e. ν R or L H) are indeed present in each particular case. This is in principle possible in models with a known CFT description but could be difficult in practice for the Gepner models here described.\n\n57 Instantons can also generate other superpotentials with interesting physical applications. One important example is the generation of a Higgs bilinear (i.e. a µ-term) in MSSM-like models [4, 3]. Thus, e.g., one could perform a systematic search for instantons (boundary states) generating a µ-term in the class of CFT Gepner orientifolds considered in the present article. Other possible application is the search for instantons inducing superpotential couplings involving only closed string moduli. The latter may be useful for the moduli-fixing problem, or for non-perturbative corrections to perturbatively allowed couplings [56].\n\nFinally, it would be important to search for analogous instanton effects inducing neutrino masses in other string constructions (heterotic, M-theory etc.). A necessary condition is that the anomaly free U(1) B-L gauge boson should become massive due to a Stückelberg term.\n\nThe importance of neutrino masses in physics beyond the Standard Model is unquestionable. We have shown that string theory instantons provide an elegant and simple mechanism to implement them in semi-realistic MSSM-like string vacua, and a powerful constraint in model building. In our opinion, the conditions of the existence of appropriate instantons to generate neutrino masses should be an important guide in a search for a string description of the Standard Model.\n\nWe thank M. Bertolini, R. Blumenhagen, S. Franco, M. Frau, S. Kachru, E. Kiritsis, A. Lerda, D. Lüst, F. Marchesano, T. Weigand for useful discussions. A.M.U. thanks M. González for encouragement and support. The research of A.N. Schellekens was funded in part by program FP 57 of the Foundation for Fundamental Research of Matter (FOM), and Research Project FPA2005-05046 of de Ministerio de Educacion y Ciencia, Spain. The research by L.E. Ibáñez and A.M. Uranga has been supported by the European Commission under RTN European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT (Spain), and the Comunidad de Madrid under project HEPHACOS P-ESP-00346.\n\nAppendix A CFT Notation\n\nHere we summarize the labelling conventions for various CFT quantities. Further details and explanations can be found in [43] .\n\nIt is important to keep in mind that there are four steps in the construction, each involving choices of some quantities. The steps are\n\n• A CFT tensor product\n\n• An extension of the chiral algebra of this tensor product\n\n• The choice of a MIPF\n\n• The choice of an orientifold The second and third step are easily confused. A MIPF can itself be of extension type (although it can also be of automorphism or mixed type), meaning that it implies an extension of the chiral algebra. The crucial difference between step two and three in that case is that in step 2 all fields that are non-local with respect to the extension are projected out, and the symmetry of the extension is imposed on all states of the CFT,\n\ni.e in particular on all boundary states. The extension in step three acts as a bulk invariant, but the boundary states are not required to respect the symmetry implied by the extension.\n\nPrimary fields of N = 2 minimal models are labeled in the usual way by three integers (l, q, s). In addition to these minimal models, one building block of our CFT's is of course a set of NSR fermions in four dimensions. They can be represented by the four conjugacy classes (0), (v), (s), (c) analogous to those of a root lattice of type D. Primary fields in a tensor product of M factors are therefore labelled as\n\nI = ((x), (l 1 , q 1 , s 1 ), . . . , (l M , q M , s M )) (A.1)\n\nwhere x = 0, v, s or c.\n\nThis tensor product is extended by the alignment currents and the spin-1 field corresponding to the space-time supersymmetry generator. This organizes the tensor product fields into orbits, which can be labelled by one of the elements of the orbit. We always choose the field of minimal conformal weight (or one of them, in case there 59 are more) as the orbit representative labelling the orbit. The supersymmetry generator may have fixed points, leading to orbits appearing more than once as primary fields of the extended theory. In those cases we need an additional degeneracy label to distinguish them. It is convenient to choose for this label a character of the discrete group that is causing the degeneracy, the \"untwisted stabilizer\", which depends on I.\n\nDenoting this character as Ψ I we get then the following set of labels for the primaries of the extended CFT\n\ni = [I, Ψ I ] (A.2)\n\nwhere I has the form (A.1). If there are no degeneracies we will leave out the square brackets and the Ψ I .\n\nIn boundary CFT's two new labels appear: the labels of Ishibashi-states that propagate in the transverse channel of the annulus, and the boundary labels. In the simplest, \"Cardy\" case both sets of labels are in one-to-one correspondence with the extended CFT labels i. But if we consider non-trivial MIPFs Z ij both sets of labels are different.\n\nThe Ishibashi states are in one-to-one correspondence with the fields i with Z ii c = 0. Degeneracies can occur here if Z ii c > 0. This requires the introduction of a degeneracy label. Such degeneracies may occur if the stabilizer of i (the set of simple currents that fix i) is non-trivial. It is convenient to use elements J of the stabilizer as degeneracy labels, so that the Ishibashi labels get the following form m = (i, J) , (A.3) where i is an extended CFT label, as defined above (to be precise, in some cases a non-trivial degeneracy label is introduced even if Z ii c = 1. The details will not matter here).\n\nBoundary states correspond to orbits of the simple current group H that defines a MIPF. To label such orbits we choose a representative. There is no obvious canonical representative (one could use one of minimal conformal weight, but the conformal weight of orbit members of a boundary state does not play any rôle in the formalism, unlike the conformal weight of a primary). So in this case we just make an arbitrary choice. Once again there can be degeneracies. In this case they are due to a subgroup of the stabilizer called the \"Central Stabilizer\". It is convenient to label the boundary states by an orbit representative i and a character ψ i of the central stabilizer. If we expand the boundary state label in all of its components we get\n\na = [i, ψ i ] = [[I, Ψ I ], ψ i ] = [((x), (l 1 , q 1 , s 1 ), . . . , (l M , q M , s M )), Ψ I ], ψ i ] . (A.4) 60\n\nNote that i is just a representative of a boundary orbit, and that I is just a representative of an orbit of the extension of the CFT.\n\nIn the hidden-sector free example discussed in some detail in section 6 we have encountered Sp(2)-type instantons, the most common kind in our scan. This particular model is the one that comes closest to the required zero mode count, although the only superfluous zero modes are rather awkward. Let us assume that the effect of these superfluous universal zero-modes instantons can be avoided. Then there is still another problem we have to face, namely that the two zero modes α i and γ i are related by an Sp(2) transformation of the label i. Then we we need at least three independent instantons (with unrelated couplings) to generate three non-zero neutrino masses, as discussed in Section 3.3. Since the technology to compute the couplings is not yet available, we cannot be completely sure that the relevant couplings are distinct, or indeed that they are non-vanishing, but at least we can inspect if there are obvious symmetries relating them.\n\nThe unextended tensor product (2, 4, 22, 22) has 64 discrete symmetries: five separate charge conjugations of the factors (including the NSR space-time factor) and the interchange of the two identical k = 22 minimal models. To get space-time supersymmetry this tensor product is extended with the product of the simple current Ramond ground states of each factor. These Ramond ground states are not invariant under charge conjugation. Therefore this extension breaks the discrete symmetries to Z 2 × Z 2 , the combined charge conjugation of all five factors and the permutation of the two identical factors. The combined charge conjugation also acts non-trivially on the Ramond ground states in each factor, but the result is the charge conjugate of the space-time supersymmetry generator, which is in the chiral algebra. The combined conjugation is in fact the charge conjugation symmetry of the extended CFT. It turns out that only a Z 2 subgroup of Z 2 × Z 2 acts non-trivially on the simple currents of the extended CFT: the permutation of the k = 22 factors acts in the same way as charge conjugation. The action of these symmetries on the complete set of primary fields is more complicated. It is easy to see that the permutation acts differently than charge conjugation. In general the primary fields of the extended CFT are labelled as i = [((x), (i 1 ), . . . , (i 4 )), Ψ] (see Appendix A). The action of the permutation is to interchange i 3 and i 4 , but in cases with a non-trivial degeneracy it is not a priori 61 clear which of the degenerate states is the image of the map. This can be resolved by examining the fusion rules, which should be invariant under the permutation: N ijk = N π(i)π(j)π(k) , (B.1) where N is a fusion coefficient and π a permutation or other automorphism. In general there may be more than one way to resolve these ambiguities, resulting in additional automorphism of the CFT. The standard example of this situation is the extension of the affine algebra A 1 level 4 by the simple current. The resulting CFT has an outer automorphism, non-trivial charge conjugation, that has no counterpart in A 1 level 4.\n\nAs mentioned above, the Z 2 permutation symmetry is respected by the MIPF and the orientifold, and since charge conjugation acts in the same way on the simple currents as the permutation, charge conjugation is respected as well.\n\nIn this way we end up with (at least) a surviving Z 2 × Z 2 discrete symmetry acting on the boundary labels, or a larger discrete symmetry if that symmetry is extended by the action on the degeneracy labels of the extension. The foregoing story repeats itself for the action on the boundary labels. The boundary labels are given in terms of the CFT labels plus a second degeneracy label, the one indicated by the second entry in the square brackets in column 2 of tables (6) and (9). Once again one has to determine not only how a symmetry acts on the first entry (this is just the action of the symmetry in the extended CFT, respecting its fusion rules), but also how it acts on the degeneracy labels. In this case the precise action can be determined from the invariance of the annulus coefficients A i ab = A π(i) π(a)π(b) , (B.2)\n\nwhere π is the action on the primaries of the extended CFT (as determined above) and π is the action on the boundary labels induced by π.\n\nSince the orientifold choice is non-trivial, boundary charge conjugation does not coincide with CFT charge conjugation. Indeed, the eight instanton boundary states are invariant under boundary charge conjugation (which they must be in order to produce a \"real\" Sp(2)-type instanton). However, just as permutations, CFT charge conjugation may induce a non-trivial discrete symmetry on the boundary states.\n\nIn addition to these \"outer automorphisms\" there is the notion of boundary simple currents, introduced in the appendix of [6] . These may be thought of as remnants of the original simple currents, and imply relations between annulus amplitudes of the form A i ab ′ = A i Ja(Jb) ′ (B.3) 62 All of the aforementioned symmetries might relate instanton couplings, and hence threaten their numerical independence. However, in order to do that they have to be symmetries of the full standard model/instanton configuration, not just relate some of the eight instantons to each other. It is easy to see that the permutation of the k = 22 factors changes the standard model brane configuration. Consider brane c: it turns out that under permutation boundary state 629 it is mapped to boundary state 544 or 545 (depending on the action on the degeneracy label), which in any case is distinct. Hence even if the instanton boundaries 414 and 456 resp. 418 and 502 are mapped to each other by boundary permutation, at the same time the standard model configuration is\n\nmapped to a distinct one.\n\nThis means that we may expect at least four distinct couplings, which should be sufficient. It is of course possible to work out the discrete symmetries exactly, but in view of this argument this would not yield any additional insight.\n\nWe do know the exact boundary orbits. The orbit of instanton label 414 is (414, 415, 416, 417) , so that instantons 414 and 417 are related. But the orbit of brane c under the same action is (629, 628, 626, 627) . Hence the action that relates 414 and 417 maps 629 to 627. In fact all four standard model boundaries a,b,c,d are mapped to different ones. This implies that instantons 414 and 417 may produce different couplings as well, so that all eight instantons may contribute in a different way.\n\nThese distinctions concern the disk correlators d (r) a in (3.12). The factors exp(-Re U r ) will be related by discrete symmetries, and it seems reasonable to expect them to be identical for instantons 414, 417, 456 and 459, which is indeed correct. However there is no reason to expect the other four instantons to have the same suppression factor, and indeed they do not.\n\nNote that these symmetries imply the existence of a much larger set of standard model configurations than the 32 discussed here. However, as mentioned before, the 32 models considered here display all possible distinct spectra."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction","text":"String Theory, as the leading candidate for a unified theory of Particle Physics and Gravity, should be able to describe all observed particle phenomena. One the most valuable experimental pieces of information obtained in the last decade concerns neutrino masses. Indeed the evidence from solar, atmospheric, reactor and accelerator experiments indicates that neutrinos are massive. The simplest explanation of the smallness of neutrino masses is the see-saw mechanism [1] . The SM gauge symmetry allows for two types of operators bilinear on the neutrinos (with dimension ≤ 4) :\n\nL ν = M ab ν a R ν b R + h ab ν a R HL b (1.1)\n\nwhere ν R is the right-handed neutrino, L is the left-handed lepton doublet and H is the Higgs field. In supersymmetric theories, this term arise from a superpotential with the above structure, upon replacing fields by chiral superfields. If M ab is large, the lightest neutrino eigenvalues have masses\n\nM ν = < H > 2 h T M -1 h (1.2)\n\nFor M ∼ 10 10 -10 13 GeV and Dirac neutrino masses of order charged lepton masses, the eigenvalues are consistent with experimental results.\n\nWhat is the structure of neutrinos and their masses in string theory? In specific compactifications giving rise to the MSSM spectra singlet fields corresponding to righthanded neutrinos ν R generically appear. Dirac neutrino masses are also generically present but the required Majorana ν R masses are absent. This is because most MSSMlike models constructed to date have extra U(1) symmetries, under which the righthanded neutrinos are charged, which hence forbid such masses. In many models, such symmetries are associated to a U(1) B-L gauge boson beyond the SM. In order to argue for the existence of ν R masses, string model builders have searched for nonrenormalizable couplings of the type (ν R ν R NR NR ) with extra singlets N R . Once the latter fields get a vev, U(1) B-L is broken and a Majorana mass appears for the ν R .\n\nAlthough indeed such couplings (or similar ones with higher dimensions) exist in some semi-realistic compactifications, such a solution to the neutrino mass problem in string theory has two problems: 1) The typical ν R masses so generated tend to be too small due to the higher dimension of the involved operators and 2) The vevs for the N R fields breaks spontaneously R-parity so that dimension 4 operators potentially giving rise to fast proton decay are generated. This is in a nutshell the neutrino problem in string compactifications (see [2] for a recent discussion in heterotic setups). 1 In [3] (see also [4] ) two of the present authors pointed out that there is a builtin mechanism in string theory which may naturally give rise to Majorana masses for right-handed neutrinos. It was pointed out that string theory instantons may generate such masses through operators of the general form M string e -U ν R ν R . (1.3) Here U is a linear combination of closed string moduli whose imaginary part gets shifted under a U(1) B-L gauge transformation in such a way that the operator is fully gauge invariant. The exponential factor comes from the semi-classical contribution of a certain class of string instantons. This a pure stringy effect distinct from the familiar gauge instanton effects which give rise to couplings violating anomalous global symmetries like (B+L) in the SM. Here also (B-L) (which is anomaly-free) is violated.\n\nThis operator is generated due to existence of instanton fermionic zero modes which are charged under (B -L) and couple to the ν R chiral superfield. Although the effect can take place in different constructions, the most intuitive description may be obtained for the case of Type IIA CY orientifold compactifications with background D6-branes wrapping 3-cycles in the CY. In the simplest configurations one has four SM stacks of D6-branes labeled a, b, c, d which correspond to U(3), SU(2) (or U(2)), U(1) R and U(1) L gauge interactions respectively, which contain the SM group. One can construct compactifications with the MSSM particle spectrum in which quarks and leptons lie at the intersections of those SM D6-branes. Then the relevant instantons correspond to euclidean D2-branes wrapping 3-cycles in the CY (satisfying specific properties so as to lead to the appropriate superspace interaction). The D2-D6 intersections lie the additional fermionic zero modes which are charged under (B -L). For instantons with the appropriate number of intersections with the appropriate D6-branes, and with open string disk couplings among the zero modes and the ν R chiral multiplet (see fig.(3.1)), the operator in (1.3) is generated.\n\nThe fact that the complex modulus U transforms under U(1) B-L gauge transformations indicates that the U(1) B-L gauge boson gets a mass from a Stückelberg term.\n\nSo a crucial ingredient in the mechanism to generate non-perturbative masses for the ν R 's is that there should be massless U(1) B-L gauge boson which become massive by a Stückelberg term. It turns out that not many semi-realistic models with U(1) B-L mass from Stückelberg couplings have been constructed up to date. In the literature there are two main classes of RR tadpole free models with massive B-L. The first class are non-susy type IIa toroidal orientifold models first constructed in [5] . The second class are the type II Gepner orientifold models constructed by one of the present authors and 2 collaborators [6, 7] . The former were already considered in [3] . In the present paper we will concentrate on the RCFT Gepner model constructions, which lead to a large class of MSSM like models, more representative of the general Calabi-Yau compactifications (for a recent discussion of instanton-induced neutrino masses in a model with no RR tadpole cancellation, see [8] ).\n\nThe class of constructions in [6, 7] start with any of the 168 Type II compactifications obtained by tensoring N = 2 SCFT minimal models. In addition one can choose a number of modular invariant partition functions (MIPF), leading to a total of 5403. Then different consistent orientifold projections are performed on the different models. This yields a total of 49304 Type II orientifolds. The open string sector of the theory is defined in terms of the boundary states of the theory. Intuitively, they play the same role as D-branes wrapping cycles in the geometrical settings. Thus one associates boundary states a, b, c, d to the gauge groups giving rise to the SM. Different choices for the SM boundary states lead to different spectra. In the present paper we will make use of the data in [6] which contains 211634 different MSSM-like spectra (including also different hidden sectors). Although this number is huge, most of these models are really extensions of the MSSM, since they have either an extra U(1) B-L or SU(2) R × U(1) B-L group factor beyond the SM group. As we said, we are actually only interested in models in which the U(1) B-L gets a Stückelberg mass. Then we find that the number of MSSM-like models with these characteristics is dramatically reduced: only 0.18 percent of the models (391) have a massive U(1) B-L .\n\nAs we said, in the geometrical setting of IIA orientifolds with intersecting D6-branes [9, 10] (see [11] for reviews and [12, 13] for the IIB counterparts), instantons are associated to D2-branes wrapping 3-cycles, like the background D6-branes do. Analogously, in the RCFT setting the same class of boundary states appearing in the SM constructions are the ones corresponding to instantons. The zero modes on the instanton is computable from the overlaps of instanton brane boundary states (zero modes uncharged under the 4d gauge group) or of instanton and 4d spacefilling brane boundary states (zero modes charged under the corresponding gauge factor). We find that the criteria for a non-perturbative superpotential to be generated [14] are only fulfilled if the Chan-Paton (CP) symmetry of the instantons is O(1). For instantons with CP symmetry 1 Sp(2) or U(1) we find that there are a few extra uncharged fermionic zero modes which would preclude the formation of the searched superpotentials. On the other hand we argue that the addition of fluxes and/or possible non-perturbative 1 We adopt the convention that the fundamental representation of Sp(m) is m-dimensional.\n\nextensions of the orientifold compactifications would allow also instantons with Sp(2) and U(1) symmetries to generate such superpotentials. We thus include all O(1), Sp(2) and U(1) instantons 2 in our systematic search. The computation of charged and uncharged fermion zero modes may be easily implemented as a routine in a systematic computer search for instanton zero modes in Gepner MSSM-like orientifolds. Results of such a systematic computer search are presented in this article.\n\nWe find that out of the 391 models with massive U(1) B-L , there are very few admitting instantons with the required minimal O(1) CP symmetry, and in fact none of them without additional vector-like zero modes. On the other hand we do find 32 models admitting Sp(2) symmetric instantons with just the required charged zero mode content (and the minimal set of non-chiral fermion zero modes). In fact they are all variations of the same orientifold Gepner model based on the tensor product 2, 4, 22, 22) . These models all in fact correspond to the same MIPF and orientifold projection, they only differ on which particular boundary states corresponding to the four a, b, c, d SM 'stacks'. All models have the same chiral content, exactly that of the MSSM , with extra vectorlike chiral fields which may in principle become massive in different points of the CY moduli space. They have no hidden sector, i.e., the gauge group is just that of the SM. For each of the models there are 8 instantons with Sp(2) CP symmetry with just the correct charged zero mode structure allowing for the superpotential coupling (1.3) giving rise to ν R Majorana masses. As we said, they have extra uncharged fermion zero modes beyond the two required to generate a superpotentials. However one would expect that these unwanted zero modes might be lifted in more generic situations in which e.g. NS/RR fluxes are added.\n\n(k 1 , k 2 , k 3 , k 4 ) = (\n\nWe thus see that, starting with a 'large' landscape of 211634 MSSM-like models, and searching for instantons inducing neutrino masses, we find there are none admitting the O(1) instantons with exactly the required zero mode structure, and only few (32) examples with Sp(2) instantons with next-to-minimal uncharged zero mode structure (and exactly the correct charged zero modes). Let us emphasize though that it is the existence of massive U(1) B-L models which is rare. Starting with the subset of models with a massive U(1) B-L , finding models with correct instanton charged zero modes within that class is relatively frequent, 10 percent of the cases. Furthermore, we will see that there are further models beyond those 32 which contain extra non-chiral instanton zero modes and which may also be viable if these modes get massive by some 2 We refer to the different kinds of instanton by their Chan-Paton symmetry on their volume. Since we are not interested in gauge theory instantons, this notation should not be confusing.\n\n4 effect (like e.g. the presence of RR/NS fluxes).\n\nInstantons may generate some other interesting superpotential couplings in addition to ν R masses, some possibly beneficial and others potentially dangerous. In particular we find that in the models which contain Sp(2) instantons which might induce ν R masses, there are also other instantons which would give rise directly to the Weinberg operator [15] L W = λ M (LHLH) (1.4) Once the Higgs field gets a vev, this gives rise directly to left-handed neutrino masses.\n\nThus we find that in that class of models both the see-saw mechanism (which also gives rise to a contribution to the Weinberg operator) and an explicit Weinberg operator might contribute to the physical masses of neutrinos. Which effect dominates will depend on the relative size of the corresponding instanton actions as well as on the size of the string scale. Among potentially dangerous operators which might be generated stand the R-parity violating operators of dimension < 5, which might give rise e.g. to fast proton decay. We make an study of the possible generation of those, and find that in all models in which ν R masses might be generated R-parity is exactly conserved. This is a very encouraging result.\n\nA natural question to ask is whether one can say something about the structure of masses and mixings for neutrinos. As argued in [3] generically large mixing angles are expected, however to be more quantitative we also need to know the structure of Yukawa couplings for leptons. In principle those may be computed in CFT but in practice this type of computation has not yet been developed for CFT orientifolds.\n\nNevertheless we show that, in the case of instantons with Sp(2) CP symmetry, a certain factorization of the flavor structure takes place, which could naturally give rise to a hierarchical structure of eigenvalues for neutrino masses.\n\nThe structure of this article is as follows. In the next section we present a discussion of instanton induced superpotentials in Type II orientifolds. This discussion will apply both to Type IIA and Type IIB CY orientifolds as well as to more abstract CFT orientifolds. We discuss the structure of both uncharged and charged instanton zero modes. In particular we show that only instantons with O(1) CP symmetry have the appropriate uncharged zero mode content to induce a superpotential contribution. We also discuss how Sp(2) and U(1) might still generate superpotential contributions if extra ingredients are added to the general setting. In section 3 we apply that discussion to the specific case of the generation of ν R Majorana masses, showing what is the required zero mode structure in this case. We show how the flavor structure of the 5 Majorana mass term factorizes in the case of instantons with Sp(2) CP symmetry, leading potentially to a hierarchical structure of eigenvalues. We further discuss the generation of other B/L-violating operators including the generation of the Weinberg operator as well as R-parity violating couplings. In section 4 we review the RCFT Type II orientifold constructions in [6, 7] . A general discussion of zero fermion modes for instantons in RCFT orientifolds is presented in section 5.\n\nThe results of our general search for instantons generating ν R masses are presented in section 6. We provide a list of all Gepner orientifolds which admit instanton configurations potentially giving rise to ν R Majorana masses. We describe the structure of the models with Sp(2) instantons having the required charged zero modes for that superpotential to be generated. We also describe the boundary states of the corresponding instantons and provide the massless spectrum of the relevant MSSM-like models. Other orientifolds with zero mode structure close to the minimal one are also briefly discussed. A brief discussion about the possible generation of R-parity violating superpotentials is included. We leave section 7 for some final comments. Some notation on the CFT orientifold constructions, and a discussion of the CFT symmetries in the Sp(2) examples are provided in two appendices.\n\nAs this paper was ready for submission, we noticed [16, 17] , whose discussion of instanton zero modes partially overlaps with our analysis in Section 2.2."},{"section_type":"OTHER","section_title":"Instanton induced superpotentials in Type II orientifolds","text":"In this Section we review the generation of superpotentials involving 4d charged fields via D-brane instantons in type II compactifications. The discussion applies both to type IIA and IIB models, and to geometrical compactification as well as to more abstract internal CFT's. For recent discussions on D-brane instantons, see [4, 3, 25, 8] .\n\nBefore starting, a notational remark in in order. Our notation is adapted to working in the covering theory, namely the type II compactification, and orientifolding in a further step. Thus we describe the brane configurations as a system of branes (described by boundary states for abstract CFT's), labeled k, and their orientifold images labeled k ′ . Similarly, we denote M the brane / boundary state corresponding to the instanton brane, and M ′ its orientifold image. If a brane is mapped to itself under the orientifold action, we call it a 'real' brane / boundary, and 'complex' otherwise. 6 2.1 D-brane instantons, gauge invariance and effective operators A basic feature of type II orientifold compactifications with D-branes is the generic presence of Stückelberg couplings between the U(1) gauge fields on the D-branes, and certain 4d RR closed string 2-forms. These couplings are required by the Green-Schwarz mechanism when the U(1)'s have non-zero triangle contributions to mixed anomalies [18, 19] , but can also exist for certain non-anomalous U(1)'s [5, 20]. These couplings make the U(1) gauge bosons massive, but the symmetry remains as a global symmetry exact in perturbation theory. Since the closed string moduli involved are scalars (0-forms) in the RR sector, the natural candidate non-perturbative effects to violate these U(1) symmetries are instantons arising from euclidean D-branes coupling to these fields.\n\nIn computing the spacetime effective interaction mediated by the instanton, one needs to integrate over the instanton zero modes. In the generic case (and in particular for the case of our interest) there are no bosonic zero modes beyond the universal ones (namely, the four translational bosonic zero modes associated to the position of the instanton). On the other hand, the structure of fermion zero modes will be crucial.\n\nSince we are interested in models with non-trivial 4d gauge group, arising from a set of background 4d spacetime filling branes, we consider separately fermion zero modes which are uncharged under the 4d gauge group and those which are charged. In this paper we restrict our discussion to 4d N = 1 supersymmetric models, and this will simplify the analysis of zero modes.\n\nFermion zero modes which are uncharged under the 4d gauge group determine the kind of 4d superspace interaction which is generated by the instanton. We are interested in generating superpotential interactions, which, as is familiar, requires the instanton to have two fermion zero modes to saturate the d 2 θ superspace integration.\n\nFor this, a necessary (but not sufficient!) condition is that the D-branes are half-BPS, so these fermion zero modes are the Goldstinos of the two broken supersymmetries.\n\nIn the string description, uncharged zero modes arise from open strings in the MM sector (in our notation, the one leading to adjoint representations), which in particular contain these Goldstinos, and the MM ′ sector (in our notation, the one leading to two-index symmetric or antisymmetric tensors). Note that both are the same for 'real' branes. Hence we are primarily interested in D-branes whose MM sector contains just two fermion zero modes, and whose MM ′ sector (for 'complex' branes) does not contain additional fermion zero modes.\n\nIn analogy with the familiar case of gauge theory instantons [21], charged fermion zero modes determine the violation of perturbative global symmetries by the instantoninduced interaction. Namely, in order to saturate the integration over the charged fermions zero modes, the spacetime interaction must contain insertions of fields charged under the 4d gauge symmetry, and in particular under the global U(1) factors, which are thus violated by the D-brane instanton. Notice that this holds irrespectively of the number of uncharged fermion zero modes, namely of the kind of superspace interaction induced by the instanton. Restricting to superpotential interactions, the resulting operator in the 4d effective action has roughly the form W inst = e -U Φ 1 . . . Φ n (2.1) Here the fields Φ 1 , . . . , Φ n are 4d N = 1 chiral multiplets charged under the 4d gauge group, and in particular also under the U(1) symmetries. Note also that the instanton amplitude contains a prefactor (which in general depends on closed and open string moduli) arising from the Gaussian path integral over (massive) fluctuations of the instanton (hence described by an open string annulus partition function, see [22, 23] for related work), which we can ignore for our purposes in this paper.\n\nFor D-brane instantons, U is the closed string modulus to which the euclidean Dbrane couples. In the D-brane picture, instanton fermion zero modes charged under the gauge factor carried by the k th stack of 4d space-filling branes (and its image k ′ ) arise from open strings in the Mk and Mk ′ sectors, transforming as usual in the ( M , k ) and ( M , k ) representations, respectively (with both related in the case of 'real' branes). The (net) number of instanton fermion zero modes with such charges is given by certain multiplicities 3 I M k , I M k ′ .\n\nA D-brane instanton, irrespectively of the superspace structure of the 4d interactions it may generate, violates U(1) k charge conservation by an amount I M k -I M k ′ for 'complex' branes and I M k for 'real' branes. In particular, this is the total charge of the field theory operator Φ 1 . . . Φ n in (2.1). From the Stückelberg couplings, it is possible to derive [3] (see [23, 24, 25] for related work, also [26]) 3 In geometric type IIA compactifications with 4d spacetime-filling branes and instanton branes given by D6-and D2-branes wrapped on Special Lagrangian 3-cycles, I Mk corresponds to the intersection number between the 3-cycles corresponding to the k th D6-and the D2-brane M (and similarly for I Mk ′ ). In geometric type IIB compactifications, it corresponds to the index of a suitable Dirac operator. In general (even for abstract CFT's) it can be defined as the Witten index for the 2d theory on the open string with the boundary conditions corresponding to the two relevant branes. We will often abuse language and refer to this quantity as intersection number, even in Section 6 where we work in the non-geometric regime of type IIB compactifications.\n\nthat for 'complex' instantons, gauge transformations of the U(1) k vector multiplets V k → V k + Λ k , transform U as U → U + k N k (I M k -I M k ′ )Λ k (2.2) For 'real' brane instantons, which were not considered in [3], the shift is given by 4 U → U + k N k I M k Λ k (2.3) (this new possibility will be an important point in our instanton scan in Section 6).\n\nThe complete interaction (2.1) is invariant under the U(1) gauge symmetries. However, from the viewpoint of the 4d low-energy effective field theory viewpoint, it leads to non-perturbative violations of the perturbative U(1) global symmetries, by the amount mentioned above.\n\nIn the string theory construction there is a simple microscopic explanation for the appearance of the insertions of the 4d charged fields (related to the mechanism in [27] ). The instanton brane action in general contains cubic terms α Φ γ, involving two instanton fermions zero modes α in the ( M , k ) and γ in the ( p , M ) coupling to the 4d spacetime field Φ in the ( k , p ) of the 4d gauge group 5 . Performing the Gaussian path integral over the instanton fermion zero modes leads to an insertion of Φ in the effective spacetime interaction. In general, and for a 'complex' instanton, there are several fermion zero modes α i , γ i in the fundamental (resp. antifundamental) of the instanton gauge group, coupling to a 4d spacetime chiral operators O ij (which could possibly be elementary charged fields, or composite chiral operators). Gaussian integration over the fermion zero modes leads to an insertion of the form det O (for 'real' brane instantons, det O should be interpreted as a Pfaffian). It is straightforward to derive our above statement on the net charge violation from this microscopic mechanism.\n\nNote that the above discussions show that instantons in different topological sectors (namely with different RR charges, and different intersection numbers with the 4d spacefilling branes) contribute to different 4d spacetime operators. In particular, multiwrapped instantons, if they exist as BPS objects, contribute to operators different 4 Equivalently, one may use (2.2), but must include an additional factor of 1/2 from the reduction of the volume for a real brane (which is invariant under the orientifold action). 5 Although there is no chirality in 0+0 dimension, the fermion zero modes α and γ are distinguished by their chirality with respect to the SO(4) global symmetry of the system (which corresponds to rotations in the 4d spacetime dimensions, longitudinal to the space-filling branes, and transverse to the instanton brane). Supersymmetry of the instantons constrains the couplings on the instanton action (such as the cubic couplings above) to have a holomorphic structure.\n\n9 from the singly wrapped instanton. This implies that the instanton expansion for a fixed operator is very convergent, and could even be finite.\n\nAnother important implication of the above discussion is that, in order to generate a specific operator via an instanton process, a necessary condition is that the instanton has an appropriate number and structure of charged zero modes. However, this is not sufficient. Insertions of 4d fields appear only if the fields couple to the instanton fermion zero modes via terms at most quadratic in the zero modes. In equivalent terms, only zero modes appearing in the Gaussian part of the instanton action can be saturated by insertions of 4d fields (those to which they couple). The requirement that the zero modes have appropriate couplings to the 4d fields may be non-trivial to verify in certain constructions. This is the case for the Gepner model orientifolds in coming sections, whose couplings are computable in principle, but unknown in practice. In such cases we will assume that any coupling which is not obviously forbidden by symmetries will be non-vanishing. Unfortunately there are no arguments to estimate the actual values of such non-vanishing couplings, hence we can argue about the existence of certain instanton induced operators, but not about the coefficients of such terms."},{"section_type":"OTHER","section_title":"Zero mode structure for D-brane instantons","text":"In this section we describe more concretely different kinds of instantons and the structure of interesting and unwanted zero modes. Our discussion will be valid for general D-brane models in perturbative type II orientifolds without closed string fluxes, although we also make some comments on more general F-theory vacua and the effects of fluxes. A more specific discussion is presented in Section 5."},{"section_type":"OTHER","section_title":"Uncharged zero modes","text":"We start discussing zero modes uncharged under the 4d gauge group. These are crucial in determining the kind of superspace interaction induced by the instanton on the 4d theory. In particular, we are interested in instantons contributing to the 4d superpotential, namely those which contain just two fermion zero modes in this sector. We are also interested in instantons which may contain additional fermion zero modes, and the possible mechanisms that can be used to lift them. Let us discuss 'real' and 'complex' brane instantons in turn.\n\n• Real brane instantons Real brane instantons correspond to branes which are mapped to themselves by the 10 orientifold action, hence M = M ′ . Uncharged zero modes arise from the MM open string sector. As discussed in Section 5, there is a universal sector of zero modes, in the sense that it is present in any BPS D-brane instanton, which we now describe. Before the orientifold projection, we have a gauge group U(n) on the volume of n coincident instantons. Notice that, although there are no gauge bosons in 0 + 0 dimensions, the gauge group is still well-defined, since it acts on charged states (open string ending on the instanton brane). There are four real bosonic zero modes and four fermion zero modes in the adjoint representation. For the minimal U(1) case, the four bosons are the translational Goldstones. The four fermions arise as follows. This sector is insensitive to the extra 4d spacefilling branes, and feels an accidental 4d N = 2 supersymmetry.\n\nThe BPS D-brane instanton breaks half of this, and leads to four Goldstinos, which are the described fermions 6 .\n\nThe orientifold projection acts on this universal sector as follows (see Section 5 for further discussion). The gauge group is projected down to orthogonal or symplectic.\n\nHence instanton branes with symplectic gauge group must have even multiplicity (a related argument, in terms of the orientifold action on Chan-Paton indices, is given in Section 5). For instantons with O(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes (related by the two supercharges of 4d N = 1 supersymmetry broken by the instanton) to the two-index symmetric representation, and the other two fermion zero modes (related by the other two supercharges of the accidental 4d N = 2 in this sector) to the antisymmetric representation. Hence for O(1) instantons (namely instantons with O(1) gauge group on their volume), we have just two fermion zero modes, which are the Goldstinos of 4d N = 1 supersymmetry, and the instanton can in principle contribute to the superpotential (if no additional zero modes arise from other non-universal sectors). For instantons with Sp(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes to the two-index antisymmetric representation, and the other two fermion zero modes to the symmetric representation. Hence, even for the minimal case of Sp(2)\n\ninstantons, we have just two fermion zero modes in the triplet representation, in addition to the two 4d N = 1 Goldstinos. Hence Sp(2) instantons cannot contribute to the superpotential in the absence of additional effects which lift these zero modes (see later) 7 .\n\n6 We thank F. Marchesano for discussions on this point. 7 For D-brane instantons corresponding to gauge instantons, the additional fermion zero modes in the universal sector couple to the boson and fermion zero modes from open strings stretched between the instanton and the 4d spacefilling brane. They act as Lagrange multipliers which impose 11 In addition to this universal sector, there exist in general additional modes, whose presence and number depends on the detailed structure of the branes. Namely, on the geometry of the brane in the 6d compact space in geometric compactifications, or on the boundary state of the internal CFT in more abstract setups. They lead to a number of boson and fermion zero modes in the symmetric or antisymmetric representation. The computation of these multiplicities in terms of the precise properties of the instanton branes is postponed to Section 5. In order to generate a superpotential, one must require these modes to be absent, except for the case of antisymmetrics of O(1) instantons, which are actually trivial.\n\nAn important point is that extra fermion zero modes (including the extra triplet fermion zero modes in the universal sector of Sp(2) instantons, and any two-index tensor fermion zero mode in the non-universal sectors) are in principle not protected against acquiring non-zero masses once the model is slightly modified. In other words, such fermions are non-chiral, in terms of the SO(4) chirality in footnote 5. One such modification is motion in the closed string moduli space, which can lift the nonuniversal modes if there are non-trivial couplings between them and closed string moduli (unfortunately, such correlators are difficult to compute, even in cases where the CFT is exactly solvable, like the Gepner models). Note that extra zero modes in the universal sector of Sp(2) instantons cannot be lifted by this effect, since it does not break the accidental 4d N = 2 in this sector. A second possible modification which in general can lift extra zero modes is the addition of fluxes, generalizing the results for D3-brane instantons in geometric compactifications [29] (for non-geometric CFT compactifications, we also expect a similar effect, once fluxes are introduced following [30] ). Note that fluxes can lift extra zero modes in the universal sector as well, since they can break the accidental 4d N = 2 susy in this sector. A last mechanism arising in more general F-theory compactifications and discussed below for complex instantons, is valid for real instanton branes as well.\n\nThe bottom line is that in the absence of such extra effects, only O(1) instantons can contribute to superpotential terms. However, in modifications of the model such extra effects can easily lift the extra fermion zero modes. Hence, this kind of extra vector-like zero modes will not be considered catastrophic, and real instantons (including the O(1) and Sp(2) cases) with such zero modes are considered in our scan in Section 6.\n\nthe fermionic constraints in the ADHM construction [28] , and may not spoil the generation of a superpotential.\n\n• Complex brane instantons Zero modes uncharged under the 4d gauge group can arise from the MM and MM ′ open string sectors. Notice that the orientifold action maps the MM sector to the M ′ M ′ , hence we simply discuss the former and impose no projection. The discussion of the MM sector is as for real brane instantons before the orientifold projection. The universal sector leads to a U(n) gauge symmetry, and four bosonic and four fermionic zero modes in the adjoint representation. The bosons are translational Goldstones, while the fermions are Goldstinos of the accidental 4d N = 2 present in this sector. Hence, even in the minimal case of U(1) brane instantons there are two extra fermion zero modes, beyond the two fermion zero modes corresponding to the 4d N = 1 Goldstinos. Hence U(1) instanton (except for those corresponding to gauge instantons, see footnote 7) cannot contribute to superpotential terms in the absence of additional effects, like closed string fluxes . However, keeping in mind the possibility of additional effects lifting them in modifications of the model, we include them in the discussion. Also, in what follows we will use the U(n) notation for the different fields to keep track of the Chan-Paton index structure.\n\nThe above statement would seem in contradiction 8 with computations of nonperturbative superpotentials [14] induced by M5-branes instantons in M-theory compactifications on Calabi-Yau fourfolds, which are dual to D3-brane instantons (with world-volume U(1) gauge group) on type IIB compactifications. The resolution is that the M5-branes that contribute are intersected by different (p, q) degenerations of the elliptic fiber. This implies that U(1) D3-brane instanton only contribute if they are intersected by mutually non-local (p, q) 7-branes. The two extra fermion zero modes exist locally on the D3-brane volume, but cannot be defined globally due to the 7-brane monodromies. Hence such effect can take place only on non-perturbative type IIB compactifications including (p, q) 7-branes. Note that in perturbative compactifications, namely IIB orientifolds, the (p, q) 7-branes are hidden inside orientifold planes [31] with their monodromy encoding the orientifold projection; hence the only branes that can contribute to the superpotential are real branes, for which the projection/monodromy removes the extra fermion zero modes, as discussed above.\n\nIn addition to this universal sector, the MM sector may contain a non-universal set of fermions and bosons, in the adjoint representation (hence uncharged under U(1)).\n\nThey depend on the specific properties of the brane instanton, and will be discussed in Section 5. These additional zero modes should be absent in order for the instanton 8 We thank S. Kachru for discussions on the ideas in this paragraph.\n\nto induce a non-trivial superpotential. Notice however that these zero modes are uncharged under any gauge symmetry, and hence vector-like. Thus, there could be lifted in modifications of the model, as discussed for real instantons.\n\nThe MM ′ sector is mapped to itself under the orientifold action. Hence it leads to a number of bosons and fermions in the two index symmetric or antisymmetric representations. Notice that the two-index antisymmetric representation is trivial for U(1), so these modes are actually not present. On the other hand, fermion zero modes in the two-index symmetric representation are chiral and charged under the brane instanton gauge symmetry. Hence they cannot be lifted by any of the familiar mechanisms, and thus spoil the appearance of a non-perturbative superpotential, even if the model is modified. Such fermion zero modes are considered catastrophic and we will look for models avoiding them in our scan in Section 6."},{"section_type":"OTHER","section_title":"Charged fermion zero modes","text":"• Real brane instantons Instanton zero modes charged under the 4d gauge group arise from Mk open string sectors (and their image Mk ′ ). In the generic case, there are no scalar zero modes in these sectors. This is because in mixed Mk open string sectors the 4d spacetime part leads to DN boundary conditions, which already saturate the vacuum energy in the NS sector. Only in the special case where the internal structure of the spacetime filling brane k and the instanton brane are the same, there may be NS ground states of the internal CFT which do not contribute extra vacuum energy, hence leading to massless scalars. However, this corresponds to brane instantons which can be interpreted as instantons of the 4d gauge theory on the 4d space-filling branes. The instantons we are interested in for the generation of neutrino Majorana mass terms are not of this kind [3] (see e.g. [32, 33, 28, 25, 34] for discussions on gauge theory instantons from D-brane instantons).\n\nHence we focus on charged fermion zero modes, which are generically present in any mixed Mk sector. Let us define L M k , L M k ′ the (positive by definition) number of left-handed chiral fermion zero modes in the representations\n\n( M , k ), ( M , k ),\n\nrespectively. The net number of chiral fermion zero modes in the ( M , k ) is given\n\nby I M k = L M k ′ -L M k .\n\nThis controls the violation of the U(1) a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φ kp and/or Φ kp ′ .\n\nIn addition, there are P M k = min(L M k ′ , L M k ) vector-like pairs of fermion zero 14 modes. Since they are vector-like, they may be lifted by slight modifications of the model, like moving in the closed string moduli space, or by introducing additional ingredients, like fluxes. In addition, they may be lifted by moving in the open string moduli space of the 4d spacefilling branes, as follows. The zero modes may lead to insertions of 4d fields Φ kk , if the kk sector contains such 4d chiral multiplets (or to insertions of composite 4d operators in the adjoint of the k th 4d gauge factor), and if they couple to the zero modes. Although this may not be generically not the case, many of our models in coming section contain such adjoint fields. Hence, a nontrivial vev for the latter can lift these extra vector-like zero modes, hence leading to instanton generating the superpotential of interest. Given these diverse mechanisms to lift these zero modes, their presence of such zero modes is thus unwanted, but again not necessarily catastrophic.\n\nOne last comment, related to the concrete kind of instanton search we will be interested in. Namely, we will be searching for instantons leading to a specific operator, carrying non-trivial charges under a specific set of 4d gauge factors. Postponing the detailed discussion to Sections 3.1, 4.2 , let us denote a, b, c, d the set of branes leading to a field theory sector, denoted 'observable' (and which reproduces the SM in our examples). We will require the instanton to have a prescribed number of chiral fermion zero modes charged under these branes, namely we require the intersection numbers of the instanton with these branes I M a , . . . , I M d to have specific values (as mentioned above, in the most restrictive scan we forbid vector-like pairs of zero modes under these branes). In addition, the model in general contains an additional sector of branes, denoted 'hidden' (since there is zero net number of chiral multiplets charged under both sectors) and labeled h i , required to fulfill the RR tadpole cancellation conditions. In general there may be instanton fermion zero modes from e.g. the Mh 1 , h 2 M sectors, which would contribute to insertions of the 4d fields in the h 1 h 2 sector if there are suitable cubic couplings. These extra insertions could be avoided if such 4d fields in the hidden sector acquire vevs (note that vevs for the (vector-like) fields charged under the visible and hidden sectors would typically break hypercharge, and should be avoided), and hence lift the zero modes. Equivalently, from the 4d perspective, the unwanted extra h 1 h 2 field insertion is replaced by its vev. However, this renders the discussion very model dependent. Moreover, the possibility of hidden brane recombination was not included in the search for SM-like models in [6, 7] (namely, the possibility of allowing for chiral fields charged under the observable and hidden gauge groups, which may become non-chiral and possibly massive upon hidden brane 15 recombination). Hence we will consider these chiral fermion zero modes as unwanted (as usual, non-chiral modes are unwanted but not catastrophic, hence they are allowed for in a more relaxed search)."},{"section_type":"OTHER","section_title":"• Complex brane instantons","text":"The discussion of 'complex' brane instantons is somewhat analogous to the previous one, with the only complication that the brane M and its image M ′ lead to independent modes, leading to a more involved pattern of fermion zero modes. Instanton zero modes charged under the 4d gauge group arise from the Mk,Mk ′ and related sectors. As for 'real' brane instantons, there are generically no scalars in these sectors (and certainly not in our case of interest). Hence we focus on charged fermion zero modes, which are generically present in any mixed sector.\n\nIn contrast with 'real' brane instantons, a net combination of fermion zero modes in the ( M , k ) + ( M , k ) is not vector-like, but chiral under the instanton gauge symmetry. Such a pair cannot therefore be lifted even by modifications of the theory.\n\nIn general, if the instanton has a mismatch in the total numbers n α , n γ of fermion zero modes α i in the M and γ j in the M , the instanton amplitude automatically vanishes.\n\nNamely, the matrix of operators O ij coupling to the zero modes necessarily has rank at most min(n α , n γ ). That is , if n α > n γ there are linear combinations of the α i which do not couple, and cannot lead to insertions. Moreover, they are not liftable by the familiar mechanisms 9 , thus in our instanton search in Section 6 such excess zero modes are forbidden even in relaxed scans.\n\nLet us thus discuss a sector of fermion zero modes with equal number n α = n γ .\n\nConsidering a given 4d space-filling brane k, let us denote\n\nL M k , L M ′ k ′ , L M k ′ , L M ′ k the\n\n(positive by definition) number of left-handed chiral fermion zero modes in the repre-\n\nsentations ( M , k ), ( M , k ), ( M , k ), and ( M , k ) respectively. The net number of chiral fermion zero modes in the ( M , k ) and ( M , k ) is given by I M k = L M k ′ -L M ′ k ′ and I M k ′ = L M k -L M ′ k , respectively\n\n. This net number of fermions zero modes controls the violation of the U(1) a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp, Mk ′ , Mp ′ sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φ kp and/or Φ kp ′ .\n\n9 Note that such a mismatch is always correlated with the existence of extra chiral zero modes in the M M ′ sectors, discussed above. Denoting Q a , Q orient the vector of RR charges of the 4d spacefilling branes and orientifold planes, they satisfy the RR tadpole conditions a\n\nN a Q a + a ′ N a Q a ′ + Q orient. = 0.\n\nBy taking the 'intersection' bilinear with the RR charges Q M of the brane instanton, we have I Ma +I Ma ′ +I M,orient = 0. This implies that the number of fundamentals minus anti-fundamentals of the instanton gauge group is related to the number of two-index tensors.\n\nThe remaining fields in this sector are vector-like pairs, in the ( M , k ) + ( M , k ) or the ( M , k ) + ( M , k ), which in principle lead to the vanishing of the instanton amplitude, but which can be lifted by additional effects (motion in closed or open string moduli space, or addition of fluxes), in a way consistent with the gauge symmetries in 4d spacetime and on the instanton. Just like for 'real' brane instantons, we conclude by commenting on our concrete instanton search in models with a set of visible branes a, b, c, d and a set of hidden branes h i . The requirement that the instanton leads to an operator with specific charges under the visible branes fixes the values of the quantities I M a -I M a ′ , etc. As we described for real branes, one may still have fermion zero modes charged under the hidden sector branes, but they lead to additional insertions, hence we rather focus on\n\ninstantons with I M h i -I M h ′ j = 0.\n\nThe two kinds of conditions, on intersection numbers with visible and hidden branes, still leave the possibility of combinations of fermion zero modes of the kind\n\n( M , k ) + ( M , k ), which do not contribute to I M k , or of the kind ( M , k ) + ( M , k ), which does not contribute to I M k ′ ."},{"section_type":"OTHER","section_title":"Such combinations are","text":"automatically vector-like, and thus may be lifted in modifications of the theory. But the condition also allow combinations like\n\n( M , k ) + ( M , k ), which exploit a cancellation between I M k and I M k ′ (as also does ( M , k ) + ( M , k )\n\n). Such combinations are chiral by themselves, and in general imply a mismatch of modes in the M and the M . The total mismatch can be arranged to vanish using combinations of the kind\n\n( M , k ) + ( M , k ) and ( M , p ) + ( M , p )\n\nfor different branes. However, the only way to lift these pairs is by breaking the gauge symmetry on the 4d space-filling branes k and p. This can be done without damage to the visible sector if these are hidden branes, but this corresponds to the recombination of hidden branes that, as mentioned already, we are not going to consider. Hence only vector-like pairs with respect to each brane are considered to be liftable in simple modifications of the theory. In our instanton search, these are the only additional fermion zero modes which we allow in relaxed scans (but they are clearly not allowed for in restricted scans)"},{"section_type":"OTHER","section_title":"Instanton induced Majorana neutrino masses","text":"In this Section we discuss the possible physical effects of D-brane instantons in string models with SM-like spectrum. In particular we describe the conditions to generate right-handed neutrino Majorana masses. We also comment on other possible B and/or L violating operators that can be generated by instantons. In this section we will again\n\n17 ν Sp(2),U(2) R L L L R E L Q U , D R R W gluon U(1) U(1) U( 3\n\n)\n\nd-Leptonic a-Baryonic b-Left c-Right R Figure 1: Quarks and leptons at intersecting branes use the geometrical language of IIA intersecting D-branes but it should be clear that our discussion equally applies to general CFT orientifolds like the ones presented in the next section."},{"section_type":"OTHER","section_title":"The MSSM on the branes","text":"Let us now specify the discussion in the previous section to the case of the generation of a right-handed neutrino mass term. In order to do that we need some realistic orientifold construction with the gauge group and fermion spectrum of the Standard Model (SM). In the context of Type II orientifolds perhaps the most economical brane configuration leading to a SM spectrum is the one first considered in [5] . This consists of four stacks, labelled a, b, c, d. The gauge factor on branes a is U(3), and contains QCD and baryon number. The d factor is U(1) d , and corresponds to lepton number.\n\nStack b contains SU(2) Weak either embedded in U(2) or Sp(2). Finally brane c can either provide a U(1) or an O(2) factor. In the brane intersection language, the chiral fermions of the SM live at the intersections of these branes, as depicted in Fig. 3.1.\n\nThe U(1) Y factor of the standard model is embedded in the Chan-Paton factors of branes a,c and d as\n\nY = 1 6 Q a - 1 2 Q c - 1 2 Q d = 1 2 (Q B-L -Q R ) (3.1)\n\nwhere Q x denotes the generator of the U(1) of brane stack x (in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator).\n\nNote that in this convention the Q d generator appears with sign opposite to other conventions in the literature, e.g. in [3] . In addition to Y these models have two 18 additional U(1) gauge symmetries:\n\nQ anom = 3Q a + Q d = 9Q B + Q L Y ′ = 1 3 Q a + Q c -Q d = Q B-L + Q R (3.2)\n\nThe first is anomalous whereas the second, which we will call B -L (with a slight abuse of language, since it is in fact a linear combination of B -L and hypercharge), is anomaly free. In models in which the electroweak gauge group is embedded in U(2), rather than in Sp(2), there is a second anomalous U(1) b . The charges of the SM particles under these U(1) symmetries are given in table 1.\n\nIntersection D = 4 fields/ zero modes\n\nQ a Q c Q d Y Q M (ab),(ab') Q L 3(3, 2) 1 0 0 1/6 0 (ca) U R 3( 3, 1) -1 1 0 -2/3 0 (c'a) D R 3( 3, 1) -1 -1 0 1/3 0 (db),(db') L 3(1, 2) 0 0 1 -1/2 0 (c'd) E R 3(1, 1) 0 -1 -1 1 0 (cd) ν R 3(1, 1) 0 1 -1 0 0 (Mc) α i 2(0, 0) 0 -1 0 1/2 1 (dM) γ i 2(0, 0) 0 0 1 -1/2 -1\n\nTable 1: Standard model spectrum and U (1) charges of particles and zero modes. Q M stands for the world-volume gauge symmetry in the case of U (1) complex instantons.\n\nThe U(1) k gauge symmetries have couplings with the RR 2-forms B r of the model, as follows\n\nS BF = k,r N k (p kr -p k ′ r ) 4d B r ∧ F k (3.3) where p kr , p k ′ ,r\n\nare given by the RR charges of the D-branes. These imply that under\n\na U(1) k gauge transformation A k → A k + dΛ k the scalar a r dual to B r transforms as a r → a r + k N k (p kr -p k ′ r ) Λ k (3.4)\n\nThis has two effects: 1) The linear combination of axion fields r (p kr -p k ′ ,r )a r is eaten up by the U(1) k massless gauge boson, making it massive. 2) For anomalous U(1) k , the anomalies cancel through a 4d version of the Green-Schwarz mechanism.\n\nThis works due to the existence of appropriate a r F ∧ F couplings, involving the gauge fields in the theory.\n\nIt is obvious that all anomalous U(1)'s become massive by this mechanism. However it is important to realize [5] that gauge bosons of anomaly-free symmetries like U(1) B-L may also become massive by combining with a linear combination of axions. This is 19 interesting since it provides a mechanism to reduce the gauge symmetry of the model without needing explicit extra Higgsing. In the models in which U(1) B-L becomes massive in this way, the gauge group left over is purely that of the SM. Moreover, we will see that having (B-L) massive by this Stückelberg mechanism is crucial to allow the generation of instanton-induced Majorana neutrino masses.\n\nNote that the B ∧ F couplings may also be potentially dangerous, since in principle they could also exist for hypercharge, removing U(1) Y from the low-energy spectrum.\n\nAs we will see in our RCFT examples later on, having massless U(1) Y but massive U(1) B-L turns out to be a strong constraint in model building."},{"section_type":"OTHER","section_title":"Majorana mass term generation","text":"As discussed in the previous section, string instantons can give rise to non-perturbative superpotentials breaking explicitly the perturbative global U(1) symmetries left-over from U(1) gauge bosons made massive through the Stückelberg mechanism. The kind of operator we are interested in has the form\n\nW ≃ e -S ins ν R ν R (3.5)\n\nwhere ν R is the right-handed neutrino superfield 10 . Here S ins transforms under both U(1) B-L and U(1) R in such a way that the overall operator is gauge invariant. This operator may be created if the mixed open string sectors lead to fermionic zero modes α i , γ i , i = 1, 2, appropriately charged under the 4d gauge factors. As we discussed in the previous section, to generate a superpotential one needs instanton with O(1) Chan-Paton symmetry, in order to lead to two uncharged fermion zero modes to saturate the d 2 θ 4d superspace integration. On the other hand, as we argued, instantons with Sp(2) or U(1) CP symmetries may also induce the required superpotentials if there is some additional dynamics getting rid of the extra uncharged zero modes which in principle appear in instantons with these symmetries. We thus consider all O(1), Sp(2) and U(1) instantons in our discussion.\n\nIn order to to get a ν R bilinear, the intersection numbers of instanton M and d, c branes are as follows Sp(2) case :\n\nI M c = 1 ; I M d = -1 (3.6)\n\n(since there is an extra multiplicity from the two branes required to produce Sp(2)) O(1) case :\n\nI M c = 2 ; I M d = -2 (3.7)\n\n10 Actually we denote by ν R the left-handed ν c L field following the usual (a bit confusing) convention.\n\n20 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 D2 D6 D6 ν R γ α Figure 2: Disk amplitude coupling two charged zero modes to ν R in the geometrical Type IIA intersecting brane approach.\n\nU(1) case :\n\nI M c = 2 ; I M d = -2 or I M d ′ = 2 ; I M c ′ = -2 (3.8)\n\nFurthermore there must be cubic couplings involving the right-handed neutrino superfield ν a in the a th family and the fermionic zero modes α i , γ j\n\nL cubic ∝ d ij a (α i ν a γ j ) , a = 1, 2, 3 (3.9)\n\nIn type IIA geometric compactifications, this coupling arises from open string disk instantons, see Fig. 3.1. In general type IIA models (resp. IIB models), the coefficients d ij a depend on the Kähler (resp. complex structure) moduli, and possibly on open string moduli. In simple CFT models (like e.g. in toroidal cases) these quantities may be in principle explicitly computed.\n\nThese trilinear couplings appear in the instanton action and after integration of the fermionic zero modes α i , γ i one gets a superpotential coupling proportional to\n\nd 2 θ d 2 α d 2 γ e -d ij a (α i ν a γ j ) = d 2 θ ν a ν b ( ǫ ij ǫ kl d ik a d jl b ) (3. 10\n\n) yielding a right-handed neutrino mass term. This term is multiplied by the exponential of the instanton euclidean action so that the final result for the right-handed neutrino mass (up to a 1-loop prefactor) has the form\n\nM R ab = M s ( ǫ ij ǫ kl d ik a d jl b ) exp( - V Π M g s + i r q M,r a r ) (3.11)\n\nFor geometric compactifications V Π M is roughly related to the wrapped volume. We keep the same notation to emphasize that the effect is non-perturbative in g s . In supersymmetric models the term in the exponential is the linear combination U of complex structure moduli to which the instanton D-brane couples, as described in the previous section. As explained, the gauge U(1) c , U(1) d transformation of the bilinear 21 piece and the e -S D2 factor nicely cancel. Note that from the viewpoint of the 4d SM effective field theory, the instanton has generated a Majorana neutrino mass violating B -L. Notice also that since this symmetry is non-anomalous, its violation cannot be associated to a gauge instanton, hence this is a pure string theory instanton effect."},{"section_type":"OTHER","section_title":"Flavor and the special case of Sp(2) instantons","text":"In order to extract more specific results for the flavor structure of the obtained Majorana mass operator, one needs to know more details about the quantities d ij a coming from the disk correlators. However in the particular case of Sp(2) instantons, the labels i, j are Sp(2) doublet indices, and the symmetry requires d ij a = d a ǫ ij . The mass matrix for the three neutrinos is given by M R ab =2M s d a d b exp (-U), so that the flavour dependence on a, b = 1, 2, 3 factorizes. More generally, as we will see in our RCFT search in Section 6, there are typically several different instantons contributing to the amplitude, so that we actually have a result for the mass\n\nM R ab = 2M s r d (r) a d (r) b e -Ur (3.12)\n\nwhere the sum goes over the different contributing instantons. One thus has a structure of the form\n\nM R ∼ r e -Ur diag (d (r) 1 , d (r) 2 , d (r) 3 ) • 1 1 1 1 1 1 1 1 1 • diag (d (r) 1 , d (r) 2 , d\n\n(r) 3 ) . (3.13) This structure is very interesting. Indeed, each instanton makes one particular (instantondependent) linear combination of the neutrinos massive, leaving two linear combinations massless. Hence, for three or more instantons, one generically has a matrix with three non-zero eigenvalues. It is easy to imagine a hierarchical structure among the three eigenvalues if e.g. the exponential suppression factors exp(-Re U r ) are different for each instanton."},{"section_type":"OTHER","section_title":"Other B-and L-violating operators","text":"Our main focus in this paper is on the generation of right-handed neutrino Majorana masses. However instantons may induce other L-and B-violating operators which we briefly summarize in this subsection. 22"},{"section_type":"OTHER","section_title":"The Weinberg operator","text":"A right-handed neutrino Majorana mass term is not the only possible operator violating lepton number. Instanton effects may also give rise to dimension 5 operators not involving ν R . Specifically, the Weinberg operator L W = λ M (LHLH) . (3.14) might be generated. Once Higgs fields get a vev v this operator gives rise directly to Majorana masses for the left-handed neutrinos of order ≃ v 2 /M. Indeed, it is easy to check that in this case the required instanton M must verify Sp(2) case :\n\nI M c = -1 ; I M d = 1 (3.15) O(1) case : I M c = -2 ; I M d = 2 (3.16) U(1) case : I M c = -2 ; I M d = 2 or I M c ′ = 2 ; I M d ′ = -2 ( 3\n\n.17) (here we are assuming SU(2) weak to be embedded in an Sp(2)). Note that these intersection numbers are different to those giving rise to ν R mass terms. In particular they lead to a transformation under B -L opposite to that of ν R mass operators 11 .\n\nIn the present case there are altogether four fermionic zero modes α i ,γ i corresponding to the intersections of the instanton M with the branes c, d. These zero modes can have couplings involving the left-handed leptons L and the u-type Higgs multiplet H\n\nL disk ∝ c ij a (α i (L a H)γ j ) . (3.18)\n\nUpon integration over the fermionic zero modes one recovers the Weinberg operator.\n\nIn the present case the scale M of the Weinberg operator will be the string scale M s and the coupling λ ≃ exp(-S ins ). Again, in the particular case of Sp(2) instantons the situation simplifies (c ij a = c a ǫ ij ) and one gets left-handed neutrino Majorana masses\n\nM L ab = < H > 2 M s r 2c (r) a c (r) b e -Sr (3.19)\n\nwhere r runs over the different contributing instantons and S r is their corresponding action. The flavour structure of this left-handed neutrino mass matrix is the same as in eq.(3.13) and again may potentially lead to a hierarchical structure of left-handed neutrino masses, as is experimentally observed.\n\n11 Instantons with these intersection numbers will be denoted with a plus sign in the instanton search later on 23 In a given model both this kind of instanton and the one giving rise to righthanded neutrino masses (which is different) may be present. This contribution to the left-handed neutrino Majorana mass is in principle sub-leading compared to the see-saw contribution\n\nM L ab (see-saw) = < H > 2 2M s h T D ( r d (r) a d (r) b e -Sr ) -1 h D (3.20)\n\nwhere is the ordinary Yukawa coupling constant h ab D (ν a R HL b ). In principle the former is doubly suppressed both by 1/M s and the exponential factor. On the other hand if the exponential suppression is not too large this mechanism involving directly the Weinberg operator may be the most relevant source of neutrino masses. This is because the see-saw contribution coming from ν R exchange is proportional to the square of the ordinary Yukawa couplings h ab D which could be small. One could even think of having just the Weinberg operator as the unique source of the observed left-handed neutrino masses. Note however that in string vacua like this, in which the ν R 's are present and massless at the perturbative level, having just the Weinberg operator would not be phenomenologically correct, and instantons of the first class are still needed so that the ν R 's get a sufficiently large mass."},{"section_type":"OTHER","section_title":"R-parity violating operators","text":"In the case of N = 1 SUSY models like the MSSM there might be operators of dimension 3 and 4 violating lepton and/or baryon number. These are the superpotential couplings\n\nW Rp = µ L a L a H + λ abc Q a D b L c + λ ′ abc U a D b D c + λ ′′ abc L a L b E c (3.21)\n\nin standard notation. Unlike the neutrino operators mentioned above, these operators violate B -L in one unit (rather than 2). It is well known that the standard R-parity of the MSSM may be identified with a Z 2 subgroup of U(1) B-L , so these terms are odd under R-parity. The simultaneous presence of all these couplings is phenomenologically unacceptable. Indeed, the third coupling violates baryon number, and the other three violate lepton number. Together they lead to proton decay at an unacceptably large rate. On the other hand couplings violating either B or L are phenomenologically allowed.\n\nIt is an interesting question whether any of these operators may be induced by string instanton effects. A first point to note is that instantons with Sp(2) Chan-Paton symmetry can never generate operators of this type. The reason is that all charged zero modes will necessarily come in Sp(2) doublets and hence the charged operators induced will always involve an even number of charged D = 4 fields and R-parity is 24 automatically preserved. On the other hand O(1) and U(1) instantons may generate R-parity violating operators. In particular, the LH bilinear is essentially the square root of the Weinberg operator, and may be induced if a U(1) or O(1) instanton M exists with\n\nI M c = -1 ; I M d = 1 or I M c ′ = 1 ; I M d ′ = -1 . (3.22)\n\n(in the O(1) case the second option is not independent from the first). Again, if the appropriate disk couplings are non-vanishing a term with µ a L ∼ M s exp(-S ins ) is generated. The rest of the operators in W Rp may also be generated. Possible instanton zero modes which may induce them are shown in table 2. For example, the QDL operator may be induced if a U(1) instanton M with intersection numbers\n\nI M b = -1 ; I M c ′ = 1 ; I M d = 1 (3.23)\n\nis present and in addition couplings\n\nL disk ∝ c ab (α(U a Q b j )γ j ) + c ′ a (βL a j γ j ) (3.24)\n\nexist. Here α, β, γ are zero modes corresponding to (Mc ′ ), (Md) and (bM) intersections and a, b(j) are flavor(SU(2) L ) indices. Analogous trilinear or quartic disk amplitudes involving two charged zero modes should exist to generate the rest of the R-parity violating amplitudes in table 2.\n\nD = 4 Operator I Ma I Ma ′ I Mb I Mc I Mc ′ I Md I Md ′ ν R ν R 0 0 0 2 0 -2 0 L HL H 0 0 0 -2 0 2 0 L H 0 0 0 -1 0 1 0 QDL 0 0 -1 0 1 1 0 U DD -1 0 0 1 2 0 0 LLE 0 0 -1 0 1 1 0 QQQL 1 0 -2 0 0 1 0 U U DE -1 0 0 2 2 -1 0\n\nTable 2: Zero modes required to generate Lepton/Baryon-number violating superpotential operators. Sp(2) instantons cannot give rise to R-parity violating operators whereas 1) instantons may in principle contribute to all of them. In the case of U (1) instantons there are additional zero mode possibilities which are obtained by exchanging\n\nO(1),U (\n\nI M x ↔ -I M x ′ . 25"},{"section_type":"OTHER","section_title":"Dimension 5 proton decay operators","text":"There are also superpotential dimension-5 operators violating B and L which may be constructed from the MSSM matter superfields. Indeed the dimension 5 operators ( 1 M ) QQQL ; ( 1 M ) UUDE (3.25) are in fact the leading source of proton decay in SUSY GUT models with R-parity.\n\nUnlike the other operators considered here these ones preserve B -L (hence R-parity) but not B + L. These operators do not contribute directly to a proton decay but need to be 'dressed' by a one loop exchange of some fermionic SUSY particle. This makes that, even although they are suppressed only by one power of the relevant fundamental scale, the loop factor and the corresponding couplings make the overall rate in SUSY-GUTS (barely) consistent with present experimental bounds for M of order the GUT scale or larger.\n\nThese dimension 5 operators may also be induced in D-brane models of the class here considered by the presence of instantons with appropriate intersection numbers.\n\nFor instance, the first operator may be induced through O(1) or U(1) instantons M with\n\nI M b = I M b ′ = -2 ; I M a = 1 ; I M d = 1 (3.26)\n\nAgain Sp(2) instantons cannot induced this operator, since the the Ma intersection would yield 6 (rather than 3) colored fermionic zero modes. The proton decay rate obtained from these operators depend on the ratio exp(-S ins ) × 1/M s . For M s of order 10 16 GeV, the rate is consistent with present bounds if exp(-S ins ) provides a suppression of a few orders of magnitude. On the other hand, models with a low string scale may be in danger unless the exponential suppression is sufficiently large (or such particular instantons are absent).\n\nAs a general conclusion, these phenomenological aspects of instanton induced operators very much depend on the action of the instanton, e.g. the volume of the wrapped D2-instanton in the intersecting D-brane constructions. In any event it is clear that the instantons here considered may indeed induce proton decay at a model-dependent rate. However in certain models R-parity will be preserved and prevent too rapid proton decay. Indeed, this is what we find in our instanton search in Gepner orientifolds.\n\nAs we said Sp(2) instantons automatically preserve R-parity. More generally, models that violate R-parity are rare, and the corresponding instantons actually generate very high dimensional operators, so R-parity breaking effects seems quite suppressed. In 26 fact in our search within MSSM-like models in Gepner model orientifolds we do not find instantons with just the correct charged zero modes to generate the low dimensional couplings discussed above. So, at least within our class of RCFT constructions, R-parity preservation is quite a common feature."},{"section_type":"OTHER","section_title":"CFT orientifolds","text":"In this section we describe the 4d string models we consider, namely orientifolds of type IIB Gepner model compactifications. This is a very large class, on which one can carry out large scans for certain desired properties. And moreover at present the only known class of (SUSY) models with massive B -L."},{"section_type":"OTHER","section_title":"Construction of the models","text":"In general, RCFT orientifolds are orientifold projections of closed string theories constructed using rational conformal field theory. Although this includes in principle rational tori and orbifolds, the real interest lies in cases where the two-dimensional CFT is interacting, because such theories are hard to access by other methods. A disadvantage of the use of RCFT is that this method is algebraic, and not geometric in nature, so that one cannot easily explore small deformations of a certain string theory.\n\nIt is best thought of as a rational scan of moduli spaces.\n\nThe most easily accessible examples are the orientifolds of tensor products of minimal N = 2 conformal field theories (\"Gepner models\") forming a type IIB closed string theory. During the last decade, examples in this class have been studied by many authors (see [35] 42] ), and searched systematically in [6] and [7] .\n\n[36][37][38][39][40][41][\n\nAlthough the Gepner models form only a small subset of RCFT's, they already offer a large number of possibilities. The total number of tensor products with the required central charge c = 9 is 168. On top of this, one can choose a large number of distinct modular invariant partition functions on the torus. The orientifold formalism is not available for all of them, but it has been completely worked out [43] for all simple current invariants (based on the charge conjugation invariant). This yields a total of 5403 distinct MIPFs. On top of this, we may choose various orientifold projections. Here the only known possibilities are a class of simple-current based choices [44][45] [46] [47] .\n\nThis then yields a total of 49304 orientifolds.\n\nFor each orientifold choice, the full open string partition function is 1 2 a,b,i\n\nN a N b A i ab χ i ( τ 2 ) + a,i N a M i a χi ( τ 2 + 1 2 ) (4.1)\n\nHere A i ab are the annulus coefficients, M i a the Moebius coefficients, N a the Chan-Paton multiplicities and χ(τ ) are the closed string characters, and χi (τ ) = T -1/2 χ i (τ ). The set of integers i is simply the set of primary fields of the closed string CFT, and depends only on the tensor product. The integers a, b are the boundary labels; this set depends on the MIPF. Our notation and labelling conventions for these CFT quantities are explained in Appendix A. The integers A i ab and M i a depend in addition also on the orientifold choice; in the case of A i ab the latter dependence is very simple: all distinct annuli can be written as\n\nA Ω,i ab = c A i c a C Ω cb ,\n\nwhere Ω is the orientifold choice (which we usually do not specify explicitly) and C Ω cb is the boundary conjugation matrix, which acts as an involution on the set of boundaries. Suppressing some details (which can be found in [43] ) we may write these integers as\n\nA Ω,i ab = m,J,K S i m R a,(m,J) g Ω,m JK R b,(m,K) S 0m (4.2) M Ω,i a = m,J,K P i m R a,(m,J) g Ω,m JK U Ω (m,K) S 0m (4.3)\n\nHere m is the label of an Ishibashi-state (the set of states that propagates in the transverse (or closed string) channel of the the annulus or Moebius diagrams). It is a subset of the set of closed string labels i, but in general there are degeneracies, so that more than one distinct Ishibashi state belongs to a given closed string label.\n\nThese degeneracies are distinguished by the labels J, K (see Appendix A). The complex numbers R and U are respectively the boundary and crosscap coefficients. Note that the latter depend on the orientifold choice, but the former do not. The only dependence of the annulus coefficients on the orientifold choice is through the Ishibashi metric g Ω JK , which is a matrix on each Ishibashi degeneracy space, and which can be a sign if there are no degeneracies. Finally, the matrix P is given by P = √ T ST 2 S √ T , where S and T are the generators of the modular group of the torus. Similar expressions exist for the Klein bottle multiplicities defining the unoriented closed sector, but they will not be needed in this paper.\n\nThe boundary labels a, b, . . . refer to all boundaries that respect the bulk symmetries of the CFT. This includes the individual N = 2 chiral algebras of the factors in the tensor product, the alignment currents 12 that ensure the proper definition of world-sheet supersymmetry and the space-time supersymmetry generator that imposes a generalized GSO-projection on the spectrum. The latter implies that all characters χ i respect (at least) N = 1 space-time supersymmetry. By construction, the boundary states are then supersymmetric as well. Both conditions (boundary and bulk space-time supersymmetry) can in principle be relaxed within the formalism, but this leads to a much larger set of bulk and boundary states. The precise labelling of the boundaries is explained in Appendix A and involves a subset of the closed string labels i and a degeneracy label, distinct from the one used for the Ishibashi states. The set of boundary labels is complete in the sense of [45] . This means that no additional boundary states exist that respect all the aforementioned symmetries. It also means that the matrices R are square matrices (although their rows and columns are defined in terms of different index sets). It is in principle possible to write down additional boundary states that break some of the world-sheet symmetries. This is an important possibility to keep in mind, but we will not consider it here.\n\nThe massless spectrum is obtained by restricting the characters χ i to massless states. Since the characters are supersymmetric those massless states are either vector multiplets or chiral multiplets. The latter can be restricted to one chirality (e.g lefthanded); the other choice merely produces the CPT conjugates. Boundaries are called real if a = a ′ , where the conjugate boundary a ′ is defined by C Ω a,a ′ = 1, and complex otherwise. The Chan-Paton multiplicities N a give rise to gauge groups U(N a ) for complex boundaries and SO(N a ) or Sp(N a ) for real ones. In the latter case N a must be even. To count bi-fundamentals we define\n\nL ab ≡ i A i ab χ i ( τ 2 ) massless,L . (4.4)\n\nNote that because of the factor 1 2 in (4.1) and the fact that L ab is symmetric, the value of L ab is indeed precisely the number of bi-fundamentals in the representation (N a , N b ).\n\nThese are spin-3 currents consisting of products of the world-sheet supercurrents of the factors in the tensor product, including the NSR space-time factor.\n\nIt is convenient to introduce the intersection matrix 13\n\nI ab ≡ L ab ′ -L a ′ b , (4.5)\n\nwhich is manifestly antisymmetric in a and b. Note that for a pair of complex boundaries a, b with conjugates a ′ , b ′ one can define four quantities that are relevant for the massless spectrum, two of which are chiral, namely I ab and I ab ′ .\n\nIt is often convenient to associate a geometric picture to these integers. Thus we will often refer to the boundary labels and their multiplicities as \"stacks of branes\", and view the integers I ab as brane intersection numbers. This is only done for convenience and does not imply a concrete brane realization; indeed, it does not make sense to say that a given boundary label corresponds to a Dp-brane for some give p. Such an interpretation might be valid in a large radius limit, assuming such a limit exists.\n\nIn general, for a choice of Chan-Paton multiplicities N a there will be tadpoles in the one-point closed string amplitudes on the disk and the crosscap. These have to be cancelled in order to make the theory consistent (since we work with supersymmetric strings we do not have the option of cancelling RR and NS-NS tadpoles separately).\n\nThis leads to a condition on the Chan-Paton multiplicities:\n\na N a R a,(m,J) = 4η m U m,J (4.6)\n\nwhere η 0 = 1 and all other η's are -1; there is such a condition for any Ishibashi label (m, J) that leads to a massless scalar in the transverse channel. The one for m = 0 (which is non-degenerate) is the dilaton tadpole condition. It has the special feature that all coefficients R a0 are real and positive. The crosscap coefficient U 0 is also real and can be chosen positive (in the CFT both signs are acceptable). If U 0 = 0 (4.6) limits the Chan-Paton multiplicities; if U 0 = 0 the only solution is N a = 0 for all a, which rules out any realization of the Standard Model. This reduces the number of usable orientifolds to 33012.\n\nTadpole cancellation condition implies cancellation of RR-charges coupling to longrange fields, and absence of local anomalies. There is a second condition that has 13 Note that L ab is a symmetric matrix giving the number of chiral multiplets in the ( a , b ) bifundamental. This is a natural quantity in unoriented CFT's, where a symmetric definition for the annulus amplitude exists. In oriented CFT the annulus is, in general, not symmetric, but on the other hand it is possible to choose the branes in such a way that only ( , ) bi-fundamentals appear.\n\nThis has become the customary way of counting states in the intersecting brane literature, even for orientifold models. The quantity I ab is defined in such a way that it is anti-symmetric in a and b. This is why boundary conjugations appear in the right hand side. This has the additional advantage of making I a more familiar quantity for readers used to the standard intersection brane conventions.\n\nto be taken into account, which has to do with Z 2 charges that do not couple to long-range fields, usually referred to as \"K-theory charges\" in geometric constructions.\n\nUncancelled K-theory charges may lead to global anomalies in symplectic factors of the gauge group. But even if this symptom is absent, the disease may still exist. A much more general way to probe for uncancelled K-theory charges is to require the absence of global anomalies not only in the Chan-Paton gauge group but also on all symplectic brane-anti-brane pairs that can be added to it as \"probe-branes\" [48] . Presently this is the most general constraint that be imposed in these models, but it is not known if additional ones are required. This probe brane constraint leads to a large number of mod-2 constraint and is potentially very restrictive, but almost harmless in practice [49] . It is satisfied by all models we consider in the present paper."},{"section_type":"OTHER","section_title":"Search for SM-like models","text":"The complete set of solutions to these conditions is finite but huge, but the vast majority is of no phenomenological interest. In the last few years systematic searches have been carried out for models that contain the Standard Model. The models that were considered have the property that the set of Chan-Paton labels can be split into two subsets, the observable and the hidden sector. The former has been limited, for practical reasons, to at most four complex brane stacks, required to contain the Standard Model gauge group and the right intersections to yield three families of quarks and leptons, plus (in general) some non-chiral (vector-like) additional matter. The hidden sector is only constrained by the requirement that there be no net number of chiral multiplets charged under both the observable and hidden sector, and by practical computational limitations. The main purpose of the hidden sector in these models is to provide variables that can be used to satisfy the tadpole and global anomaly conditions, since the multiplicities in the observable sector are already fixed. In some cases the observable sector already satisfies the constraints by itself, and there is no hidden sector.\n\nThe observable sector can be realized in many different ways if one only imposes the constraint that the standard model should be contained in it. These possibilities were recently explored in [7] . We will focus on the realization described in Section 3.1, first considered in [5] . There are four stacks, namely a (containing QCD and baryon number as U(3)), b (containing electroweak SU(2) embedded as U(2) or Sp(2)), c (providing a U(1) or an O(2) factor 14 , and d (providing another U(1) factor). 14 In [6] also Sp(2) was considered, but this requires an additional Higgs mechanism.\n\nThe standard model hypercharge generator is , defined in (3.1):\n\nY = 1 6 Q a - 1 2 Q c - 1 2 Q d (4.7)\n\nwhere Q x denotes the generator of the U(1) of brane stack x; in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator. In addition to Y these models have two or three additional U(1) gauge symmetries (the latter case if electroweak SU(2) arises from U(2)). These (except the combination B -L) are anomalous, with anomaly cancelled by the Green-Schwarz mechanism, implying the existence of a B ∧ F coupling making them massive. In fact, as already mentioned, such Stückelberg couplings may be present for non-anomalous U(1)'s as well. We are interested in models where the hypercharge gauge boson does not have such couplings (otherwise the model would be phenomenologically unacceptable), but\n\nwhere the B -L gauge boson is massive by such couplings (both in order that the gauge group reduces to the SM one, and that neutrino Majorana masses may be induced by string instantons, as discussed in previous sections).\n\nThe combined requirements of having a massive B -L and a massless Y turn out to be difficult to satisfy. In fact, if the group on brane c is O(2) they are impossible to satisfy simultaneously, because the O(2) component of the vector boson does not couple to any axions, and hence the B -L and Y bosons have the same mass. But even in models with a U(1) group on brane c it happens rather rarely that both constraints are satisfied simultaneously, at least in the searches that have been done so far.\n\nWe will make use here of the data presented in [6, 7] , which are available in slightly improved form on the website www.nikhef.nl/ ∼t58/filtersols.php. This database consist of 211634 distinct spectra. Here \"distinct\" means that they are physically different for a given MIPF 15 if the hidden sector is ignored. Hence the differences can be the number of vector-like states of various kinds or the dilaton couplings of branes a, b, c, d. Geometrically, these spectra may originate from the same moduli space, but then in any case from different points on this moduli space. The improvements in comparison with the data presented in [6] consist of taking into account the full global anomaly conditions from probe branes. In some cases this required nothing more than checking these conditions for an existing solution of the tadpole conditions, but in other cases a new solution had to be found. As a result, a few models disappeared from the original database, but due to improved algorithms a few new ones could be added. The net re-15 Rare cases of identical spectra and couplings originating from different MIPFs are treated as distinct.\n\n32 sult is some small but inconsequential changes in the total number of models of various kinds. The numbers we will mention below are based on the improved database.\n\nThe total number of models in that database with a Chan-Paton group U(3) × Sp(2) × U(1) × U(1) is 10587. Of these, 391 (about 4%) have a massive B -L vector boson. For U(3) × U(2) × U(1) × U(1) these numbers are, respectively, 51 and 0. Hence no examples of the latter type were found, although they were found with 1,2 and 4 families (in a limited search), in a few percent of the total number of models. It seems therefore reasonable to expect that U(3) × U(2) × U(1) × U(1) with massive B -L do exist, and that their absence is just a matter of statistics. Just for comparison, the total number of U(3) × Sp(2) × O(2) × U(1) models is 56627."},{"section_type":"OTHER","section_title":"Fermion zero modes for instantons on RCFT's","text":"In this section we discuss D-brane instantons for general compactifications, including abstract CFT ones. We also provide the spectrum of zero modes on an instanton brane, using the information about their internal structure i.e. in the compactified dimension in geometric models, or of the internal CFT in more abstract setups like in previous section. We will be interested in the latter case.\n\nA first question that should be addressed is what this internal structure is. For instance, in type IIA geometric compactifications, it corresponds to a supersymmetric (i.e. special lagrangian) 3-cycle. Notice that these are the same kind of 3-cycles already used to wrap the D6-branes that give rise to the 4d gauge symmetry of such models.\n\nFor general CFT's, D-branes are described as boundary states. To describe instantons, one can simply use the same boundary state of the internal CFT to describe the 4d space-filling branes present in the model and the instanton branes. The only difference is that boundaries satisfy Neumann conditions in the 4d space-filling case, and Dirichlet in the instanton case. This exploits the fact that whenever a boundary state of the internal CFT, and with Neumann boundary conditions in the 4d space is an acceptable state of the full CFT, the same boundary state of the internal CFT, combined with Dirichlet boundary conditions in the 4d space also gives an acceptable state of the full CFT. For geometric compactifications this is related to Bott periodicity of the K-theory classes associated to the D-brane charges, but it is possible to show it in general.\n\nSince instanton D-branes can thus be naturally associated to the boundary states of 4d space-filling branes, it is convenient to express the spectrum of zero modes of the former in terms of the massless states of the latter. This is particularly useful, since 33 the computation of the spectra on 4d space-filling branes for Gepner model orientifolds has already been described (although the arguments below are valid also for geometric compactifications). Hence, let us denote by M a 4d space-filling brane associated with the same boundary state of the internal CFT as the instanton brane M of interest.\n\nNote that the 4d space-filling brane M is an auxiliary tool, and need not be (and, for our instantons of interest, will not be) one of the 4d space-filling branes present in the model."},{"section_type":"OTHER","section_title":"'Real' brane instantons","text":"Let us first consider the case of 'real' brane instantons. Consider a set of m 4d spacefilling branes M, and focus first on the massless spectrum in the MM sector. Before the orientifold projection, it leads to a universal 4d N = 1 U(m) vector multiplet, and a number L MM of adjoint chiral multiplets. The orientifold operation maps this sector to itself, acting on the Chan-Paton with a matrix γ Ω,M . This matrix satisfies\n\nγ T Ω,M γ -1 Ω,M = ±1 m (5.1)\n\nThe two possibilities can be chosen to correspond to\n\nγ Ω,M = 1 m or γ Ω,M = ǫ m , with ǫ m = 0 1 r -1 r 0\n\n, and m = 2r hence necessarily even in the latter case. They correspond to the SO and Sp projections, respectively.\n\nThe orientifold projection on the N = 1 vector multiplet Chan-Paton matrices is given by\n\nλ = -γ Ω,M λ T γ -1 Ω,M (5.2)\n\nand leads to SO(m) or Sp(m) vector multiplets for the SO or Sp projection (hence the name). Concerning the N = 1 chiral multiplets, they fall in two classes of p -, p +\n\n(with p -+ p + = L MM ) which suffer the projections λ = ±γ Ω,M λ T γ -1 Ω,M (5.3)\n\nFor the SO projection, this leads to p + , p -chiral multiplets in the , representation.\n\nFor the Sp projection, there are p + , p -chiral multiplets in the , representation.\n\nThe sectors Ma (where a is a 4d space-filling branes present in the model) are mapped to sectors Ma ′ , so it is enough to focus on the former. After the orientifold projection one gets L Ma , L Ma ′ chiral multiplets in the ( M , a ), ( M , a ).\n\nLet us now obtain the zero modes for a set of m instanton branes M in terms of the above spectrum. The MM sector is closely related to the MM sector, by changing the NN boundary conditions in 4d spacetime to DD boundary conditions 34 (which can be done in a covariant formalism, but not in the light-cone gauge). Before the orientifold projection, one obtains the same set of states (since moddings for NN and DD boundary conditions are identical, both in the NS and R sector), but with different world-volume interpretation. Also, the change in boundary conditions implies that some polarization states which are unphysical for the 4d spacefilling brane are physical in the instanton brane. Hence, the U(m) gauge bosons on the 4d space-filling brane M correspond to four adjoint real scalars in the instanton brane M. Similarly, the 4d spinors in M, correspond to four fermion zero modes on M, transforming as two spinors of opposite chiralities θ α , θ α of the SO(4) rotation group in transverse space. The orientifold projection maps the MM sector to itself, acting on Chan-Paton indices with a matrix γ Ω,M . In close analogy with the argument in [50] for the familiar D5-D9-brane system in type I (see [51, 52] for related derivations), one can show that the condition (5.1) flips sign upon changing four NN boundary conditions to DD, hence\n\nγ T Ω,M γ -1 Ω,M = ∓1 m (5.4)\n\nNamely, the instanton brane has Sp(m) gauge group when the 4d space-filling brane (with same internal boundary state) has gauge group O(m), and vice-versa. We still refer to these projections as SO and Sp, hoping no confusion arises. Note that, as mentioned in Section 2.2, although there are no gauge bosons in 0 + 0 dimensions, the gauge group is present on the instantons in that it acts on open string endpoints.\n\nLet us consider the effect of the orientifold projection on the MM states, as compared with the effect on MM states. Again, following arguments familiar in the D5-D9 brane system in type I, one can show that the signs in conditions like (5.2), (5.3) remain unchanged upon changing four NN dimensions to DD, except for bosonic modes polarized along the directions longitudinal to these four dimensions (and for fermions related to them by the unbroken susy of the total system). To be concrete, considering the four MM adjoint bosons, and two MM adjoint fermions θ α associated to the universal MM vector multiplets, they suffer the projection\n\nλ = +γ Ω,M λ T γ -1 M (5.5)\n\nHence they transform in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection. On the other hand, for the two fermion zero modes θ α, the projection is\n\nλ = -γ Ω,M λ T γ -1 M (5.6) 35\n\nand leads to two fermion zero modes in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection.\n\nThis implies that in order to obtain two fermion zero modes from this universal multiplet, in order to generate a superpotential, one should consider instantons with orthogonal gauge group and multiplicity one (O(1) instantons). For instantons with symplectic gauge group and multiplicity two (Sp(2) instantons), there are two additional fermion zero modes in the triplet representation. As mentioned, we will continue to consider such instantons in our relaxed scan. Multiple instantons, i.e. boundary states with higher multiplicity, lead to a larger amount of additional fermion zero modes (due to the larger gauge representations for the fermions), and do not contribute to superpotentials; we will not consider such cases even in relaxed scans, since they also very often lead to too many charged fermion zero modes and cannot contribute to the operators of interest (except possibly for O(2) and U(2) instantons with low intersections, which are kept in our scan as a curiosity).\n\nSimilarly, for the p ± sets of MM scalars and fermions associated to the MM 4d chiral multiplets, the projection is\n\nλ = ±γ Ω,M λ T γ -1 Ω,M (5.7)\n\nwith the same sign choice as in (5.3). The different structure of γ Ω implies that, for the SO projection we get p + , p -sets of scalars and fermions in the , , while for the Sp projection there are p + , p -sets of scalars and fermions in the , .\n\nThis concludes the discussion of the MM sector. Let us not consider the Ma sectors, from the information from the Ma sectors. Notice that this implies changing four NN boundary conditions to DN, which have different moddings. Hence the states are different in both situations, but the information on the multiplicities is preserved.\n\nSpecifically, in the NS sector the DN boundary condition introduce an additional vacuum energy which generically makes all states massive. Hence there are no massless scalar zero modes in generic Ma sectors. In the R sector, the change in the moddings reduces the dimension of the massless ground state, leading to a single (chiral) fermionic degree of freedom. Since the orientifold action maps the Ma sector to Ma ′ sectors, there are no subtleties in the orientifold projection. The end result is L Ma , L Ma ′ fermion zero modes in the ( M , a ), ( M , a ). The net number of chiral fermion zero modes in the ( M , a ) is given by I Ma = L Ma ′ -L Ma , i.e. the net number of chiral multiplets in the related Ma sector.\n\nThe results for orientifold projections for real branes are shown in table 3. 36 Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) SO N = 1 vect. mult. U(m) O(m) Sp(m)\n\n2 f + 2 f + 4 b N = 1 ch. mult. (p + + p -) Ad p + + p - 2p + ( f + b ) + 2p -( f + b ) Sp N = 1 vect. mult. U(m) Sp(m) O(m) 2 f + 2 f + 4 b N = 1 ch. mult. (p + + p -) Ad p + + p - 2p + ( f + b ) + 2p -( f + b ) Any N = 1 ch. mult. L Ma ′ ( M , a )+ L Ma ′ ( M , a )+ L M a ′ ( M , a ) f L Ma ( M , a ) L Ma ( M , a ) L M a ( M , a ) f net I Ma ( M , a ) net I M a ( M , a ) f\n\nTable 3: Orientifold projection for real branes: Massless modes of the 4d space-filling branes M (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes)"},{"section_type":"OTHER","section_title":"Complex brane instantons","text":"We now consider the case of complex brane instantons. The arguments are very similar, hence the discussion is more sketchy. Consider m 4d spacefilling branes M, associated to the internal boundary state of the instanton brane M of interest. The MM leads to a 4d N = 1 U(m) vector multiplet and a number L MM ′ of adjoint chiral multiplets. The orientifold action maps it to the M ′ M ′ sector, hence we may keep just the former and impose no projection. The MM ′ sector is mapped to itself under the orientifold projection. Denoting by γ Ω,M the action on Chan-Paton indices, the MM ′\n\nmodes split into sets L ± M M , L ± M ′ M ′ , which suffer a projection λ = ±γ Ω,M λ T γ -1 Ω,M (5.8) leading, for γ Ω,M = 1 m , to L + MM , L - MM chiral multiplets in the , , and L + M ′ M ′ , L - M ′ M ′ chiral\n\nmultiplets in the , . The net number of chiral multiplets in the , is\n\nI + MM ′ = L + M,M -L + M ′ M ′ , I - MM ′ = L - M,M -L - M ′ M ′ .\n\nAnd oppositely for γ Ω,M = ǫ m . Finally, the Ma, Ma ′ and related sectors lead, after the orientifold projection, to\n\nL Ma ′ , L Ma , L M ′ a ′ , L M ′ a chiral multiplets in the ( M , a ), ( M , a ), ( M , a ), ( M , a ).\n\nIn order to simplify notation, we replace M → M in these expressions in our discussions of instanton zero modes.\n\nLet us now consider m brane instantons M and compute their zero mode spectrum in terms of the above. In the MM (and its image M ′ M ′ ) sector there are four scalar modes and four fermions in the adjoint of the U(m) gauge symmetry group; these are related to the 4d vector multiplet in the MM sector. In addition, there are L M M ′ sets of scalars and fermions in the adjoint, related to the L MM ′ non-universal chiral multiplets in the MM sector. The MM ′ sector is mapped to itself, and one has to impose the orientifold projection (recalling that the matrix γ Ω,M differs from γ Ω,M ).\n\nFor γ Ω,M = 1, hence γ Ω,M = ǫ, we obtain L + M M , L - M M chiral multiplets in the , , and L + M ′ M ′ , L - M ′ M ′ chiral multiplets in the , . The net number of chiral multiplets in the ,\n\nis I + M M ′ = L + M M -L + M ′ M ′ , I - M M ′ = L - M M -L - M ′ M ′ .\n\nAnd oppositely for γ Ω,M = ǫ hence γ Ω,M = 1.\n\nIn the Ma, Ma ′ and related sectors, there are generically no bosonic zero modes, and there are and ( M , a ) respectively. The net number of chiral fermion zero modes in the ( M , a ) and\n\nL Ma , L M ′ a ′ , L Ma ′ , L M ′ a chiral fermion zero modes in the ( M , a ), ( M , a ), ( M , a ),\n\n( M , a ) is given by I Ma = L Ma ′ -L M ′ a ′ and I Ma ′ = L Ma -L M ′ a . In\n\norder to simplify notation, we replace M → M in these expressions in our discussions of instanton zero modes.\n\nThe results for orientifold projections for real branes are shown in table 4."},{"section_type":"OTHER","section_title":"Search for M instantons","text":"In this section we perform a search of models which admit an instanton inducing a right-handed neutrino Majorana mass operator. Namely, for each model with the chiral content of the SM in the classification described in Section 4.2, we first scan over boundary states, searching for all instantons with the required uncharged and charged fermion zero mode structure to yield neutrino masses. We then relax our criteria a bit and allow for instantons with correct charged zero mode structure but having extra non-chiral zero modes (both charged and uncharged). The idea is that these non-chiral zero modes could be lifted by diverse effects, as discussed.\n\nIt is important to recall that the cubic couplings between instanton zero modes and 4d chiral multiplets are difficult to compute in Gepner model orientifolds. Hence, we will simply assume that such couplings are non-zero if there is no symmetry forbidding them. 38 Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) Any N = 1 vect. mult. U(m) × U(m) ′ U(m) U(m) 4 Ad f + 4 Ad b N = 1 ch. mult. p adj Ad + p adj Ad ′ p adj Ad 2p adj ( Ad f + Ad b ) SO N = 1 ch.mult.\n\nL MM ( M , M ′ ) L + MM M + L - MM M 2L + M M b,f + 2L - M M b,f L M ′ M ′ ( M , M ′ ) L + M ′ M ′ M + L - M ′ M ′ M 2L + M ′ M ′ b,f + 2L - M ′ M ′ b,f Sp N = 1 ch.mult. L MM ( M , M ′ ) L + MM M + L - MM M 2L + M M b,f + 2L - M M b,f L M ′ M ′ ( M , M ′ ) L + M ′ M ′ M + L - M ′ M ′ M L + M ′ M ′ b,f + L - M ′ M ′ b,f Any N = 1 ch. mult. L Ma ′ ( M , a )+ L Ma ′ ( M , a ) L M a ′ ( M , a ) f . . . L Ma ( M , a ) L M a ( M , a ) f . . . L M ′ a ′ ( M , a ) L M ′ a ′ ( M , a ) f . . . L M ′ a ( M , a ) L M ′ a ( M , a ) f net I Ma ( M , a ) net I M a ( M , a ) f net I Ma ′ ( M , a ) net I M a ′ ( M , a ) f\n\nTable 4: Orientifold projection for complex branes: Massless modes of the 4d space-filling branes M (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes)"},{"section_type":"OTHER","section_title":"The instanton scan","text":"Our detailed strategy will become clear along the description of the results. Given a set of a,b,c,d standard model branes, we must look for additional boundary states M that satisfy the requirements of a (B -L)-violating instanton. From the internal CFT point of view this is just another boundary state, differing from 4d spacefilling branes only in the fully localized 4d spacetime structure. The minimal requirement for such a boundary state is B -L violation, which means explicitly\n\nI M a -I M a ′ -I M d + I M d ′ = 0 (6.1)\n\nIt is easy to see that the existence of such an instanton implies (and hence requires) the existence of a Stückelberg coupling making B -L massive. To see this, consider adding to the Standard Model configuration a 4d spacefilling brane M (in fact used in Section 5) associated to the boundary state M (RR tadpoles can be avoided by simultaneously including M antibranes, which will not change the argument). The new sector in the chiral spectrum charged under the branes M can be obtained by reversing the argument in Section 5, and is controlled by the intersection numbers of M. From 39 the above condition it follows that the complete system has mixed U(1) B-L × (G M ) 2 anomalies, where G M is the Chan-Paton-factor of brane M. These anomalies are cancelled by a Green-Schwarz mechanism involving a (B -L)-axion bilinear coupling, which ends up giving a mass to B -L via the Stückelberg mechanism. This coupling is in fact not sensitive to the presence of the brane M, hence it must have been present already in the initial model (without M).\n\nHence the existence of a boundary label M that satisfies (6.1) implies that B -L is massive. Unfortunately the converse is not true: even if B -L has a Stückelberg mass, this still does not imply the existence of suitable instantons satisfying (6.1) 16 Indeed, in several models we found not a single boundary state satisfying (6.1).\n\nNote that, since hypercharge must be massless, one can use the reverse argument and obtain that\n\nI M a -I M a ′ -I M c + I M c ′ -I M d + I M d ′ = 0 (6.2)\n\nin all models. We verified this for all models we considered as a check on the computations.\n\nAs already discussed in Section 4.2, in the search for SM constructions in Gepner orientifold, there are 391 models with massless hypercharge and massive B-L. In these models we found a total of 29680 instantons with B -L violation, i.e. with intersection numbers satisfying (6.1). Of course, in order to serve our purpose of generating a Majorana mass superpotential, the instantons have to satisfy some more conditions.\n\nLet us consider them in order of importance, and start with the conditions on the net number of chiral fermion zero modes charged under the 4d observable sector. Clearly we need I M a = I M a ′ and I M b = I M b ′ . The latter condition is automatically satisfied in this case, because the b-brane is real in all 391 models. The chiral conditions on the zero modes charged under the branes c and d are as in [3] 17 and are given in equations (3.6), (3.7) (3.8) of the present paper. These are the instantons of most interest, and on which we mainly focus. However, as discussed in Section 3.4, other important Band/or L-violating operators (such as the Weinberg operator or the LH operator) can 16 From intuition in geometric compactifications, one expects that there may always exist a D-brane with the appropriate topological pairings, but there is no guarantee that there is a supersymmetric representative in that topological sector, and even less that it would have no additional fermion zero modes. Note also that even if such D-brane instantons exists, there is no guarantee that it will fall in the scan over RCFT boundary states. 17 Note that there is a sign change in the contribution of the U (1) d generator to Y in comparison to [3] 40 be generated by instantons with similar intersection numbers, up to a factor of 2 and a sign, see table 2 . For this reason we also allow at this stage any instanton which has the correct number of charged zero modes to generate them. Imposing these conditions reduces the number of candidate instantons potentially contributing to neutrino masses in any of the models to 1315.\n\nAll instantons satisfying these requirements are summarized in the table 5. In columns 1,2 and 3 we list the tensor combination, MIPF and orientifold choice for which the model occurred. The latter two numbers codify simple current data that describe respectively a MIPF and an orientifold. MIPFs are in general defined by means of a subgroup H of the simple current group G, plus a certain matrix X of rational numbers [55] . Orientifolds are defined by a simple current and a set of signs [43] . In previous work [6] we have enumerated these quantities (up to permutation symmetries) and assigned integer labels to them for future reference. We only refer to these numbers here, but further details are available upon request. Usually for each MIPF and orientifold which contains the standard model there are several choices a,b,c,d for which it is obtained. For a given choice of tensor combination, MIPF and orientifold and SM branes there may be several instantons. For clarity we put all such instantons together in the information in table 5 . In column 4 we indicate which type of instanton branes were found. Five types are distinguished: O1, O2, S2, U1 and U2, corresponding to O(1), O(2), Sp(2), U(1) and U(2) Chan-Paton symmetry on the instanton volume. The number indicates the instanton brane multiplicity that gives the correct number of instanton charged zero modes from the a, b, c, d branes, to lead to right-handed neutrino Majorana masses. The number of zero modes is in general the product of the instanton brane multiplicity and 'intersection number' with the corresponding 4d spacefilling brane. As discussed in Section 5, for symplectic branes the smallest possible brane multiplicity is 2. As we discussed there, only O1 instantons may have the required universal minimal set of two zero modes in the uncharged sector.\n\nStill we look for all O(1), Sp(2) and U(1) instantons which may yield a superpotential if the extra uncharged fermion zero modes. In this vein we also include a search for O2 and U2 instantons. Note also that such O2 or U2 instantons imply the existence of other instantons involving the same boundary state, but with multiplicity 1, which may lead to the R-parity violating operator LH. We will discuss the generation of R-parity violating operators at the end of this section. The third character (+ or -)\n\nin the instanton in table 5 is the sign of I M c ′ -I M c . For the instantons giving rise to right-handed neutrino Majorana masses this sign should be negative, whereas it should 41 be positive for instantons giving rise to the Weinberg operator (or the LH operator), see table 2 .\n\nThe 1315 instantons are divided in the following way over the different types: 3 of types O1+ and O1-, 46 of type U1+, 24 of type U1-, 550 S2+, 627 S2-, 27 of types U2+ and U2-and four of types O2+ and O2-. Notice that the vast majority (97.5%) of the instanton solutions are of type S2+ and S2-. This is encouraging given the nice properties of such instantons, concerning e.g. R-parity conservation. Note also that in almost all cases both S-and S+ are simultaneously present, 18 so both sources of physical neutrino Majorana masses (from the see-saw mechanism or the Weinberg operator) are present. The other instanton classes possibly generating right-handed neutrino masses are O1-and U1-, which are much less abundant. There is just one orientifold with O1-instantons, for which one can obtain cancellation of RR tadpoles, see below. On the other hand we have found no orientifold with U1-instantons and cancellation of tadpoles, see below.\n\nTable 5: Summary of instanton branes. (1, 16, 16, 16) (2, 4, 12, 82) 2, 4, 12, 82) (2, 4, 12, 82) (2,4,14,46) 10 0 (2,4,14,46) 16 0 (2,4,16,34) 15 0 (2,4,16,34) 15 1 (2, 4, 16, 34) (2,4,16,34) 16 1 (2,4,16,34) (2, 4, 16, 34) (2, 4, 16, 34)"},{"section_type":"OTHER","section_title":"Tensor MIPF Orientifold Instanton Solution","text":"12 0 S2 + , S2 - Yes\n\n19 0 S2 -! ? (\n\n19 0 U2 + !, U2 -! No\n\n19 0 U1 + , U1 - No\n\n16 0 S2 + , S2 - Yes\n\n18 0 S2 - Yes\n\n18 0 U1 + , U1 -, U2 + , U2 -No\n\n49 0 U2 + , S2 -!, U1 + Yes\n\nContinued on next page 18 In some models contributing many instantons there is an exact symmetry between S-and S+.\n\nThis explains the approximate symmetry in the full set. In some cases this symmetry can be understood in terms of flipping the degeneracy labels of boundary states. We regard it as accidental, since it is not found in all models.\n\nTable 5 -continued from previous page Tensor MIPF Orientifold Instanton Solution (2,4,16,34) 49 0 U1 - No (2,4,18,28) 17 0 2, 4, 22, 22) 2, 4, 22, 22) (2, 4, 22, 22) (2, 4, 22, 22) (2, 4, 22, 22) (2,4,22,22) 20 0 (2,4,22,22) 46 0 (2,4,22,22) (2, 6, 14, 14) 1 1 U1 + No (2,6,14,14) 22 2 (2,6,14,14) 60 2 (2,6,14,14) 64 0 (2,6,14,14) 65 0 (2,6,10,22) 22 2 2,6,8,38) 16 0 (2,8,8,18) (2, 8, 8, 18) (2, 10, 10, 10) (4,6,6,10) 41 0 (4,4,6,22) 43 0 (6,6,6,6) 18 0 Most models have a hidden sector containing extra boundary states beyond the SM ones. In the same spirit of imposing chiral conditions first, we should require\n\n(\n\n13 3 S2 + !, S2 -! Yes! (\n\n13 2 S2 + !, S2 -! Yes\n\n13 1 S2 + , S2 - No\n\n13 0 S2 + , S2 - Yes\n\n31 1 U1 + , U1 - No\n\n49 1 O2 + , O2 -, O1 + , O1 -Yes\n\n(\n\n14 2 S2 + !, S2 -! Yes\n\n14 0 S2 + !, S2 -! No\n\n52 0 U1 + , U1 - No\n\nthat I M h = I M h ′ ,\n\nwhere h is a hidden sector brane. This is to guarantee that the generated superpotential does not violate some hidden sector gauge symmetry which would require the presence of hidden sector fields along with the ν R bilinear. The latter condition is not imposed on the previously known hidden sector (i.e. the one in [6, 7] ), but instead a new search for tadpole solutions was performed, for each M, restricting the candidate hidden sector branes to those satisfying I M h = I M h ′ (as discussed in 43 Section 5). This is because in general the known hidden sector in [6, 7] is just a sample out of a huge number of possibilities.\n\nIn column 5 we indicate for which instantons it was possible to satisfy the tadpole conditions with this additional constraint. With regard to observable-hidden matter we use the same condition as in [6] , namely that it is allowed only if it is vector-like. Such a solution could be found for 879 of the 1315 instantons, with ten cases inconclusive (i.e it was computationally too difficult to decide if a solution does or does not exist).\n\nThe latter are indicated with a question mark in column 5 (for most of the undecidable cases there is a tadpole solution for a different instanton with the same characteristics;\n\nfor that reason just one question mark appears).\n\nIndependently of the RR tadpole condition (since there may be alternative sources for its cancellation, or hidden sectors which fall beyond the reach of RCFT), we can also consider the further constraint that the number of charged fermion zero modes is exactly right, not just in the chiral sense. This means\n\nI M a = I M a ′ = I M b = I M b ′ = 0, I M c = 2, I M c ′ = 0 and I M d = -2, I M d ′ = 0 or vice-versa.\n\nFurthermore we require that there are no adjoint or rank-2 tensor zero-modes (note that the latter could be chiral if the instanton brane is complex, and indeed they are in some of the 1315 cases).\n\nThis reduces the 1315 instantons to 263. In column 4 we indicate those cases with an exclamation mark. It is noteworthy that the success rate for solving the tadpole conditions is highest for these instantons: 254 of the 263 allow a solution (with 3 undecided). If an exclamation mark appears in column 4, this only indicates that some of the instantons are free of the aforementioned zero modes, not that all of them are. But in all cases, if there are tadpole solutions, they exist in particular for the configurations with an exclamation mark. Finally we may impose the condition that I M h and I M h ′ are separately zero. This is indicated with and exclamation mark in column 5. This turns out to be very restrictive. The only cases where this happens have no hidden sector at all.\n\nIt is worth remarking that the only instantons having exactly the correct set of charged zero modes and cancelling tadpoles are of S2 ± type. Also those instantons are the only cases marked with an exclamation mark in column 4 and 5. These examples, which will be discussed below in some detail, also have just the minimal set of fermion zero modes, except for the universal sector (which for Sp(2) instantons contains two extra triplets).\n\nThe main conclusion about this scan is that we did not find any instantons with exactly the zero mode fermions to generate the neutrino mass superpotential. However 44 we have found a number of examples which come very close to that, with exactly the required charged zero modes and a very reduced set of extra uncharged zero modes from the universal sector. These extra zero modes are non-chiral and hence one expects that e.g. RR/NS fluxes or other effects may easily lift them, as we discussed in section 2. Concerning O(1) instantons, which have just the two required fermion zero modes in the universal sector, we have found one example, with the appropriate net structure of charged zero modes. However, it has plenty of other extra zero modes. We discuss examples of O(1) and Sp(2) instantons in the following subsections."},{"section_type":"OTHER","section_title":"An O1 example","text":"Let us first discuss the case of O(1) instantons. In principle they would be the more attractive since they have no undesirable universal zero modes at all. Unfortunately this type of instanton is rare within the set we scanned, and we found just one example with a solution to the tadpole equations without any unwanted chiral zero-modes. The instanton however has a very large number of uncharged and charged vector-like zero modes.\n\nThe standard model brane configuration occurs for tensor product (2, 4, 22, 22), MIPF 49, orientifold 1, boundaries (a,b,c,d) = (487, 1365, 576, 486) . As usual we only provide this information in order to locate this model in the database. Further details are available on request.\n\nThe bi-fundamental fermion spectrum of this model in the (a,b,c,d) sector is fairly close to the MSSM: there is an extra up-quark mirror pair, two mirror pairs of leptoquarks with down quark charges and one with up-quark charges, plus two extra righthanded neutrinos (i.e. a total of five right-handed neutrinos). There are three MSSM Higgs pairs. The tensor spectrum is far less appealing, in particular for brane c: this has 25 adjoints and 7 vector-like pairs of anti-symmetric tensors.\n\nAs we said, there is just one instanton brane of type O1-. It has exactly the right number of zero-modes with brane d, but five superfluous pairs of vector-like zeromodes with brane c, plus one vector-like pair with brane a. In addition there are four symmetric tensor zero-modes on the instanton brane (which of course are vector-like, since it is a real brane): the parameter p + in table 3 is equal to 2.\n\nThe tadpole solution that is (chirally speaking) compatible with this instanton has a large hidden sector: O(1)×O(2) 4 ×O(3)×U(1) 2 ×Sp(2) 2 ×U(3) (there are other possibilities, but no simple ones). This hidden sector introduces more undesirable features: vector-like observable/hidden matter, vector-like instanton/hidden sector modes, plus 45 chiral and non-chiral matter within the hidden sector. Finally the coupling ratios are as follows: α 3 /α 2 = .54, sin 2 θ w = .094, and the instanton coupling is 3.4 times weaker than the QCD coupling (α 3 /α Instanton = 3.4).\n\nDespite these unappealing features this model does demonstrate the existence of this kind of solution."},{"section_type":"OTHER","section_title":"The S2 models","text":"As we have mentioned, these are the examples which come closer to the minimal set of fermion zero modes. As we see in Table 5 , all such instantons satisfying the criteria on the zero mode structure (except for the extra universal zero modes) appear for models based on the same CFT orientifold. It is the one obtained from the (2, 4, 22, 22) Gepner model with MIPF 13 and orientifold 3 in the table. The model is obtained as follows."},{"section_type":"OTHER","section_title":"The closed string sector","text":"We start with the tensor product (2, 4, 22, 22) . This yields a CFT with 12060 primary fields, 48 of which are simple currents, forming a discrete group G = Z 12 ×Z 2 ×Z 2 . After taking into account the permutation symmetry of the last two factors, we find that this tensor product has 54 symmetric MIPFs, and we choose one of them to build the model of interest. For convenience we specify all quantities in terms of a standard minimal model notation, but also in terms of the labelling of the computer program \"kac\" that generates the spectrum. This particular MIPF is nr. 13. To build it we choose a subgroup of G, which is isomorphic to H = Z 12 × Z 2 . The generator of the Z 12 factor is primary field nr. 1, (0, 0, 0, {24, -24, 0}, {24, 20, 0}), and the Z 2 factor is generated by primary field nr. 24, (0, 0, 0, 0, {24, 20, 2}). The representations are specified on a basis (NSR, k = 2, k = 4, k = 22, k = 22), i.e. the boundary conditions of the NSRfermions and the four minimal models in the tensor product. Here 0 indicates the CFT vacuum, and for all other states we use the familiar (l, q, s) notation for the N = 2 minimal models. The first generator has conformal weight h = 11 12 and has ground state dimension 1. The second has weight h = 11 2 and has ground state dimension 2: the ground state contains both (0, 0, 0, 0, {24, 20, 2}) and (0, 0, 0, {24, 20, 2}, 0) . The matrix X defining the MIPF according to the prescription given in [53] [54][55]\n\nis 1 12 0 0 1 2 (6.3) 46\n\nThis simple current modification is applied to the charge conjugation invariant of the tensor product. This defines a MIPF that corresponds to an automorphism of the fusion rules, and that pairs all the primaries in the CFT off-diagonally. The number of Ishibashi states, and hence the number of boundary states is 1080. The MIPF is invariant under exchange of the two k = 22 factors: this maps current 24 to itself, and current 1 to current 11, which is also in H. Hence this symmetry of the tensor product maps H into itself, and it also preserves the matrix X.\n\nTo define an orientifold, we must specify a \"Klein bottle current\" plus two signs defined on the basis of the simple current group. For the current K we use the generator of the second Z 2 in G, primary field nr. 12. This is the representation (0, 0, {4, -4, 0}, {(24, 16, 2)}, {(24, -12, 2)}) which is degenerate with nine other states, all of dimension 1 and conformal weight 7. The crosscap signs are chosen, on the aforementioned basis of H as (+, -). This results in a crosscap coefficient of 0.0464731, and it is orientifold nr. 3 of a total of 8. The orientifold is also invariant under permutation of the identical factors.\n\nThe closed string spectrum contains 14 vector multiplets and 60 chiral multiplets."},{"section_type":"OTHER","section_title":"The standard model branes","text":"To build a standard model configuration we have to specify the boundary state labels.\n\nIt turns out that we have four choices for label a and b, one for c and two for d.\n\nThis leads to a total of 32 possibilities. Among these 32 there are 22 have distinct spectra (distinguished by the number of vector-like states), but for all 32 choices one obtains the same set of dilaton couplings. It seems plausible that these choices simply correspond to putting the a, b and d branes in slightly different positions, so that we move the configuration in brane moduli space. The choices are as follows (these are boundary labels assigned by the computer program, and can be decomposed in terms of minimal model representations; this will be explained in table 6 below) a : 10, 22, 130, 142 b : 210, 282, 290, 291 c : 629 d : 712, 797 There are additional possibilities, but they do not give rise to additional distinct spectra. 47 Table 6: Branes appearing in standard model configurations Label Orbit/Deg. Reps Weight Dimension 10 240 (0, 0, 0, 0, {10, 0, 0}) 5/4 1 130 2760 (0, 0, 0, {10, 0, 0}, 0) 5/4 1 22 [528,0] (0, 0, 0, {1, -1, 0}, {11, 1, 0}) 3/2 1 (0, 0, 0, {1, 1, 0}, {11, -1, 0}) 3/2 1 142 [3048,0] (0, 0, 0, {11, -1, 0}, {1, 1, 0}) 3/2 1 (0, 0, 0, {11, 1, 0}, {1, -1, 0}) 3/2 1 210 4248 (0, 0, {3, 3, 0}, {3, -3, 0}, {9, -9, 0}) 1/2 1 282 5760 (0, 0, {3, 3, 0}{9, -9, 0}{3, -3, 0}) 1/2 1 290 [5952,0] (0, 0, {1, 1, 0}{9, 7, 0}{11, -11, 0}) 5/6 1 291 [5952,24] (0, 0, {1,\n\n1, 0}{9, 7, 0}{11, -11, 0}) 5/6 1 629 [9348,30] (0, (1, -1, 0), 0, {9, 9, 0}{5, -3, 0} 7/12 1 712 [9852,0] (0, {1, 1, 0}{3, -3, 0}{1, 1, 0}{5, 5, 0}) 1/2 2 (0, {1, 1, 0}{1, -1, 0}{1, 1, 0}{5, -3, 0}) 1/2 2 797 [10356,30] (0, {1, 1, 0}{3, -3, 0}{5, 5, 0}{1, 1, 0}) 1/2 2 (0, {1, 1, 0}{1, -1, 0}{5, -3, 0}{1, 1, 0}) 1/2 2\n\nThe second column gives the boundary labels in terms of a primary field label and a degeneracy label (boundaries not indicated by square brackets are not degenerate).\n\nThe labels appearing in columns 1 and 2 are assigned by the computer program, and are listed here only for the purpose of reproducing the results using that program.\n\nIn column 2, the boundary labels are expressed in terms of primary field labels, as in formula (A.4). If a single number appears, this is a representative of an H-orbit corresponding to the boundary. If square brackets are used, this means that the Horbit has fixed points, and that it corresponds to more than one boundary label. The second entry in the square brackets is the degeneracy label, and refers to a character of the \"Central Stabilizer\" defined in [43] ; the details of the definition and the labelling will not be important here. In this case the first entry within the square brackets refers to an orbit representative.\n\nThese orbit representatives can also be expressed in a standard form for minimal model tensor products. This is done in column 3. This is basically the same expansion shown in (A.4), except that the degeneracy label Ψ I turns out to be trivial in all cases, 48 both for the standard model and for the instanton branes shown below (although the theory does contain primaries with non-trivial Ψ's). In columns 4 and 5 we specify the weight and ground state dimension of the corresponding highest weight representation.\n\nThese data are not directly relevant for the boundary state, but helps in identifying it.\n\nSince boundaries are specified by orbit representatives, it is not straightforward to compare them, since the standard choice (the one listed in column 2) is arbitrary. For this reason we have used another representative in columns 3, 4 and 5, selected by an objective criterion: we choose the one of minimal dimension and minimal conformal weight (in that order). If there is more than one representative satisfying these criteria we list all."},{"section_type":"OTHER","section_title":"The open string spectrum","text":"In Table 7 we summarize the spectra of the 32 models. The first four columns list the a,b,c,d brane labels. The last eight columns specify the total number of multiplets of types Q (quark doublet), U (up quark singlet), D (down quark singlet), L (lepton doublet), E (charged lepton singlet), N (neutrino singlet), Y (lepto-quark) and H (Higgs).\n\nThe numbers given are for the total number of lefthanded fermions in the representation, plus their complex conjugates. So for example a 7 in column \"Q\" means that there are 5 quark doublets in the usual representation (3, 2, 1 6 ), plus two in the complex conjugate representation (3 * , 2, -1 6 ). This yields the required three families of quark doublets, plus two mirror pairs.\n\nHence the smallest number that can occur in the six columns QUDLEN is three, if there are no mirrors (note that cubic anomaly cancellation requires three right-handed neutrinos in this class of models). The lepto-quarks Y are all in the same representation as the down-quarks (D), or the conjugate thereof, and they occur only as vector-like mirror pairs. They differ from D-type mirror quarks because they carry lepton number, because they come from open strings ending on the d-brane instead of the c-brane.\n\nIn general, there can also exist U-type lepto-quarks, but in these models they do not occur. Finally the numbers 10, 18 and 26 in column 'H' mean that there are 5, 9 or 13 MSSM Higgs pairs H + H. It is worth noticing that right-handed quarks U, D and neutrinos N = ν R do not have vectorlike copies. On the other hand right-handed\n\nleptons E always have one and the left-handed fields Q, L may have up to 3 vector-like copies.\n\nTable 7: Spectrum all 32 configurations. U(3) Sp(2) U(1) U(1) Q U D L E N Y H 10 210 629 712 7 3 3 9 5 3 6 10 22 210 629 712 7 3 3 9 5 3 6 10 130 210 629 712 3 3 3 9 5 3 2 10 142 210 629 712 3 3 3 9 5 3 2 10 10 282 629 712 3 3 3 5 5 3 6 26 22 282 629 712 3 3 3 5 5 3 6 26 130 282 629 712 7 3 3 5 5 3 2 26 142 282 629 712 7 3 3 5 5 3 2 26 10 290 629 712 3 3 3 3 5 3 6 18 22 290 629 712 3 3 3 3 5 3 6 18 130 290 629 712 3 3 3 3 5 3 2 18 142 290 629 712 3 3 3 3 5 3 2 18 10 291 629 712 3 3 3 3 5 3 6 18 22 291 629 712 5 3 3 3 5 3 6 18 130 291 629 712 3 3 3 3 5 3 2 18 142 291 629 712 3 3 3 3 5 3 2 18 10 210 629 797 7 3 3 5 5 3 2 10 22 210 629 797 7 3 3 5 5 3 2 10 130 210 629 797 3 3 3 5 5 3 6 10 142 210 629 797 3 3 3 5 5 3 6 10 10 282 629 797 3 3 3 9 5 3 2 26 22 282 629 797 3 3 3 9 5 3 2 26 130 282 629 797 7 3 3 9 5 3 6 26 142 282 629 797 7 3 3 9 5 3 6 26 10 290 629 797 3 3 3 3 5 3 2 18 22 290 629 797 3 3 3 3 5 3 2 18 130 290 629 797 3 3 3 3 5 3 6 18 142 290 629 797 3 3 3 3 5 3 6 18 10 291 629 797 3 3 3 3 5 3 2 18 22 291 629 797 5 3 3 3 5 3 2 18 130 291 629 797 3 3 3 3 5 3 6 18 Continued on next page 50 Table 7 -continued from previous page U(3) Sp(2) U(1) U(1) Q U D L E N Y H 142 291 629 797 3 3 3 3 5 3 6 18 In the following table we list the multiplicities L aa and L aa ′ of the branes that occur in these models, leading to vector-like sets of adjoints and rank-2 tensors. Since brane b is symplectic, the number of adjoints is equal to the number of symmetric tensors.\n\nTable 8: 4d matter from the aa and aa ′ sectors.\n\nBoundary Adjoints Anti-symm. Symm. a(10) 2 2 6 a(22) 2 2 2 a(130) 2 2 6 a(142) 2 2 2 b(210) -14 10 b(282) -14 10 b(290) -14 6 b(291) -14 6 c(629) 9 -14 d(712) 3 -6 d(797) 3 -6 It should be emphasized that CFT constructions generically correspond to particular points in moduli space of CY orientifolds. Due to this, they usually have an 'enhanced' massless particle content with extra vector-like matter and closed string gauge interactions. Thus one would expect that many of the massless vector-like chiral fields present in this class of models could gain masses while moving to a nearby point in moduli space. 51 6.3.4 The instantons Each of these 32 Standard Model compactifications admits 8 instantons. The instanton labels are identical for all the 32 models. They are listed in Table 9 . The first five columns use the same notation as for the standard model boundary labels. In column 6 we list the numerical value of the dilaton coupling to the instanton brane. This quantity is proportional to 1 g 2 . It is instructive to compare these couplings to the gauge couplings, in order to gain intuition on the suppression factor for our instantons. In these models the U(3) dilaton couplings are 0.00622, so that the instantons are more strongly coupled than QCD 19 On the other hand in this particular model the ratio α 3 /α 2 at the string scale is 3.23 (the value of sin 2 θ w at the string scale is 0.527). All of these couplings are subject to renormalization group running, and there are plenty of vector-like states to contribute to this, if one assumes that they acquire masses at a sufficiently low scale. One should perform a detailed renormalization group analysis to check whether one may obtain consistency with the gauge couplings measured at low-energies. Let us emphasize however that one expects that moving in moduli space many of these vector-like states will gain masses and also the values of the different gauge couplings will also generically vary.\n\nSince the value of the Type II dilaton is a free parameter at this level, one can get the appropriate (intermediate) mass scale for the right-handed neutrino Majorana masses by choosing an appropriate value for the dilaton. In this context, it is satisfactory to verify that the instanton couplings are unrelated to the gauge couplings, as expected since they do not correspond to gauge instantons [3], and are in fact less suppressed than the latter.\n\nNote that the 8 instantons fall into two distinct classes (evidently not related by any discrete symmetry, since the conformal weight on the boundary orbit is distinct, and the coupling is different as well). Within each class, the orbits of the four instanton boundaries appear to be related by the Z 2 symmetries of interchange of the last two tensor factors, and simultaneous inversion of the charge q of the minimal model. However, one has to be very careful in reading off symmetries directly from the labels in columns 3 of Tables ( 6) and (9) for a number of reasons. First of all the entries in column 3 are representatives of boundary orbits, and these representatives themselves are merely representatives of extension orbits. Secondly the action of any discrete 19 Note that the Type II dilaton in this compactifications is an arbitrary parameter which can always be chosen so that we consistently work at weak coupling. It is the relative value of gauge couplings which we are comparing here.\n\nTable 9: Instantons for all 32 configurations Lbl. Orbit/Deg. Reps Weight Dim. coupling 414 [8064,0] (0, {1, 1, 0}, 0, {22, -22, 0}, {20, 16, 0}) 5/2 1 0.0016993 417 [8076,30] (0, {1, -1, 0}, 0, {22, 22, 0}, {20, -16, 0}) 5/2 1 0.0016993 456 [8316,0] (0, {1, 1, 0}, 0, {20, 16, 0}, {22, -22, 0}) 5/2 1 0.0016993 459 [8328,30] (0, {1, -1, 0}, 0, {20, -16, 0}, {22, 22, 0}) 5/2 1 0.0016993 418 [8088,0] (0, {1, 1, 0}, 0, {22, -22, 0}, {18, 16, 0}) 5/3 1 0.0027033 420 [8100,0] (0, {1, -1, 0}, 0, {22, 22, 0}, {18, -16, 0}) 5/3 1 0.0027033 502 [8592,0] (0, {1, 1, 0}, 0, {18, 16, 0}, {22, -22, 0}) 5/3 1 0.0027033 505 [8604,30] (0, {1, -1, 0}, 0, {18, -16, 0}, {22, 22, 0}) 5/3 1 0.0027033 symmetry on the degeneracy labels can be non-trivial. In appendix B we discuss these symmetries in more detail."},{"section_type":"OTHER","section_title":"Other examples","text":"The Sp(2) instanton examples just discussed are the ones which get closer to the required minimal set of fermion zero modes. Under slightly weaker conditions, we find many more solutions. In all these cases some additional mechanism beyond exact RCFT will be needed to lift some undesirable zero modes.\n\nThe simplest such case is the following. The tensor product is (2, 8, 8, 18) , MIPF nr. 14, orientifold 2 (the precise spectra may be found using this information in the database www.nikhef.nl/ ∼t58/filtersols.php). There are three distinct brane configurations for which almost perfect instantons exist, namely (a, b, c, d) = (64, 562, 389, 67) and (64, 577, 389, 67) and (65, 560, 189, 66) . Each has six instantons, three of type S2+ and three of type S2-. As in the foregoing example, the six instantons are identical for the three standard model configurations. In this example, they have three different dilaton coupling strengths: .00254, .00665 and .0108 (each value occurs once for S2+ and once for S2-). By comparison, the U(3)-brane dilaton coupling strength is 0.0119338, so that the instanton brane coupling is quite a bit stronger than the QCD coupling. This is again an interesting point if we want that ν R masses are not too much 53 suppressed. Furthermore in this example there are three distinct instanton couplings, so that one may expect three non-zero eigenvalues (with a hierarchy) in the mass matrix. As in the previous examples there is not gauge coupling unification, one rather has α 3 /α 2 = .4813 and sin 2 (θ w ) = .183 at the string scale. Again a full renormalization group analysis should be performed in order to check consistency with the measured low-energy gauge coupling values.\n\nThese models all have a hidden sector consisting of a single Sp(2) factor. They have respectively 3, 1 and 3 susy Higgs pairs, and a spectrum of bi-fundamentals that is closer to that of the standard model than the previously discussed Sp(2) examples: quarks and leptons do not have vector-like copies (there are only some vector-like leptoquarks), and even one of the three models have the minimal set of Higgs fields of the MSSM. The rest of the spectrum is purely vector-like, and contains a number of rank-2 tensors, including eight or six adjoints of U(3). Furthermore there is vectorlike observable-hidden matter. The only undesirable instanton zero-mode is a single bi-fundamental between the hidden sector Sp(2) brane and the instanton brane. Still, these SM brane configurations without the hidden sector, provide interesting and very simple local models of D-brane sectors admitting instantons generating neutrino masses (with the additional ingredients required to eliminate the extra universal triplets of fermion zero modes)."},{"section_type":"OTHER","section_title":"R-parity violation","text":"We now turn to the generation of other possible superpotentials violating B -L. An instanton violates R-parity if the amount of B -L violation,\n\nI M a -I M a ′ -I M d + I M d ′ (6.4)\n\nis odd. Examples of instantons with that property were found in the following tensor product/MIPF/orientifold combinations: [(1, 16, 16, 16) , 12, 0], [(2, 4, 16, 34) , 49, 0], [(2, 4, 12, 82) , 19, 0] [(2, 4, 22, 22), 49, 0] and [(2, 4, 16, 34), 18, 0] . Note that all cases for which O2 or U2 instantons were found necessarily have R-parity violating instantons as well: the corresponding O1 and U1 instantons have I M d or I M d ′ equal to ±1, whereas the intersection with the a is non-chiral. In principle, there are many more ways to obtain R-parity violating instantons (either due to non-vanishing contributions to\n\nI M a -I M a ′ or higher values of I M d -I M d ′ )\n\n, and indeed, many such instantons turn out to exist. But the number of tensor product/MIPF/orientifold combinations where they occur hardly increases: only in the case [(1, 16, 16, 16), 12, 0] we found R-parity 54 violating instantons, but no U1 or O1 instantons. This suggests that in the other cases R-parity is a true symmetry of the model. Unfortunately we have no way of rigorously ruling out any other non-perturbative effects, but at least the set we can examine respects R-parity. This includes in particular the models without hidden sector (found for [(2, 4, 22, 22), 13, 3] ) discussed above.\n\nThe following table list the total number of instantons with the chiral intersections listed in table 2. The total number of instantons (boundaries violating the sum rule, as defined in (6.1)) is 29680, for all standard model configurations combined. The last four columns indicate how many unitary instantons satisfy the sum rule exactly as listed in table (6.1), how many satisfy it with I M x ↔ -I M x ′ (the column U'), and how many O-type and S-type instantons there are. Here 'S' refers to boundaries with a symplectic Chan-Paton group if the boundary is used as an instanton brane. All intersection numbers for type S have been multiplied by 2 before comparing with table 2. For real branes, the relevant quantities used in the comparison are\n\nI M a -I M a ′ , I M c -I M c ′ and I M d -I M d ′ , while I M b = 0.\n\nThere are fewer unitary instantons possibly generating Majorana masses then the numbers mentioned above because the conditions we use here are stricter: we require here that I M x and I M x ′ match exactly, not just their difference. Note however that this still allows additional vector-like zero-modes. If we only wish to consider cases without any spurious zero-modes, we may limit ourselves to the O-type instantons in the last column. There are very few to inspect, and all of them turn out to have a few non-universal zero modes.\n\nD = 4 Operator U U' S O ν R ν R 1 2 627 3 L HL H 0 5 550 3 L H 3 3 0 4 QDL 8 4 0 4 UDD 0 0 0 4 LLE 8 4 0 4 QQQL 0 4 0 3892 UUDE 4 0 0 3880\n\nTable 10: Number of instantons in our search which may induce neutrino masses (first 2 rows), R-parity violation (next 4 rows) or proton decay operators (last 2 rows).\n\nThe last two cases are B -L preserving dimension five operators, and obviously do not come from the set of 29680 B -L violating instantons. They were searched for 55 separately, but the search was limited to the same 391 models we used in the rest of the paper. Obviously, one could equally well look for such instantons in the full database, since their existence does not require a massive B -L.\n\nIt is interesting to note that in the classes of MSSM-like models discussed earlier in this section with the closest to minimal zero mode structure, there are no instantons al all generating either R-parity violating or the B -L dim=5 operators in the table.\n\nThis makes them particularly attractive.\n\nNote that all numbers in table 10 refer to the occurrence of instantons in the set of 391 tadpole-free models with massive B-L, but without checking the presence of zero-modes between the hidden sector and the instanton. It makes little sense to use the hidden sector in the database for such a check, since this is just one sample from a (usually) large number of possibilities. A meaningful question would be: can one find a hidden sector that has no zero-modes with the instanton. We have done such a search for the B -L violating instantons (see the exclamation marks in the last column of table (5)), but not for the B -L preserving instantons."},{"section_type":"CONCLUSION","section_title":"Conclusions and outlook","text":"In this paper we have presented a systematic search for MSSM-like Type II Gepner orientifold models allowing for boundary states associated to instantons giving rise to neutrino Majorana masses. This search is very well motivated since neutrino masses are not easily accommodated in the semi-realistic compactifications constructed up to now. String instanton induced Majorana masses provides a novel and promising way to understand the origin of neutrino masses in the string theory context.\n\nThe string instantons under discussion are not gauge instantons. Thus, for example, they not only break B + L symmetry (like 't Hooft instantons do) but also B -L, allowing for Majorana neutrino mass generation. The obtained mass terms are of order M s exp(-V /g 2 ) but this suppression is unrelated to the exponential suppression of e.g. electroweak instantons and may be mild. In fact we find in our most interesting examples that the instanton action is typically substantially smaller than that of QCD or electroweak instantons, and hence these effects are much less suppressed than those coming from gauge theory instantons.\n\nTo perform our instanton search we have analyzed the structure of the zero modes that these instantons must have in order to induce the required superpotential. This analysis goes beyond the particular context of Gepner orientifolds and has general 56 validity for Type II CY orientifolds. We have found that instantons with O(1) CP symmetry have the required universal sector of just two fermionic zero modes for the superpotential to be generated. Instantons with Sp(2) and U(1) CP symmetries have extra unwanted universal fermionic zero modes, which however may be lifted in a variety of ways in more general setups, as we discuss in the text. In fact we find in our search that around 98 % of the instantons with the correct structure of charged zero modes have Sp(2) CP symmetry. Indeed, from a number of viewpoints the Sp(2)\n\ninstantons are specially interesting. The instantons we find with the simplest structure of fermionic zero modes are Sp(2) instantons which are also the ones which are present more frequently in the MSSM-like class of Gepner constructions considered. They have also some interesting features from the phenomenological point of view. Indeed, due to the non-Abelian structure of the CP symmetry, the structure in flavor space of the neutrino Majorana masses factorizes. This makes that, irrespective of what particular compactification is considered, Sp(2) instantons may easily lead to a hierarchical structure of neutrino masses. It would be important to further study the possible phenomenological applications of the present neutrino mass generating mechanism.\n\nString instanton effects can also give rise to other B-or L-violating operators. Of particular interest is the dimension 5 Weinberg operator giving direct Majorana masses to the left-handed neutrinos. We find that in the most interesting cases, different instantons giving rise to the Weinberg operator and to ν R Majorana masses are both simultaneously present. Which effect is the dominant one in the generation of the physical light neutrino masses depends on the values of the instanton actions and amplitudes as well as on the value of the string scale. Instantons may also generate dim< 5 operators violating R-parity. We find however that instantons inducing such operators are extremely rare, and in fact are completely absent in the Gepner models with the simplest Sp(2) instantons inducing neutrino masses.\n\nThere are many avenues yet to be explored. It would be important to understand better the possible sources (moving in moduli space, addition of RR/NS backgrounds etc.) of uplifting for the extra uncharged fermionic zero modes in the most favoured Sp(2) instantons. A second important question is that we have concentrated on checking the existence of instanton zero modes appropriate to generate neutrino masses; one should further check that the required couplings among the fermionic zero modes and the relevant 4d superfields (i.e. ν R or L H) are indeed present in each particular case. This is in principle possible in models with a known CFT description but could be difficult in practice for the Gepner models here described.\n\n57 Instantons can also generate other superpotentials with interesting physical applications. One important example is the generation of a Higgs bilinear (i.e. a µ-term) in MSSM-like models [4, 3]. Thus, e.g., one could perform a systematic search for instantons (boundary states) generating a µ-term in the class of CFT Gepner orientifolds considered in the present article. Other possible application is the search for instantons inducing superpotential couplings involving only closed string moduli. The latter may be useful for the moduli-fixing problem, or for non-perturbative corrections to perturbatively allowed couplings [56].\n\nFinally, it would be important to search for analogous instanton effects inducing neutrino masses in other string constructions (heterotic, M-theory etc.). A necessary condition is that the anomaly free U(1) B-L gauge boson should become massive due to a Stückelberg term.\n\nThe importance of neutrino masses in physics beyond the Standard Model is unquestionable. We have shown that string theory instantons provide an elegant and simple mechanism to implement them in semi-realistic MSSM-like string vacua, and a powerful constraint in model building. In our opinion, the conditions of the existence of appropriate instantons to generate neutrino masses should be an important guide in a search for a string description of the Standard Model."},{"section_type":"OTHER","section_title":"Acknowledgements","text":"We thank M. Bertolini, R. Blumenhagen, S. Franco, M. Frau, S. Kachru, E. Kiritsis, A. Lerda, D. Lüst, F. Marchesano, T. Weigand for useful discussions. A.M.U. thanks M. González for encouragement and support. The research of A.N. Schellekens was funded in part by program FP 57 of the Foundation for Fundamental Research of Matter (FOM), and Research Project FPA2005-05046 of de Ministerio de Educacion y Ciencia, Spain. The research by L.E. Ibáñez and A.M. Uranga has been supported by the European Commission under RTN European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT (Spain), and the Comunidad de Madrid under project HEPHACOS P-ESP-00346.\n\nAppendix A CFT Notation\n\nHere we summarize the labelling conventions for various CFT quantities. Further details and explanations can be found in [43] .\n\nIt is important to keep in mind that there are four steps in the construction, each involving choices of some quantities. The steps are\n\n• A CFT tensor product\n\n• An extension of the chiral algebra of this tensor product\n\n• The choice of a MIPF\n\n• The choice of an orientifold The second and third step are easily confused. A MIPF can itself be of extension type (although it can also be of automorphism or mixed type), meaning that it implies an extension of the chiral algebra. The crucial difference between step two and three in that case is that in step 2 all fields that are non-local with respect to the extension are projected out, and the symmetry of the extension is imposed on all states of the CFT,\n\ni.e in particular on all boundary states. The extension in step three acts as a bulk invariant, but the boundary states are not required to respect the symmetry implied by the extension.\n\nPrimary fields of N = 2 minimal models are labeled in the usual way by three integers (l, q, s). In addition to these minimal models, one building block of our CFT's is of course a set of NSR fermions in four dimensions. They can be represented by the four conjugacy classes (0), (v), (s), (c) analogous to those of a root lattice of type D. Primary fields in a tensor product of M factors are therefore labelled as\n\nI = ((x), (l 1 , q 1 , s 1 ), . . . , (l M , q M , s M )) (A.1)\n\nwhere x = 0, v, s or c.\n\nThis tensor product is extended by the alignment currents and the spin-1 field corresponding to the space-time supersymmetry generator. This organizes the tensor product fields into orbits, which can be labelled by one of the elements of the orbit. We always choose the field of minimal conformal weight (or one of them, in case there 59 are more) as the orbit representative labelling the orbit. The supersymmetry generator may have fixed points, leading to orbits appearing more than once as primary fields of the extended theory. In those cases we need an additional degeneracy label to distinguish them. It is convenient to choose for this label a character of the discrete group that is causing the degeneracy, the \"untwisted stabilizer\", which depends on I.\n\nDenoting this character as Ψ I we get then the following set of labels for the primaries of the extended CFT\n\ni = [I, Ψ I ] (A.2)\n\nwhere I has the form (A.1). If there are no degeneracies we will leave out the square brackets and the Ψ I .\n\nIn boundary CFT's two new labels appear: the labels of Ishibashi-states that propagate in the transverse channel of the annulus, and the boundary labels. In the simplest, \"Cardy\" case both sets of labels are in one-to-one correspondence with the extended CFT labels i. But if we consider non-trivial MIPFs Z ij both sets of labels are different.\n\nThe Ishibashi states are in one-to-one correspondence with the fields i with Z ii c = 0. Degeneracies can occur here if Z ii c > 0. This requires the introduction of a degeneracy label. Such degeneracies may occur if the stabilizer of i (the set of simple currents that fix i) is non-trivial. It is convenient to use elements J of the stabilizer as degeneracy labels, so that the Ishibashi labels get the following form m = (i, J) , (A.3) where i is an extended CFT label, as defined above (to be precise, in some cases a non-trivial degeneracy label is introduced even if Z ii c = 1. The details will not matter here).\n\nBoundary states correspond to orbits of the simple current group H that defines a MIPF. To label such orbits we choose a representative. There is no obvious canonical representative (one could use one of minimal conformal weight, but the conformal weight of orbit members of a boundary state does not play any rôle in the formalism, unlike the conformal weight of a primary). So in this case we just make an arbitrary choice. Once again there can be degeneracies. In this case they are due to a subgroup of the stabilizer called the \"Central Stabilizer\". It is convenient to label the boundary states by an orbit representative i and a character ψ i of the central stabilizer. If we expand the boundary state label in all of its components we get\n\na = [i, ψ i ] = [[I, Ψ I ], ψ i ] = [((x), (l 1 , q 1 , s 1 ), . . . , (l M , q M , s M )), Ψ I ], ψ i ] . (A.4) 60\n\nNote that i is just a representative of a boundary orbit, and that I is just a representative of an orbit of the extension of the CFT."},{"section_type":"OTHER","section_title":"B Instanton boundary symmetries","text":"In the hidden-sector free example discussed in some detail in section 6 we have encountered Sp(2)-type instantons, the most common kind in our scan. This particular model is the one that comes closest to the required zero mode count, although the only superfluous zero modes are rather awkward. Let us assume that the effect of these superfluous universal zero-modes instantons can be avoided. Then there is still another problem we have to face, namely that the two zero modes α i and γ i are related by an Sp(2) transformation of the label i. Then we we need at least three independent instantons (with unrelated couplings) to generate three non-zero neutrino masses, as discussed in Section 3.3. Since the technology to compute the couplings is not yet available, we cannot be completely sure that the relevant couplings are distinct, or indeed that they are non-vanishing, but at least we can inspect if there are obvious symmetries relating them.\n\nThe unextended tensor product (2, 4, 22, 22) has 64 discrete symmetries: five separate charge conjugations of the factors (including the NSR space-time factor) and the interchange of the two identical k = 22 minimal models. To get space-time supersymmetry this tensor product is extended with the product of the simple current Ramond ground states of each factor. These Ramond ground states are not invariant under charge conjugation. Therefore this extension breaks the discrete symmetries to Z 2 × Z 2 , the combined charge conjugation of all five factors and the permutation of the two identical factors. The combined charge conjugation also acts non-trivially on the Ramond ground states in each factor, but the result is the charge conjugate of the space-time supersymmetry generator, which is in the chiral algebra. The combined conjugation is in fact the charge conjugation symmetry of the extended CFT. It turns out that only a Z 2 subgroup of Z 2 × Z 2 acts non-trivially on the simple currents of the extended CFT: the permutation of the k = 22 factors acts in the same way as charge conjugation. The action of these symmetries on the complete set of primary fields is more complicated. It is easy to see that the permutation acts differently than charge conjugation. In general the primary fields of the extended CFT are labelled as i = [((x), (i 1 ), . . . , (i 4 )), Ψ] (see Appendix A). The action of the permutation is to interchange i 3 and i 4 , but in cases with a non-trivial degeneracy it is not a priori 61 clear which of the degenerate states is the image of the map. This can be resolved by examining the fusion rules, which should be invariant under the permutation: N ijk = N π(i)π(j)π(k) , (B.1) where N is a fusion coefficient and π a permutation or other automorphism. In general there may be more than one way to resolve these ambiguities, resulting in additional automorphism of the CFT. The standard example of this situation is the extension of the affine algebra A 1 level 4 by the simple current. The resulting CFT has an outer automorphism, non-trivial charge conjugation, that has no counterpart in A 1 level 4.\n\nAs mentioned above, the Z 2 permutation symmetry is respected by the MIPF and the orientifold, and since charge conjugation acts in the same way on the simple currents as the permutation, charge conjugation is respected as well.\n\nIn this way we end up with (at least) a surviving Z 2 × Z 2 discrete symmetry acting on the boundary labels, or a larger discrete symmetry if that symmetry is extended by the action on the degeneracy labels of the extension. The foregoing story repeats itself for the action on the boundary labels. The boundary labels are given in terms of the CFT labels plus a second degeneracy label, the one indicated by the second entry in the square brackets in column 2 of tables (6) and (9). Once again one has to determine not only how a symmetry acts on the first entry (this is just the action of the symmetry in the extended CFT, respecting its fusion rules), but also how it acts on the degeneracy labels. In this case the precise action can be determined from the invariance of the annulus coefficients A i ab = A π(i) π(a)π(b) , (B.2)\n\nwhere π is the action on the primaries of the extended CFT (as determined above) and π is the action on the boundary labels induced by π.\n\nSince the orientifold choice is non-trivial, boundary charge conjugation does not coincide with CFT charge conjugation. Indeed, the eight instanton boundary states are invariant under boundary charge conjugation (which they must be in order to produce a \"real\" Sp(2)-type instanton). However, just as permutations, CFT charge conjugation may induce a non-trivial discrete symmetry on the boundary states.\n\nIn addition to these \"outer automorphisms\" there is the notion of boundary simple currents, introduced in the appendix of [6] . These may be thought of as remnants of the original simple currents, and imply relations between annulus amplitudes of the form A i ab ′ = A i Ja(Jb) ′ (B.3) 62 All of the aforementioned symmetries might relate instanton couplings, and hence threaten their numerical independence. However, in order to do that they have to be symmetries of the full standard model/instanton configuration, not just relate some of the eight instantons to each other. It is easy to see that the permutation of the k = 22 factors changes the standard model brane configuration. Consider brane c: it turns out that under permutation boundary state 629 it is mapped to boundary state 544 or 545 (depending on the action on the degeneracy label), which in any case is distinct. Hence even if the instanton boundaries 414 and 456 resp. 418 and 502 are mapped to each other by boundary permutation, at the same time the standard model configuration is\n\nmapped to a distinct one.\n\nThis means that we may expect at least four distinct couplings, which should be sufficient. It is of course possible to work out the discrete symmetries exactly, but in view of this argument this would not yield any additional insight.\n\nWe do know the exact boundary orbits. The orbit of instanton label 414 is (414, 415, 416, 417) , so that instantons 414 and 417 are related. But the orbit of brane c under the same action is (629, 628, 626, 627) . Hence the action that relates 414 and 417 maps 629 to 627. In fact all four standard model boundaries a,b,c,d are mapped to different ones. This implies that instantons 414 and 417 may produce different couplings as well, so that all eight instantons may contribute in a different way.\n\nThese distinctions concern the disk correlators d (r) a in (3.12). The factors exp(-Re U r ) will be related by discrete symmetries, and it seems reasonable to expect them to be identical for instantons 414, 417, 456 and 459, which is indeed correct. However there is no reason to expect the other four instantons to have the same suppression factor, and indeed they do not.\n\nNote that these symmetries imply the existence of a much larger set of standard model configurations than the 32 discussed here. However, as mentioned before, the 32 models considered here display all possible distinct spectra."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"String Theory, as the leading candidate for a unified theory of Particle Physics and Gravity, should be able to describe all observed particle phenomena. One the most valuable experimental pieces of information obtained in the last decade concerns neutrino masses. Indeed the evidence from solar, atmospheric, reactor and accelerator experiments indicates that neutrinos are massive. The simplest explanation of the smallness of neutrino masses is the see-saw mechanism [1] . The SM gauge symmetry allows for two types of operators bilinear on the neutrinos (with dimension ≤ 4) :\\n\\nL ν = M ab ν a R ν b R + h ab ν a R HL b (1.1)\\n\\nwhere ν R is the right-handed neutrino, L is the left-handed lepton doublet and H is the Higgs field. In supersymmetric theories, this term arise from a superpotential with the above structure, upon replacing fields by chiral superfields. If M ab is large, the lightest neutrino eigenvalues have masses\\n\\nM ν = < H > 2 h T M -1 h (1.2)\\n\\nFor M ∼ 10 10 -10 13 GeV and Dirac neutrino masses of order charged lepton masses, the eigenvalues are consistent with experimental results.\\n\\nWhat is the structure of neutrinos and their masses in string theory? In specific compactifications giving rise to the MSSM spectra singlet fields corresponding to righthanded neutrinos ν R generically appear. Dirac neutrino masses are also generically present but the required Majorana ν R masses are absent. This is because most MSSMlike models constructed to date have extra U(1) symmetries, under which the righthanded neutrinos are charged, which hence forbid such masses. In many models, such symmetries are associated to a U(1) B-L gauge boson beyond the SM. In order to argue for the existence of ν R masses, string model builders have searched for nonrenormalizable couplings of the type (ν R ν R NR NR ) with extra singlets N R . Once the latter fields get a vev, U(1) B-L is broken and a Majorana mass appears for the ν R .\\n\\nAlthough indeed such couplings (or similar ones with higher dimensions) exist in some semi-realistic compactifications, such a solution to the neutrino mass problem in string theory has two problems: 1) The typical ν R masses so generated tend to be too small due to the higher dimension of the involved operators and 2) The vevs for the N R fields breaks spontaneously R-parity so that dimension 4 operators potentially giving rise to fast proton decay are generated. This is in a nutshell the neutrino problem in string compactifications (see [2] for a recent discussion in heterotic setups). 1 In [3] (see also [4] ) two of the present authors pointed out that there is a builtin mechanism in string theory which may naturally give rise to Majorana masses for right-handed neutrinos. It was pointed out that string theory instantons may generate such masses through operators of the general form M string e -U ν R ν R . (1.3) Here U is a linear combination of closed string moduli whose imaginary part gets shifted under a U(1) B-L gauge transformation in such a way that the operator is fully gauge invariant. The exponential factor comes from the semi-classical contribution of a certain class of string instantons. This a pure stringy effect distinct from the familiar gauge instanton effects which give rise to couplings violating anomalous global symmetries like (B+L) in the SM. Here also (B-L) (which is anomaly-free) is violated.\\n\\nThis operator is generated due to existence of instanton fermionic zero modes which are charged under (B -L) and couple to the ν R chiral superfield. Although the effect can take place in different constructions, the most intuitive description may be obtained for the case of Type IIA CY orientifold compactifications with background D6-branes wrapping 3-cycles in the CY. In the simplest configurations one has four SM stacks of D6-branes labeled a, b, c, d which correspond to U(3), SU(2) (or U(2)), U(1) R and U(1) L gauge interactions respectively, which contain the SM group. One can construct compactifications with the MSSM particle spectrum in which quarks and leptons lie at the intersections of those SM D6-branes. Then the relevant instantons correspond to euclidean D2-branes wrapping 3-cycles in the CY (satisfying specific properties so as to lead to the appropriate superspace interaction). The D2-D6 intersections lie the additional fermionic zero modes which are charged under (B -L). For instantons with the appropriate number of intersections with the appropriate D6-branes, and with open string disk couplings among the zero modes and the ν R chiral multiplet (see fig.(3.1)), the operator in (1.3) is generated.\\n\\nThe fact that the complex modulus U transforms under U(1) B-L gauge transformations indicates that the U(1) B-L gauge boson gets a mass from a Stückelberg term.\\n\\nSo a crucial ingredient in the mechanism to generate non-perturbative masses for the ν R 's is that there should be massless U(1) B-L gauge boson which become massive by a Stückelberg term. It turns out that not many semi-realistic models with U(1) B-L mass from Stückelberg couplings have been constructed up to date. In the literature there are two main classes of RR tadpole free models with massive B-L. The first class are non-susy type IIa toroidal orientifold models first constructed in [5] . The second class are the type II Gepner orientifold models constructed by one of the present authors and 2 collaborators [6, 7] . The former were already considered in [3] . In the present paper we will concentrate on the RCFT Gepner model constructions, which lead to a large class of MSSM like models, more representative of the general Calabi-Yau compactifications (for a recent discussion of instanton-induced neutrino masses in a model with no RR tadpole cancellation, see [8] ).\\n\\nThe class of constructions in [6, 7] start with any of the 168 Type II compactifications obtained by tensoring N = 2 SCFT minimal models. In addition one can choose a number of modular invariant partition functions (MIPF), leading to a total of 5403. Then different consistent orientifold projections are performed on the different models. This yields a total of 49304 Type II orientifolds. The open string sector of the theory is defined in terms of the boundary states of the theory. Intuitively, they play the same role as D-branes wrapping cycles in the geometrical settings. Thus one associates boundary states a, b, c, d to the gauge groups giving rise to the SM. Different choices for the SM boundary states lead to different spectra. In the present paper we will make use of the data in [6] which contains 211634 different MSSM-like spectra (including also different hidden sectors). Although this number is huge, most of these models are really extensions of the MSSM, since they have either an extra U(1) B-L or SU(2) R × U(1) B-L group factor beyond the SM group. As we said, we are actually only interested in models in which the U(1) B-L gets a Stückelberg mass. Then we find that the number of MSSM-like models with these characteristics is dramatically reduced: only 0.18 percent of the models (391) have a massive U(1) B-L .\\n\\nAs we said, in the geometrical setting of IIA orientifolds with intersecting D6-branes [9, 10] (see [11] for reviews and [12, 13] for the IIB counterparts), instantons are associated to D2-branes wrapping 3-cycles, like the background D6-branes do. Analogously, in the RCFT setting the same class of boundary states appearing in the SM constructions are the ones corresponding to instantons. The zero modes on the instanton is computable from the overlaps of instanton brane boundary states (zero modes uncharged under the 4d gauge group) or of instanton and 4d spacefilling brane boundary states (zero modes charged under the corresponding gauge factor). We find that the criteria for a non-perturbative superpotential to be generated [14] are only fulfilled if the Chan-Paton (CP) symmetry of the instantons is O(1). For instantons with CP symmetry 1 Sp(2) or U(1) we find that there are a few extra uncharged fermionic zero modes which would preclude the formation of the searched superpotentials. On the other hand we argue that the addition of fluxes and/or possible non-perturbative 1 We adopt the convention that the fundamental representation of Sp(m) is m-dimensional.\\n\\nextensions of the orientifold compactifications would allow also instantons with Sp(2) and U(1) symmetries to generate such superpotentials. We thus include all O(1), Sp(2) and U(1) instantons 2 in our systematic search. The computation of charged and uncharged fermion zero modes may be easily implemented as a routine in a systematic computer search for instanton zero modes in Gepner MSSM-like orientifolds. Results of such a systematic computer search are presented in this article.\\n\\nWe find that out of the 391 models with massive U(1) B-L , there are very few admitting instantons with the required minimal O(1) CP symmetry, and in fact none of them without additional vector-like zero modes. On the other hand we do find 32 models admitting Sp(2) symmetric instantons with just the required charged zero mode content (and the minimal set of non-chiral fermion zero modes). In fact they are all variations of the same orientifold Gepner model based on the tensor product 2, 4, 22, 22) . These models all in fact correspond to the same MIPF and orientifold projection, they only differ on which particular boundary states corresponding to the four a, b, c, d SM 'stacks'. All models have the same chiral content, exactly that of the MSSM , with extra vectorlike chiral fields which may in principle become massive in different points of the CY moduli space. They have no hidden sector, i.e., the gauge group is just that of the SM. For each of the models there are 8 instantons with Sp(2) CP symmetry with just the correct charged zero mode structure allowing for the superpotential coupling (1.3) giving rise to ν R Majorana masses. As we said, they have extra uncharged fermion zero modes beyond the two required to generate a superpotentials. However one would expect that these unwanted zero modes might be lifted in more generic situations in which e.g. NS/RR fluxes are added.\\n\\n(k 1 , k 2 , k 3 , k 4 ) = (\\n\\nWe thus see that, starting with a 'large' landscape of 211634 MSSM-like models, and searching for instantons inducing neutrino masses, we find there are none admitting the O(1) instantons with exactly the required zero mode structure, and only few (32) examples with Sp(2) instantons with next-to-minimal uncharged zero mode structure (and exactly the correct charged zero modes). Let us emphasize though that it is the existence of massive U(1) B-L models which is rare. Starting with the subset of models with a massive U(1) B-L , finding models with correct instanton charged zero modes within that class is relatively frequent, 10 percent of the cases. Furthermore, we will see that there are further models beyond those 32 which contain extra non-chiral instanton zero modes and which may also be viable if these modes get massive by some 2 We refer to the different kinds of instanton by their Chan-Paton symmetry on their volume. Since we are not interested in gauge theory instantons, this notation should not be confusing.\\n\\n4 effect (like e.g. the presence of RR/NS fluxes).\\n\\nInstantons may generate some other interesting superpotential couplings in addition to ν R masses, some possibly beneficial and others potentially dangerous. In particular we find that in the models which contain Sp(2) instantons which might induce ν R masses, there are also other instantons which would give rise directly to the Weinberg operator [15] L W = λ M (LHLH) (1.4) Once the Higgs field gets a vev, this gives rise directly to left-handed neutrino masses.\\n\\nThus we find that in that class of models both the see-saw mechanism (which also gives rise to a contribution to the Weinberg operator) and an explicit Weinberg operator might contribute to the physical masses of neutrinos. Which effect dominates will depend on the relative size of the corresponding instanton actions as well as on the size of the string scale. Among potentially dangerous operators which might be generated stand the R-parity violating operators of dimension < 5, which might give rise e.g. to fast proton decay. We make an study of the possible generation of those, and find that in all models in which ν R masses might be generated R-parity is exactly conserved. This is a very encouraging result.\\n\\nA natural question to ask is whether one can say something about the structure of masses and mixings for neutrinos. As argued in [3] generically large mixing angles are expected, however to be more quantitative we also need to know the structure of Yukawa couplings for leptons. In principle those may be computed in CFT but in practice this type of computation has not yet been developed for CFT orientifolds.\\n\\nNevertheless we show that, in the case of instantons with Sp(2) CP symmetry, a certain factorization of the flavor structure takes place, which could naturally give rise to a hierarchical structure of eigenvalues for neutrino masses.\\n\\nThe structure of this article is as follows. In the next section we present a discussion of instanton induced superpotentials in Type II orientifolds. This discussion will apply both to Type IIA and Type IIB CY orientifolds as well as to more abstract CFT orientifolds. We discuss the structure of both uncharged and charged instanton zero modes. In particular we show that only instantons with O(1) CP symmetry have the appropriate uncharged zero mode content to induce a superpotential contribution. We also discuss how Sp(2) and U(1) might still generate superpotential contributions if extra ingredients are added to the general setting. In section 3 we apply that discussion to the specific case of the generation of ν R Majorana masses, showing what is the required zero mode structure in this case. We show how the flavor structure of the 5 Majorana mass term factorizes in the case of instantons with Sp(2) CP symmetry, leading potentially to a hierarchical structure of eigenvalues. We further discuss the generation of other B/L-violating operators including the generation of the Weinberg operator as well as R-parity violating couplings. In section 4 we review the RCFT Type II orientifold constructions in [6, 7] . A general discussion of zero fermion modes for instantons in RCFT orientifolds is presented in section 5.\\n\\nThe results of our general search for instantons generating ν R masses are presented in section 6. We provide a list of all Gepner orientifolds which admit instanton configurations potentially giving rise to ν R Majorana masses. We describe the structure of the models with Sp(2) instantons having the required charged zero modes for that superpotential to be generated. We also describe the boundary states of the corresponding instantons and provide the massless spectrum of the relevant MSSM-like models. Other orientifolds with zero mode structure close to the minimal one are also briefly discussed. A brief discussion about the possible generation of R-parity violating superpotentials is included. We leave section 7 for some final comments. Some notation on the CFT orientifold constructions, and a discussion of the CFT symmetries in the Sp(2) examples are provided in two appendices.\\n\\nAs this paper was ready for submission, we noticed [16, 17] , whose discussion of instanton zero modes partially overlaps with our analysis in Section 2.2.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Instanton induced superpotentials in Type II orientifolds\",\n \"text\": \"In this Section we review the generation of superpotentials involving 4d charged fields via D-brane instantons in type II compactifications. The discussion applies both to type IIA and IIB models, and to geometrical compactification as well as to more abstract internal CFT's. For recent discussions on D-brane instantons, see [4, 3, 25, 8] .\\n\\nBefore starting, a notational remark in in order. Our notation is adapted to working in the covering theory, namely the type II compactification, and orientifolding in a further step. Thus we describe the brane configurations as a system of branes (described by boundary states for abstract CFT's), labeled k, and their orientifold images labeled k ′ . Similarly, we denote M the brane / boundary state corresponding to the instanton brane, and M ′ its orientifold image. If a brane is mapped to itself under the orientifold action, we call it a 'real' brane / boundary, and 'complex' otherwise. 6 2.1 D-brane instantons, gauge invariance and effective operators A basic feature of type II orientifold compactifications with D-branes is the generic presence of Stückelberg couplings between the U(1) gauge fields on the D-branes, and certain 4d RR closed string 2-forms. These couplings are required by the Green-Schwarz mechanism when the U(1)'s have non-zero triangle contributions to mixed anomalies [18, 19] , but can also exist for certain non-anomalous U(1)'s [5, 20]. These couplings make the U(1) gauge bosons massive, but the symmetry remains as a global symmetry exact in perturbation theory. Since the closed string moduli involved are scalars (0-forms) in the RR sector, the natural candidate non-perturbative effects to violate these U(1) symmetries are instantons arising from euclidean D-branes coupling to these fields.\\n\\nIn computing the spacetime effective interaction mediated by the instanton, one needs to integrate over the instanton zero modes. In the generic case (and in particular for the case of our interest) there are no bosonic zero modes beyond the universal ones (namely, the four translational bosonic zero modes associated to the position of the instanton). On the other hand, the structure of fermion zero modes will be crucial.\\n\\nSince we are interested in models with non-trivial 4d gauge group, arising from a set of background 4d spacetime filling branes, we consider separately fermion zero modes which are uncharged under the 4d gauge group and those which are charged. In this paper we restrict our discussion to 4d N = 1 supersymmetric models, and this will simplify the analysis of zero modes.\\n\\nFermion zero modes which are uncharged under the 4d gauge group determine the kind of 4d superspace interaction which is generated by the instanton. We are interested in generating superpotential interactions, which, as is familiar, requires the instanton to have two fermion zero modes to saturate the d 2 θ superspace integration.\\n\\nFor this, a necessary (but not sufficient!) condition is that the D-branes are half-BPS, so these fermion zero modes are the Goldstinos of the two broken supersymmetries.\\n\\nIn the string description, uncharged zero modes arise from open strings in the MM sector (in our notation, the one leading to adjoint representations), which in particular contain these Goldstinos, and the MM ′ sector (in our notation, the one leading to two-index symmetric or antisymmetric tensors). Note that both are the same for 'real' branes. Hence we are primarily interested in D-branes whose MM sector contains just two fermion zero modes, and whose MM ′ sector (for 'complex' branes) does not contain additional fermion zero modes.\\n\\nIn analogy with the familiar case of gauge theory instantons [21], charged fermion zero modes determine the violation of perturbative global symmetries by the instantoninduced interaction. Namely, in order to saturate the integration over the charged fermions zero modes, the spacetime interaction must contain insertions of fields charged under the 4d gauge symmetry, and in particular under the global U(1) factors, which are thus violated by the D-brane instanton. Notice that this holds irrespectively of the number of uncharged fermion zero modes, namely of the kind of superspace interaction induced by the instanton. Restricting to superpotential interactions, the resulting operator in the 4d effective action has roughly the form W inst = e -U Φ 1 . . . Φ n (2.1) Here the fields Φ 1 , . . . , Φ n are 4d N = 1 chiral multiplets charged under the 4d gauge group, and in particular also under the U(1) symmetries. Note also that the instanton amplitude contains a prefactor (which in general depends on closed and open string moduli) arising from the Gaussian path integral over (massive) fluctuations of the instanton (hence described by an open string annulus partition function, see [22, 23] for related work), which we can ignore for our purposes in this paper.\\n\\nFor D-brane instantons, U is the closed string modulus to which the euclidean Dbrane couples. In the D-brane picture, instanton fermion zero modes charged under the gauge factor carried by the k th stack of 4d space-filling branes (and its image k ′ ) arise from open strings in the Mk and Mk ′ sectors, transforming as usual in the ( M , k ) and ( M , k ) representations, respectively (with both related in the case of 'real' branes). The (net) number of instanton fermion zero modes with such charges is given by certain multiplicities 3 I M k , I M k ′ .\\n\\nA D-brane instanton, irrespectively of the superspace structure of the 4d interactions it may generate, violates U(1) k charge conservation by an amount I M k -I M k ′ for 'complex' branes and I M k for 'real' branes. In particular, this is the total charge of the field theory operator Φ 1 . . . Φ n in (2.1). From the Stückelberg couplings, it is possible to derive [3] (see [23, 24, 25] for related work, also [26]) 3 In geometric type IIA compactifications with 4d spacetime-filling branes and instanton branes given by D6-and D2-branes wrapped on Special Lagrangian 3-cycles, I Mk corresponds to the intersection number between the 3-cycles corresponding to the k th D6-and the D2-brane M (and similarly for I Mk ′ ). In geometric type IIB compactifications, it corresponds to the index of a suitable Dirac operator. In general (even for abstract CFT's) it can be defined as the Witten index for the 2d theory on the open string with the boundary conditions corresponding to the two relevant branes. We will often abuse language and refer to this quantity as intersection number, even in Section 6 where we work in the non-geometric regime of type IIB compactifications.\\n\\nthat for 'complex' instantons, gauge transformations of the U(1) k vector multiplets V k → V k + Λ k , transform U as U → U + k N k (I M k -I M k ′ )Λ k (2.2) For 'real' brane instantons, which were not considered in [3], the shift is given by 4 U → U + k N k I M k Λ k (2.3) (this new possibility will be an important point in our instanton scan in Section 6).\\n\\nThe complete interaction (2.1) is invariant under the U(1) gauge symmetries. However, from the viewpoint of the 4d low-energy effective field theory viewpoint, it leads to non-perturbative violations of the perturbative U(1) global symmetries, by the amount mentioned above.\\n\\nIn the string theory construction there is a simple microscopic explanation for the appearance of the insertions of the 4d charged fields (related to the mechanism in [27] ). The instanton brane action in general contains cubic terms α Φ γ, involving two instanton fermions zero modes α in the ( M , k ) and γ in the ( p , M ) coupling to the 4d spacetime field Φ in the ( k , p ) of the 4d gauge group 5 . Performing the Gaussian path integral over the instanton fermion zero modes leads to an insertion of Φ in the effective spacetime interaction. In general, and for a 'complex' instanton, there are several fermion zero modes α i , γ i in the fundamental (resp. antifundamental) of the instanton gauge group, coupling to a 4d spacetime chiral operators O ij (which could possibly be elementary charged fields, or composite chiral operators). Gaussian integration over the fermion zero modes leads to an insertion of the form det O (for 'real' brane instantons, det O should be interpreted as a Pfaffian). It is straightforward to derive our above statement on the net charge violation from this microscopic mechanism.\\n\\nNote that the above discussions show that instantons in different topological sectors (namely with different RR charges, and different intersection numbers with the 4d spacefilling branes) contribute to different 4d spacetime operators. In particular, multiwrapped instantons, if they exist as BPS objects, contribute to operators different 4 Equivalently, one may use (2.2), but must include an additional factor of 1/2 from the reduction of the volume for a real brane (which is invariant under the orientifold action). 5 Although there is no chirality in 0+0 dimension, the fermion zero modes α and γ are distinguished by their chirality with respect to the SO(4) global symmetry of the system (which corresponds to rotations in the 4d spacetime dimensions, longitudinal to the space-filling branes, and transverse to the instanton brane). Supersymmetry of the instantons constrains the couplings on the instanton action (such as the cubic couplings above) to have a holomorphic structure.\\n\\n9 from the singly wrapped instanton. This implies that the instanton expansion for a fixed operator is very convergent, and could even be finite.\\n\\nAnother important implication of the above discussion is that, in order to generate a specific operator via an instanton process, a necessary condition is that the instanton has an appropriate number and structure of charged zero modes. However, this is not sufficient. Insertions of 4d fields appear only if the fields couple to the instanton fermion zero modes via terms at most quadratic in the zero modes. In equivalent terms, only zero modes appearing in the Gaussian part of the instanton action can be saturated by insertions of 4d fields (those to which they couple). The requirement that the zero modes have appropriate couplings to the 4d fields may be non-trivial to verify in certain constructions. This is the case for the Gepner model orientifolds in coming sections, whose couplings are computable in principle, but unknown in practice. In such cases we will assume that any coupling which is not obviously forbidden by symmetries will be non-vanishing. Unfortunately there are no arguments to estimate the actual values of such non-vanishing couplings, hence we can argue about the existence of certain instanton induced operators, but not about the coefficients of such terms.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Zero mode structure for D-brane instantons\",\n \"text\": \"In this section we describe more concretely different kinds of instantons and the structure of interesting and unwanted zero modes. Our discussion will be valid for general D-brane models in perturbative type II orientifolds without closed string fluxes, although we also make some comments on more general F-theory vacua and the effects of fluxes. A more specific discussion is presented in Section 5.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Uncharged zero modes\",\n \"text\": \"We start discussing zero modes uncharged under the 4d gauge group. These are crucial in determining the kind of superspace interaction induced by the instanton on the 4d theory. In particular, we are interested in instantons contributing to the 4d superpotential, namely those which contain just two fermion zero modes in this sector. We are also interested in instantons which may contain additional fermion zero modes, and the possible mechanisms that can be used to lift them. Let us discuss 'real' and 'complex' brane instantons in turn.\\n\\n• Real brane instantons Real brane instantons correspond to branes which are mapped to themselves by the 10 orientifold action, hence M = M ′ . Uncharged zero modes arise from the MM open string sector. As discussed in Section 5, there is a universal sector of zero modes, in the sense that it is present in any BPS D-brane instanton, which we now describe. Before the orientifold projection, we have a gauge group U(n) on the volume of n coincident instantons. Notice that, although there are no gauge bosons in 0 + 0 dimensions, the gauge group is still well-defined, since it acts on charged states (open string ending on the instanton brane). There are four real bosonic zero modes and four fermion zero modes in the adjoint representation. For the minimal U(1) case, the four bosons are the translational Goldstones. The four fermions arise as follows. This sector is insensitive to the extra 4d spacefilling branes, and feels an accidental 4d N = 2 supersymmetry.\\n\\nThe BPS D-brane instanton breaks half of this, and leads to four Goldstinos, which are the described fermions 6 .\\n\\nThe orientifold projection acts on this universal sector as follows (see Section 5 for further discussion). The gauge group is projected down to orthogonal or symplectic.\\n\\nHence instanton branes with symplectic gauge group must have even multiplicity (a related argument, in terms of the orientifold action on Chan-Paton indices, is given in Section 5). For instantons with O(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes (related by the two supercharges of 4d N = 1 supersymmetry broken by the instanton) to the two-index symmetric representation, and the other two fermion zero modes (related by the other two supercharges of the accidental 4d N = 2 in this sector) to the antisymmetric representation. Hence for O(1) instantons (namely instantons with O(1) gauge group on their volume), we have just two fermion zero modes, which are the Goldstinos of 4d N = 1 supersymmetry, and the instanton can in principle contribute to the superpotential (if no additional zero modes arise from other non-universal sectors). For instantons with Sp(n) gauge symmetry, the orientifold projects the four bosonic zero modes and two fermion zero modes to the two-index antisymmetric representation, and the other two fermion zero modes to the symmetric representation. Hence, even for the minimal case of Sp(2)\\n\\ninstantons, we have just two fermion zero modes in the triplet representation, in addition to the two 4d N = 1 Goldstinos. Hence Sp(2) instantons cannot contribute to the superpotential in the absence of additional effects which lift these zero modes (see later) 7 .\\n\\n6 We thank F. Marchesano for discussions on this point. 7 For D-brane instantons corresponding to gauge instantons, the additional fermion zero modes in the universal sector couple to the boson and fermion zero modes from open strings stretched between the instanton and the 4d spacefilling brane. They act as Lagrange multipliers which impose 11 In addition to this universal sector, there exist in general additional modes, whose presence and number depends on the detailed structure of the branes. Namely, on the geometry of the brane in the 6d compact space in geometric compactifications, or on the boundary state of the internal CFT in more abstract setups. They lead to a number of boson and fermion zero modes in the symmetric or antisymmetric representation. The computation of these multiplicities in terms of the precise properties of the instanton branes is postponed to Section 5. In order to generate a superpotential, one must require these modes to be absent, except for the case of antisymmetrics of O(1) instantons, which are actually trivial.\\n\\nAn important point is that extra fermion zero modes (including the extra triplet fermion zero modes in the universal sector of Sp(2) instantons, and any two-index tensor fermion zero mode in the non-universal sectors) are in principle not protected against acquiring non-zero masses once the model is slightly modified. In other words, such fermions are non-chiral, in terms of the SO(4) chirality in footnote 5. One such modification is motion in the closed string moduli space, which can lift the nonuniversal modes if there are non-trivial couplings between them and closed string moduli (unfortunately, such correlators are difficult to compute, even in cases where the CFT is exactly solvable, like the Gepner models). Note that extra zero modes in the universal sector of Sp(2) instantons cannot be lifted by this effect, since it does not break the accidental 4d N = 2 in this sector. A second possible modification which in general can lift extra zero modes is the addition of fluxes, generalizing the results for D3-brane instantons in geometric compactifications [29] (for non-geometric CFT compactifications, we also expect a similar effect, once fluxes are introduced following [30] ). Note that fluxes can lift extra zero modes in the universal sector as well, since they can break the accidental 4d N = 2 susy in this sector. A last mechanism arising in more general F-theory compactifications and discussed below for complex instantons, is valid for real instanton branes as well.\\n\\nThe bottom line is that in the absence of such extra effects, only O(1) instantons can contribute to superpotential terms. However, in modifications of the model such extra effects can easily lift the extra fermion zero modes. Hence, this kind of extra vector-like zero modes will not be considered catastrophic, and real instantons (including the O(1) and Sp(2) cases) with such zero modes are considered in our scan in Section 6.\\n\\nthe fermionic constraints in the ADHM construction [28] , and may not spoil the generation of a superpotential.\\n\\n• Complex brane instantons Zero modes uncharged under the 4d gauge group can arise from the MM and MM ′ open string sectors. Notice that the orientifold action maps the MM sector to the M ′ M ′ , hence we simply discuss the former and impose no projection. The discussion of the MM sector is as for real brane instantons before the orientifold projection. The universal sector leads to a U(n) gauge symmetry, and four bosonic and four fermionic zero modes in the adjoint representation. The bosons are translational Goldstones, while the fermions are Goldstinos of the accidental 4d N = 2 present in this sector. Hence, even in the minimal case of U(1) brane instantons there are two extra fermion zero modes, beyond the two fermion zero modes corresponding to the 4d N = 1 Goldstinos. Hence U(1) instanton (except for those corresponding to gauge instantons, see footnote 7) cannot contribute to superpotential terms in the absence of additional effects, like closed string fluxes . However, keeping in mind the possibility of additional effects lifting them in modifications of the model, we include them in the discussion. Also, in what follows we will use the U(n) notation for the different fields to keep track of the Chan-Paton index structure.\\n\\nThe above statement would seem in contradiction 8 with computations of nonperturbative superpotentials [14] induced by M5-branes instantons in M-theory compactifications on Calabi-Yau fourfolds, which are dual to D3-brane instantons (with world-volume U(1) gauge group) on type IIB compactifications. The resolution is that the M5-branes that contribute are intersected by different (p, q) degenerations of the elliptic fiber. This implies that U(1) D3-brane instanton only contribute if they are intersected by mutually non-local (p, q) 7-branes. The two extra fermion zero modes exist locally on the D3-brane volume, but cannot be defined globally due to the 7-brane monodromies. Hence such effect can take place only on non-perturbative type IIB compactifications including (p, q) 7-branes. Note that in perturbative compactifications, namely IIB orientifolds, the (p, q) 7-branes are hidden inside orientifold planes [31] with their monodromy encoding the orientifold projection; hence the only branes that can contribute to the superpotential are real branes, for which the projection/monodromy removes the extra fermion zero modes, as discussed above.\\n\\nIn addition to this universal sector, the MM sector may contain a non-universal set of fermions and bosons, in the adjoint representation (hence uncharged under U(1)).\\n\\nThey depend on the specific properties of the brane instanton, and will be discussed in Section 5. These additional zero modes should be absent in order for the instanton 8 We thank S. Kachru for discussions on the ideas in this paragraph.\\n\\nto induce a non-trivial superpotential. Notice however that these zero modes are uncharged under any gauge symmetry, and hence vector-like. Thus, there could be lifted in modifications of the model, as discussed for real instantons.\\n\\nThe MM ′ sector is mapped to itself under the orientifold action. Hence it leads to a number of bosons and fermions in the two index symmetric or antisymmetric representations. Notice that the two-index antisymmetric representation is trivial for U(1), so these modes are actually not present. On the other hand, fermion zero modes in the two-index symmetric representation are chiral and charged under the brane instanton gauge symmetry. Hence they cannot be lifted by any of the familiar mechanisms, and thus spoil the appearance of a non-perturbative superpotential, even if the model is modified. Such fermion zero modes are considered catastrophic and we will look for models avoiding them in our scan in Section 6.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Charged fermion zero modes\",\n \"text\": \"• Real brane instantons Instanton zero modes charged under the 4d gauge group arise from Mk open string sectors (and their image Mk ′ ). In the generic case, there are no scalar zero modes in these sectors. This is because in mixed Mk open string sectors the 4d spacetime part leads to DN boundary conditions, which already saturate the vacuum energy in the NS sector. Only in the special case where the internal structure of the spacetime filling brane k and the instanton brane are the same, there may be NS ground states of the internal CFT which do not contribute extra vacuum energy, hence leading to massless scalars. However, this corresponds to brane instantons which can be interpreted as instantons of the 4d gauge theory on the 4d space-filling branes. The instantons we are interested in for the generation of neutrino Majorana mass terms are not of this kind [3] (see e.g. [32, 33, 28, 25, 34] for discussions on gauge theory instantons from D-brane instantons).\\n\\nHence we focus on charged fermion zero modes, which are generically present in any mixed Mk sector. Let us define L M k , L M k ′ the (positive by definition) number of left-handed chiral fermion zero modes in the representations\\n\\n( M , k ), ( M , k ),\\n\\nrespectively. The net number of chiral fermion zero modes in the ( M , k ) is given\\n\\nby I M k = L M k ′ -L M k .\\n\\nThis controls the violation of the U(1) a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φ kp and/or Φ kp ′ .\\n\\nIn addition, there are P M k = min(L M k ′ , L M k ) vector-like pairs of fermion zero 14 modes. Since they are vector-like, they may be lifted by slight modifications of the model, like moving in the closed string moduli space, or by introducing additional ingredients, like fluxes. In addition, they may be lifted by moving in the open string moduli space of the 4d spacefilling branes, as follows. The zero modes may lead to insertions of 4d fields Φ kk , if the kk sector contains such 4d chiral multiplets (or to insertions of composite 4d operators in the adjoint of the k th 4d gauge factor), and if they couple to the zero modes. Although this may not be generically not the case, many of our models in coming section contain such adjoint fields. Hence, a nontrivial vev for the latter can lift these extra vector-like zero modes, hence leading to instanton generating the superpotential of interest. Given these diverse mechanisms to lift these zero modes, their presence of such zero modes is thus unwanted, but again not necessarily catastrophic.\\n\\nOne last comment, related to the concrete kind of instanton search we will be interested in. Namely, we will be searching for instantons leading to a specific operator, carrying non-trivial charges under a specific set of 4d gauge factors. Postponing the detailed discussion to Sections 3.1, 4.2 , let us denote a, b, c, d the set of branes leading to a field theory sector, denoted 'observable' (and which reproduces the SM in our examples). We will require the instanton to have a prescribed number of chiral fermion zero modes charged under these branes, namely we require the intersection numbers of the instanton with these branes I M a , . . . , I M d to have specific values (as mentioned above, in the most restrictive scan we forbid vector-like pairs of zero modes under these branes). In addition, the model in general contains an additional sector of branes, denoted 'hidden' (since there is zero net number of chiral multiplets charged under both sectors) and labeled h i , required to fulfill the RR tadpole cancellation conditions. In general there may be instanton fermion zero modes from e.g. the Mh 1 , h 2 M sectors, which would contribute to insertions of the 4d fields in the h 1 h 2 sector if there are suitable cubic couplings. These extra insertions could be avoided if such 4d fields in the hidden sector acquire vevs (note that vevs for the (vector-like) fields charged under the visible and hidden sectors would typically break hypercharge, and should be avoided), and hence lift the zero modes. Equivalently, from the 4d perspective, the unwanted extra h 1 h 2 field insertion is replaced by its vev. However, this renders the discussion very model dependent. Moreover, the possibility of hidden brane recombination was not included in the search for SM-like models in [6, 7] (namely, the possibility of allowing for chiral fields charged under the observable and hidden gauge groups, which may become non-chiral and possibly massive upon hidden brane 15 recombination). Hence we will consider these chiral fermion zero modes as unwanted (as usual, non-chiral modes are unwanted but not catastrophic, hence they are allowed for in a more relaxed search).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"• Complex brane instantons\",\n \"text\": \"The discussion of 'complex' brane instantons is somewhat analogous to the previous one, with the only complication that the brane M and its image M ′ lead to independent modes, leading to a more involved pattern of fermion zero modes. Instanton zero modes charged under the 4d gauge group arise from the Mk,Mk ′ and related sectors. As for 'real' brane instantons, there are generically no scalars in these sectors (and certainly not in our case of interest). Hence we focus on charged fermion zero modes, which are generically present in any mixed sector.\\n\\nIn contrast with 'real' brane instantons, a net combination of fermion zero modes in the ( M , k ) + ( M , k ) is not vector-like, but chiral under the instanton gauge symmetry. Such a pair cannot therefore be lifted even by modifications of the theory.\\n\\nIn general, if the instanton has a mismatch in the total numbers n α , n γ of fermion zero modes α i in the M and γ j in the M , the instanton amplitude automatically vanishes.\\n\\nNamely, the matrix of operators O ij coupling to the zero modes necessarily has rank at most min(n α , n γ ). That is , if n α > n γ there are linear combinations of the α i which do not couple, and cannot lead to insertions. Moreover, they are not liftable by the familiar mechanisms 9 , thus in our instanton search in Section 6 such excess zero modes are forbidden even in relaxed scans.\\n\\nLet us thus discuss a sector of fermion zero modes with equal number n α = n γ .\\n\\nConsidering a given 4d space-filling brane k, let us denote\\n\\nL M k , L M ′ k ′ , L M k ′ , L M ′ k the\\n\\n(positive by definition) number of left-handed chiral fermion zero modes in the repre-\\n\\nsentations ( M , k ), ( M , k ), ( M , k ), and ( M , k ) respectively. The net number of chiral fermion zero modes in the ( M , k ) and ( M , k ) is given by I M k = L M k ′ -L M ′ k ′ and I M k ′ = L M k -L M ′ k , respectively\\n\\n. This net number of fermions zero modes controls the violation of the U(1) a global charge by the instanton. Namely, such fermion zero modes in the Mk, Mp, Mk ′ , Mp ′ sectors lead (if suitable couplings are present) to the insertion of 4d charged fields Φ kp and/or Φ kp ′ .\\n\\n9 Note that such a mismatch is always correlated with the existence of extra chiral zero modes in the M M ′ sectors, discussed above. Denoting Q a , Q orient the vector of RR charges of the 4d spacefilling branes and orientifold planes, they satisfy the RR tadpole conditions a\\n\\nN a Q a + a ′ N a Q a ′ + Q orient. = 0.\\n\\nBy taking the 'intersection' bilinear with the RR charges Q M of the brane instanton, we have I Ma +I Ma ′ +I M,orient = 0. This implies that the number of fundamentals minus anti-fundamentals of the instanton gauge group is related to the number of two-index tensors.\\n\\nThe remaining fields in this sector are vector-like pairs, in the ( M , k ) + ( M , k ) or the ( M , k ) + ( M , k ), which in principle lead to the vanishing of the instanton amplitude, but which can be lifted by additional effects (motion in closed or open string moduli space, or addition of fluxes), in a way consistent with the gauge symmetries in 4d spacetime and on the instanton. Just like for 'real' brane instantons, we conclude by commenting on our concrete instanton search in models with a set of visible branes a, b, c, d and a set of hidden branes h i . The requirement that the instanton leads to an operator with specific charges under the visible branes fixes the values of the quantities I M a -I M a ′ , etc. As we described for real branes, one may still have fermion zero modes charged under the hidden sector branes, but they lead to additional insertions, hence we rather focus on\\n\\ninstantons with I M h i -I M h ′ j = 0.\\n\\nThe two kinds of conditions, on intersection numbers with visible and hidden branes, still leave the possibility of combinations of fermion zero modes of the kind\\n\\n( M , k ) + ( M , k ), which do not contribute to I M k , or of the kind ( M , k ) + ( M , k ), which does not contribute to I M k ′ .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Such combinations are\",\n \"text\": \"automatically vector-like, and thus may be lifted in modifications of the theory. But the condition also allow combinations like\\n\\n( M , k ) + ( M , k ), which exploit a cancellation between I M k and I M k ′ (as also does ( M , k ) + ( M , k )\\n\\n). Such combinations are chiral by themselves, and in general imply a mismatch of modes in the M and the M . The total mismatch can be arranged to vanish using combinations of the kind\\n\\n( M , k ) + ( M , k ) and ( M , p ) + ( M , p )\\n\\nfor different branes. However, the only way to lift these pairs is by breaking the gauge symmetry on the 4d space-filling branes k and p. This can be done without damage to the visible sector if these are hidden branes, but this corresponds to the recombination of hidden branes that, as mentioned already, we are not going to consider. Hence only vector-like pairs with respect to each brane are considered to be liftable in simple modifications of the theory. In our instanton search, these are the only additional fermion zero modes which we allow in relaxed scans (but they are clearly not allowed for in restricted scans)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Instanton induced Majorana neutrino masses\",\n \"text\": \"In this Section we discuss the possible physical effects of D-brane instantons in string models with SM-like spectrum. In particular we describe the conditions to generate right-handed neutrino Majorana masses. We also comment on other possible B and/or L violating operators that can be generated by instantons. In this section we will again\\n\\n17 ν Sp(2),U(2) R L L L R E L Q U , D R R W gluon U(1) U(1) U( 3\\n\\n)\\n\\nd-Leptonic a-Baryonic b-Left c-Right R Figure 1: Quarks and leptons at intersecting branes use the geometrical language of IIA intersecting D-branes but it should be clear that our discussion equally applies to general CFT orientifolds like the ones presented in the next section.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The MSSM on the branes\",\n \"text\": \"Let us now specify the discussion in the previous section to the case of the generation of a right-handed neutrino mass term. In order to do that we need some realistic orientifold construction with the gauge group and fermion spectrum of the Standard Model (SM). In the context of Type II orientifolds perhaps the most economical brane configuration leading to a SM spectrum is the one first considered in [5] . This consists of four stacks, labelled a, b, c, d. The gauge factor on branes a is U(3), and contains QCD and baryon number. The d factor is U(1) d , and corresponds to lepton number.\\n\\nStack b contains SU(2) Weak either embedded in U(2) or Sp(2). Finally brane c can either provide a U(1) or an O(2) factor. In the brane intersection language, the chiral fermions of the SM live at the intersections of these branes, as depicted in Fig. 3.1.\\n\\nThe U(1) Y factor of the standard model is embedded in the Chan-Paton factors of branes a,c and d as\\n\\nY = 1 6 Q a - 1 2 Q c - 1 2 Q d = 1 2 (Q B-L -Q R ) (3.1)\\n\\nwhere Q x denotes the generator of the U(1) of brane stack x (in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator).\\n\\nNote that in this convention the Q d generator appears with sign opposite to other conventions in the literature, e.g. in [3] . In addition to Y these models have two 18 additional U(1) gauge symmetries:\\n\\nQ anom = 3Q a + Q d = 9Q B + Q L Y ′ = 1 3 Q a + Q c -Q d = Q B-L + Q R (3.2)\\n\\nThe first is anomalous whereas the second, which we will call B -L (with a slight abuse of language, since it is in fact a linear combination of B -L and hypercharge), is anomaly free. In models in which the electroweak gauge group is embedded in U(2), rather than in Sp(2), there is a second anomalous U(1) b . The charges of the SM particles under these U(1) symmetries are given in table 1.\\n\\nIntersection D = 4 fields/ zero modes\\n\\nQ a Q c Q d Y Q M (ab),(ab') Q L 3(3, 2) 1 0 0 1/6 0 (ca) U R 3( 3, 1) -1 1 0 -2/3 0 (c'a) D R 3( 3, 1) -1 -1 0 1/3 0 (db),(db') L 3(1, 2) 0 0 1 -1/2 0 (c'd) E R 3(1, 1) 0 -1 -1 1 0 (cd) ν R 3(1, 1) 0 1 -1 0 0 (Mc) α i 2(0, 0) 0 -1 0 1/2 1 (dM) γ i 2(0, 0) 0 0 1 -1/2 -1\\n\\nTable 1: Standard model spectrum and U (1) charges of particles and zero modes. Q M stands for the world-volume gauge symmetry in the case of U (1) complex instantons.\\n\\nThe U(1) k gauge symmetries have couplings with the RR 2-forms B r of the model, as follows\\n\\nS BF = k,r N k (p kr -p k ′ r ) 4d B r ∧ F k (3.3) where p kr , p k ′ ,r\\n\\nare given by the RR charges of the D-branes. These imply that under\\n\\na U(1) k gauge transformation A k → A k + dΛ k the scalar a r dual to B r transforms as a r → a r + k N k (p kr -p k ′ r ) Λ k (3.4)\\n\\nThis has two effects: 1) The linear combination of axion fields r (p kr -p k ′ ,r )a r is eaten up by the U(1) k massless gauge boson, making it massive. 2) For anomalous U(1) k , the anomalies cancel through a 4d version of the Green-Schwarz mechanism.\\n\\nThis works due to the existence of appropriate a r F ∧ F couplings, involving the gauge fields in the theory.\\n\\nIt is obvious that all anomalous U(1)'s become massive by this mechanism. However it is important to realize [5] that gauge bosons of anomaly-free symmetries like U(1) B-L may also become massive by combining with a linear combination of axions. This is 19 interesting since it provides a mechanism to reduce the gauge symmetry of the model without needing explicit extra Higgsing. In the models in which U(1) B-L becomes massive in this way, the gauge group left over is purely that of the SM. Moreover, we will see that having (B-L) massive by this Stückelberg mechanism is crucial to allow the generation of instanton-induced Majorana neutrino masses.\\n\\nNote that the B ∧ F couplings may also be potentially dangerous, since in principle they could also exist for hypercharge, removing U(1) Y from the low-energy spectrum.\\n\\nAs we will see in our RCFT examples later on, having massless U(1) Y but massive U(1) B-L turns out to be a strong constraint in model building.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Majorana mass term generation\",\n \"text\": \"As discussed in the previous section, string instantons can give rise to non-perturbative superpotentials breaking explicitly the perturbative global U(1) symmetries left-over from U(1) gauge bosons made massive through the Stückelberg mechanism. The kind of operator we are interested in has the form\\n\\nW ≃ e -S ins ν R ν R (3.5)\\n\\nwhere ν R is the right-handed neutrino superfield 10 . Here S ins transforms under both U(1) B-L and U(1) R in such a way that the overall operator is gauge invariant. This operator may be created if the mixed open string sectors lead to fermionic zero modes α i , γ i , i = 1, 2, appropriately charged under the 4d gauge factors. As we discussed in the previous section, to generate a superpotential one needs instanton with O(1) Chan-Paton symmetry, in order to lead to two uncharged fermion zero modes to saturate the d 2 θ 4d superspace integration. On the other hand, as we argued, instantons with Sp(2) or U(1) CP symmetries may also induce the required superpotentials if there is some additional dynamics getting rid of the extra uncharged zero modes which in principle appear in instantons with these symmetries. We thus consider all O(1), Sp(2) and U(1) instantons in our discussion.\\n\\nIn order to to get a ν R bilinear, the intersection numbers of instanton M and d, c branes are as follows Sp(2) case :\\n\\nI M c = 1 ; I M d = -1 (3.6)\\n\\n(since there is an extra multiplicity from the two branes required to produce Sp(2)) O(1) case :\\n\\nI M c = 2 ; I M d = -2 (3.7)\\n\\n10 Actually we denote by ν R the left-handed ν c L field following the usual (a bit confusing) convention.\\n\\n20 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 000000000000000 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 111111111111111 D2 D6 D6 ν R γ α Figure 2: Disk amplitude coupling two charged zero modes to ν R in the geometrical Type IIA intersecting brane approach.\\n\\nU(1) case :\\n\\nI M c = 2 ; I M d = -2 or I M d ′ = 2 ; I M c ′ = -2 (3.8)\\n\\nFurthermore there must be cubic couplings involving the right-handed neutrino superfield ν a in the a th family and the fermionic zero modes α i , γ j\\n\\nL cubic ∝ d ij a (α i ν a γ j ) , a = 1, 2, 3 (3.9)\\n\\nIn type IIA geometric compactifications, this coupling arises from open string disk instantons, see Fig. 3.1. In general type IIA models (resp. IIB models), the coefficients d ij a depend on the Kähler (resp. complex structure) moduli, and possibly on open string moduli. In simple CFT models (like e.g. in toroidal cases) these quantities may be in principle explicitly computed.\\n\\nThese trilinear couplings appear in the instanton action and after integration of the fermionic zero modes α i , γ i one gets a superpotential coupling proportional to\\n\\nd 2 θ d 2 α d 2 γ e -d ij a (α i ν a γ j ) = d 2 θ ν a ν b ( ǫ ij ǫ kl d ik a d jl b ) (3. 10\\n\\n) yielding a right-handed neutrino mass term. This term is multiplied by the exponential of the instanton euclidean action so that the final result for the right-handed neutrino mass (up to a 1-loop prefactor) has the form\\n\\nM R ab = M s ( ǫ ij ǫ kl d ik a d jl b ) exp( - V Π M g s + i r q M,r a r ) (3.11)\\n\\nFor geometric compactifications V Π M is roughly related to the wrapped volume. We keep the same notation to emphasize that the effect is non-perturbative in g s . In supersymmetric models the term in the exponential is the linear combination U of complex structure moduli to which the instanton D-brane couples, as described in the previous section. As explained, the gauge U(1) c , U(1) d transformation of the bilinear 21 piece and the e -S D2 factor nicely cancel. Note that from the viewpoint of the 4d SM effective field theory, the instanton has generated a Majorana neutrino mass violating B -L. Notice also that since this symmetry is non-anomalous, its violation cannot be associated to a gauge instanton, hence this is a pure string theory instanton effect.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Flavor and the special case of Sp(2) instantons\",\n \"text\": \"In order to extract more specific results for the flavor structure of the obtained Majorana mass operator, one needs to know more details about the quantities d ij a coming from the disk correlators. However in the particular case of Sp(2) instantons, the labels i, j are Sp(2) doublet indices, and the symmetry requires d ij a = d a ǫ ij . The mass matrix for the three neutrinos is given by M R ab =2M s d a d b exp (-U), so that the flavour dependence on a, b = 1, 2, 3 factorizes. More generally, as we will see in our RCFT search in Section 6, there are typically several different instantons contributing to the amplitude, so that we actually have a result for the mass\\n\\nM R ab = 2M s r d (r) a d (r) b e -Ur (3.12)\\n\\nwhere the sum goes over the different contributing instantons. One thus has a structure of the form\\n\\nM R ∼ r e -Ur diag (d (r) 1 , d (r) 2 , d (r) 3 ) • 1 1 1 1 1 1 1 1 1 • diag (d (r) 1 , d (r) 2 , d\\n\\n(r) 3 ) . (3.13) This structure is very interesting. Indeed, each instanton makes one particular (instantondependent) linear combination of the neutrinos massive, leaving two linear combinations massless. Hence, for three or more instantons, one generically has a matrix with three non-zero eigenvalues. It is easy to imagine a hierarchical structure among the three eigenvalues if e.g. the exponential suppression factors exp(-Re U r ) are different for each instanton.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Other B-and L-violating operators\",\n \"text\": \"Our main focus in this paper is on the generation of right-handed neutrino Majorana masses. However instantons may induce other L-and B-violating operators which we briefly summarize in this subsection. 22\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Weinberg operator\",\n \"text\": \"A right-handed neutrino Majorana mass term is not the only possible operator violating lepton number. Instanton effects may also give rise to dimension 5 operators not involving ν R . Specifically, the Weinberg operator L W = λ M (LHLH) . (3.14) might be generated. Once Higgs fields get a vev v this operator gives rise directly to Majorana masses for the left-handed neutrinos of order ≃ v 2 /M. Indeed, it is easy to check that in this case the required instanton M must verify Sp(2) case :\\n\\nI M c = -1 ; I M d = 1 (3.15) O(1) case : I M c = -2 ; I M d = 2 (3.16) U(1) case : I M c = -2 ; I M d = 2 or I M c ′ = 2 ; I M d ′ = -2 ( 3\\n\\n.17) (here we are assuming SU(2) weak to be embedded in an Sp(2)). Note that these intersection numbers are different to those giving rise to ν R mass terms. In particular they lead to a transformation under B -L opposite to that of ν R mass operators 11 .\\n\\nIn the present case there are altogether four fermionic zero modes α i ,γ i corresponding to the intersections of the instanton M with the branes c, d. These zero modes can have couplings involving the left-handed leptons L and the u-type Higgs multiplet H\\n\\nL disk ∝ c ij a (α i (L a H)γ j ) . (3.18)\\n\\nUpon integration over the fermionic zero modes one recovers the Weinberg operator.\\n\\nIn the present case the scale M of the Weinberg operator will be the string scale M s and the coupling λ ≃ exp(-S ins ). Again, in the particular case of Sp(2) instantons the situation simplifies (c ij a = c a ǫ ij ) and one gets left-handed neutrino Majorana masses\\n\\nM L ab = < H > 2 M s r 2c (r) a c (r) b e -Sr (3.19)\\n\\nwhere r runs over the different contributing instantons and S r is their corresponding action. The flavour structure of this left-handed neutrino mass matrix is the same as in eq.(3.13) and again may potentially lead to a hierarchical structure of left-handed neutrino masses, as is experimentally observed.\\n\\n11 Instantons with these intersection numbers will be denoted with a plus sign in the instanton search later on 23 In a given model both this kind of instanton and the one giving rise to righthanded neutrino masses (which is different) may be present. This contribution to the left-handed neutrino Majorana mass is in principle sub-leading compared to the see-saw contribution\\n\\nM L ab (see-saw) = < H > 2 2M s h T D ( r d (r) a d (r) b e -Sr ) -1 h D (3.20)\\n\\nwhere is the ordinary Yukawa coupling constant h ab D (ν a R HL b ). In principle the former is doubly suppressed both by 1/M s and the exponential factor. On the other hand if the exponential suppression is not too large this mechanism involving directly the Weinberg operator may be the most relevant source of neutrino masses. This is because the see-saw contribution coming from ν R exchange is proportional to the square of the ordinary Yukawa couplings h ab D which could be small. One could even think of having just the Weinberg operator as the unique source of the observed left-handed neutrino masses. Note however that in string vacua like this, in which the ν R 's are present and massless at the perturbative level, having just the Weinberg operator would not be phenomenologically correct, and instantons of the first class are still needed so that the ν R 's get a sufficiently large mass.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"R-parity violating operators\",\n \"text\": \"In the case of N = 1 SUSY models like the MSSM there might be operators of dimension 3 and 4 violating lepton and/or baryon number. These are the superpotential couplings\\n\\nW Rp = µ L a L a H + λ abc Q a D b L c + λ ′ abc U a D b D c + λ ′′ abc L a L b E c (3.21)\\n\\nin standard notation. Unlike the neutrino operators mentioned above, these operators violate B -L in one unit (rather than 2). It is well known that the standard R-parity of the MSSM may be identified with a Z 2 subgroup of U(1) B-L , so these terms are odd under R-parity. The simultaneous presence of all these couplings is phenomenologically unacceptable. Indeed, the third coupling violates baryon number, and the other three violate lepton number. Together they lead to proton decay at an unacceptably large rate. On the other hand couplings violating either B or L are phenomenologically allowed.\\n\\nIt is an interesting question whether any of these operators may be induced by string instanton effects. A first point to note is that instantons with Sp(2) Chan-Paton symmetry can never generate operators of this type. The reason is that all charged zero modes will necessarily come in Sp(2) doublets and hence the charged operators induced will always involve an even number of charged D = 4 fields and R-parity is 24 automatically preserved. On the other hand O(1) and U(1) instantons may generate R-parity violating operators. In particular, the LH bilinear is essentially the square root of the Weinberg operator, and may be induced if a U(1) or O(1) instanton M exists with\\n\\nI M c = -1 ; I M d = 1 or I M c ′ = 1 ; I M d ′ = -1 . (3.22)\\n\\n(in the O(1) case the second option is not independent from the first). Again, if the appropriate disk couplings are non-vanishing a term with µ a L ∼ M s exp(-S ins ) is generated. The rest of the operators in W Rp may also be generated. Possible instanton zero modes which may induce them are shown in table 2. For example, the QDL operator may be induced if a U(1) instanton M with intersection numbers\\n\\nI M b = -1 ; I M c ′ = 1 ; I M d = 1 (3.23)\\n\\nis present and in addition couplings\\n\\nL disk ∝ c ab (α(U a Q b j )γ j ) + c ′ a (βL a j γ j ) (3.24)\\n\\nexist. Here α, β, γ are zero modes corresponding to (Mc ′ ), (Md) and (bM) intersections and a, b(j) are flavor(SU(2) L ) indices. Analogous trilinear or quartic disk amplitudes involving two charged zero modes should exist to generate the rest of the R-parity violating amplitudes in table 2.\\n\\nD = 4 Operator I Ma I Ma ′ I Mb I Mc I Mc ′ I Md I Md ′ ν R ν R 0 0 0 2 0 -2 0 L HL H 0 0 0 -2 0 2 0 L H 0 0 0 -1 0 1 0 QDL 0 0 -1 0 1 1 0 U DD -1 0 0 1 2 0 0 LLE 0 0 -1 0 1 1 0 QQQL 1 0 -2 0 0 1 0 U U DE -1 0 0 2 2 -1 0\\n\\nTable 2: Zero modes required to generate Lepton/Baryon-number violating superpotential operators. Sp(2) instantons cannot give rise to R-parity violating operators whereas 1) instantons may in principle contribute to all of them. In the case of U (1) instantons there are additional zero mode possibilities which are obtained by exchanging\\n\\nO(1),U (\\n\\nI M x ↔ -I M x ′ . 25\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Dimension 5 proton decay operators\",\n \"text\": \"There are also superpotential dimension-5 operators violating B and L which may be constructed from the MSSM matter superfields. Indeed the dimension 5 operators ( 1 M ) QQQL ; ( 1 M ) UUDE (3.25) are in fact the leading source of proton decay in SUSY GUT models with R-parity.\\n\\nUnlike the other operators considered here these ones preserve B -L (hence R-parity) but not B + L. These operators do not contribute directly to a proton decay but need to be 'dressed' by a one loop exchange of some fermionic SUSY particle. This makes that, even although they are suppressed only by one power of the relevant fundamental scale, the loop factor and the corresponding couplings make the overall rate in SUSY-GUTS (barely) consistent with present experimental bounds for M of order the GUT scale or larger.\\n\\nThese dimension 5 operators may also be induced in D-brane models of the class here considered by the presence of instantons with appropriate intersection numbers.\\n\\nFor instance, the first operator may be induced through O(1) or U(1) instantons M with\\n\\nI M b = I M b ′ = -2 ; I M a = 1 ; I M d = 1 (3.26)\\n\\nAgain Sp(2) instantons cannot induced this operator, since the the Ma intersection would yield 6 (rather than 3) colored fermionic zero modes. The proton decay rate obtained from these operators depend on the ratio exp(-S ins ) × 1/M s . For M s of order 10 16 GeV, the rate is consistent with present bounds if exp(-S ins ) provides a suppression of a few orders of magnitude. On the other hand, models with a low string scale may be in danger unless the exponential suppression is sufficiently large (or such particular instantons are absent).\\n\\nAs a general conclusion, these phenomenological aspects of instanton induced operators very much depend on the action of the instanton, e.g. the volume of the wrapped D2-instanton in the intersecting D-brane constructions. In any event it is clear that the instantons here considered may indeed induce proton decay at a model-dependent rate. However in certain models R-parity will be preserved and prevent too rapid proton decay. Indeed, this is what we find in our instanton search in Gepner orientifolds.\\n\\nAs we said Sp(2) instantons automatically preserve R-parity. More generally, models that violate R-parity are rare, and the corresponding instantons actually generate very high dimensional operators, so R-parity breaking effects seems quite suppressed. In 26 fact in our search within MSSM-like models in Gepner model orientifolds we do not find instantons with just the correct charged zero modes to generate the low dimensional couplings discussed above. So, at least within our class of RCFT constructions, R-parity preservation is quite a common feature.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"CFT orientifolds\",\n \"text\": \"In this section we describe the 4d string models we consider, namely orientifolds of type IIB Gepner model compactifications. This is a very large class, on which one can carry out large scans for certain desired properties. And moreover at present the only known class of (SUSY) models with massive B -L.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Construction of the models\",\n \"text\": \"In general, RCFT orientifolds are orientifold projections of closed string theories constructed using rational conformal field theory. Although this includes in principle rational tori and orbifolds, the real interest lies in cases where the two-dimensional CFT is interacting, because such theories are hard to access by other methods. A disadvantage of the use of RCFT is that this method is algebraic, and not geometric in nature, so that one cannot easily explore small deformations of a certain string theory.\\n\\nIt is best thought of as a rational scan of moduli spaces.\\n\\nThe most easily accessible examples are the orientifolds of tensor products of minimal N = 2 conformal field theories (\\\"Gepner models\\\") forming a type IIB closed string theory. During the last decade, examples in this class have been studied by many authors (see [35] 42] ), and searched systematically in [6] and [7] .\\n\\n[36][37][38][39][40][41][\\n\\nAlthough the Gepner models form only a small subset of RCFT's, they already offer a large number of possibilities. The total number of tensor products with the required central charge c = 9 is 168. On top of this, one can choose a large number of distinct modular invariant partition functions on the torus. The orientifold formalism is not available for all of them, but it has been completely worked out [43] for all simple current invariants (based on the charge conjugation invariant). This yields a total of 5403 distinct MIPFs. On top of this, we may choose various orientifold projections. Here the only known possibilities are a class of simple-current based choices [44][45] [46] [47] .\\n\\nThis then yields a total of 49304 orientifolds.\\n\\nFor each orientifold choice, the full open string partition function is 1 2 a,b,i\\n\\nN a N b A i ab χ i ( τ 2 ) + a,i N a M i a χi ( τ 2 + 1 2 ) (4.1)\\n\\nHere A i ab are the annulus coefficients, M i a the Moebius coefficients, N a the Chan-Paton multiplicities and χ(τ ) are the closed string characters, and χi (τ ) = T -1/2 χ i (τ ). The set of integers i is simply the set of primary fields of the closed string CFT, and depends only on the tensor product. The integers a, b are the boundary labels; this set depends on the MIPF. Our notation and labelling conventions for these CFT quantities are explained in Appendix A. The integers A i ab and M i a depend in addition also on the orientifold choice; in the case of A i ab the latter dependence is very simple: all distinct annuli can be written as\\n\\nA Ω,i ab = c A i c a C Ω cb ,\\n\\nwhere Ω is the orientifold choice (which we usually do not specify explicitly) and C Ω cb is the boundary conjugation matrix, which acts as an involution on the set of boundaries. Suppressing some details (which can be found in [43] ) we may write these integers as\\n\\nA Ω,i ab = m,J,K S i m R a,(m,J) g Ω,m JK R b,(m,K) S 0m (4.2) M Ω,i a = m,J,K P i m R a,(m,J) g Ω,m JK U Ω (m,K) S 0m (4.3)\\n\\nHere m is the label of an Ishibashi-state (the set of states that propagates in the transverse (or closed string) channel of the the annulus or Moebius diagrams). It is a subset of the set of closed string labels i, but in general there are degeneracies, so that more than one distinct Ishibashi state belongs to a given closed string label.\\n\\nThese degeneracies are distinguished by the labels J, K (see Appendix A). The complex numbers R and U are respectively the boundary and crosscap coefficients. Note that the latter depend on the orientifold choice, but the former do not. The only dependence of the annulus coefficients on the orientifold choice is through the Ishibashi metric g Ω JK , which is a matrix on each Ishibashi degeneracy space, and which can be a sign if there are no degeneracies. Finally, the matrix P is given by P = √ T ST 2 S √ T , where S and T are the generators of the modular group of the torus. Similar expressions exist for the Klein bottle multiplicities defining the unoriented closed sector, but they will not be needed in this paper.\\n\\nThe boundary labels a, b, . . . refer to all boundaries that respect the bulk symmetries of the CFT. This includes the individual N = 2 chiral algebras of the factors in the tensor product, the alignment currents 12 that ensure the proper definition of world-sheet supersymmetry and the space-time supersymmetry generator that imposes a generalized GSO-projection on the spectrum. The latter implies that all characters χ i respect (at least) N = 1 space-time supersymmetry. By construction, the boundary states are then supersymmetric as well. Both conditions (boundary and bulk space-time supersymmetry) can in principle be relaxed within the formalism, but this leads to a much larger set of bulk and boundary states. The precise labelling of the boundaries is explained in Appendix A and involves a subset of the closed string labels i and a degeneracy label, distinct from the one used for the Ishibashi states. The set of boundary labels is complete in the sense of [45] . This means that no additional boundary states exist that respect all the aforementioned symmetries. It also means that the matrices R are square matrices (although their rows and columns are defined in terms of different index sets). It is in principle possible to write down additional boundary states that break some of the world-sheet symmetries. This is an important possibility to keep in mind, but we will not consider it here.\\n\\nThe massless spectrum is obtained by restricting the characters χ i to massless states. Since the characters are supersymmetric those massless states are either vector multiplets or chiral multiplets. The latter can be restricted to one chirality (e.g lefthanded); the other choice merely produces the CPT conjugates. Boundaries are called real if a = a ′ , where the conjugate boundary a ′ is defined by C Ω a,a ′ = 1, and complex otherwise. The Chan-Paton multiplicities N a give rise to gauge groups U(N a ) for complex boundaries and SO(N a ) or Sp(N a ) for real ones. In the latter case N a must be even. To count bi-fundamentals we define\\n\\nL ab ≡ i A i ab χ i ( τ 2 ) massless,L . (4.4)\\n\\nNote that because of the factor 1 2 in (4.1) and the fact that L ab is symmetric, the value of L ab is indeed precisely the number of bi-fundamentals in the representation (N a , N b ).\\n\\nThese are spin-3 currents consisting of products of the world-sheet supercurrents of the factors in the tensor product, including the NSR space-time factor.\\n\\nIt is convenient to introduce the intersection matrix 13\\n\\nI ab ≡ L ab ′ -L a ′ b , (4.5)\\n\\nwhich is manifestly antisymmetric in a and b. Note that for a pair of complex boundaries a, b with conjugates a ′ , b ′ one can define four quantities that are relevant for the massless spectrum, two of which are chiral, namely I ab and I ab ′ .\\n\\nIt is often convenient to associate a geometric picture to these integers. Thus we will often refer to the boundary labels and their multiplicities as \\\"stacks of branes\\\", and view the integers I ab as brane intersection numbers. This is only done for convenience and does not imply a concrete brane realization; indeed, it does not make sense to say that a given boundary label corresponds to a Dp-brane for some give p. Such an interpretation might be valid in a large radius limit, assuming such a limit exists.\\n\\nIn general, for a choice of Chan-Paton multiplicities N a there will be tadpoles in the one-point closed string amplitudes on the disk and the crosscap. These have to be cancelled in order to make the theory consistent (since we work with supersymmetric strings we do not have the option of cancelling RR and NS-NS tadpoles separately).\\n\\nThis leads to a condition on the Chan-Paton multiplicities:\\n\\na N a R a,(m,J) = 4η m U m,J (4.6)\\n\\nwhere η 0 = 1 and all other η's are -1; there is such a condition for any Ishibashi label (m, J) that leads to a massless scalar in the transverse channel. The one for m = 0 (which is non-degenerate) is the dilaton tadpole condition. It has the special feature that all coefficients R a0 are real and positive. The crosscap coefficient U 0 is also real and can be chosen positive (in the CFT both signs are acceptable). If U 0 = 0 (4.6) limits the Chan-Paton multiplicities; if U 0 = 0 the only solution is N a = 0 for all a, which rules out any realization of the Standard Model. This reduces the number of usable orientifolds to 33012.\\n\\nTadpole cancellation condition implies cancellation of RR-charges coupling to longrange fields, and absence of local anomalies. There is a second condition that has 13 Note that L ab is a symmetric matrix giving the number of chiral multiplets in the ( a , b ) bifundamental. This is a natural quantity in unoriented CFT's, where a symmetric definition for the annulus amplitude exists. In oriented CFT the annulus is, in general, not symmetric, but on the other hand it is possible to choose the branes in such a way that only ( , ) bi-fundamentals appear.\\n\\nThis has become the customary way of counting states in the intersecting brane literature, even for orientifold models. The quantity I ab is defined in such a way that it is anti-symmetric in a and b. This is why boundary conjugations appear in the right hand side. This has the additional advantage of making I a more familiar quantity for readers used to the standard intersection brane conventions.\\n\\nto be taken into account, which has to do with Z 2 charges that do not couple to long-range fields, usually referred to as \\\"K-theory charges\\\" in geometric constructions.\\n\\nUncancelled K-theory charges may lead to global anomalies in symplectic factors of the gauge group. But even if this symptom is absent, the disease may still exist. A much more general way to probe for uncancelled K-theory charges is to require the absence of global anomalies not only in the Chan-Paton gauge group but also on all symplectic brane-anti-brane pairs that can be added to it as \\\"probe-branes\\\" [48] . Presently this is the most general constraint that be imposed in these models, but it is not known if additional ones are required. This probe brane constraint leads to a large number of mod-2 constraint and is potentially very restrictive, but almost harmless in practice [49] . It is satisfied by all models we consider in the present paper.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Search for SM-like models\",\n \"text\": \"The complete set of solutions to these conditions is finite but huge, but the vast majority is of no phenomenological interest. In the last few years systematic searches have been carried out for models that contain the Standard Model. The models that were considered have the property that the set of Chan-Paton labels can be split into two subsets, the observable and the hidden sector. The former has been limited, for practical reasons, to at most four complex brane stacks, required to contain the Standard Model gauge group and the right intersections to yield three families of quarks and leptons, plus (in general) some non-chiral (vector-like) additional matter. The hidden sector is only constrained by the requirement that there be no net number of chiral multiplets charged under both the observable and hidden sector, and by practical computational limitations. The main purpose of the hidden sector in these models is to provide variables that can be used to satisfy the tadpole and global anomaly conditions, since the multiplicities in the observable sector are already fixed. In some cases the observable sector already satisfies the constraints by itself, and there is no hidden sector.\\n\\nThe observable sector can be realized in many different ways if one only imposes the constraint that the standard model should be contained in it. These possibilities were recently explored in [7] . We will focus on the realization described in Section 3.1, first considered in [5] . There are four stacks, namely a (containing QCD and baryon number as U(3)), b (containing electroweak SU(2) embedded as U(2) or Sp(2)), c (providing a U(1) or an O(2) factor 14 , and d (providing another U(1) factor). 14 In [6] also Sp(2) was considered, but this requires an additional Higgs mechanism.\\n\\nThe standard model hypercharge generator is , defined in (3.1):\\n\\nY = 1 6 Q a - 1 2 Q c - 1 2 Q d (4.7)\\n\\nwhere Q x denotes the generator of the U(1) of brane stack x; in case the Chan-Paton factor of brane c is O(2) one should use the properly normalized O(2) generator. In addition to Y these models have two or three additional U(1) gauge symmetries (the latter case if electroweak SU(2) arises from U(2)). These (except the combination B -L) are anomalous, with anomaly cancelled by the Green-Schwarz mechanism, implying the existence of a B ∧ F coupling making them massive. In fact, as already mentioned, such Stückelberg couplings may be present for non-anomalous U(1)'s as well. We are interested in models where the hypercharge gauge boson does not have such couplings (otherwise the model would be phenomenologically unacceptable), but\\n\\nwhere the B -L gauge boson is massive by such couplings (both in order that the gauge group reduces to the SM one, and that neutrino Majorana masses may be induced by string instantons, as discussed in previous sections).\\n\\nThe combined requirements of having a massive B -L and a massless Y turn out to be difficult to satisfy. In fact, if the group on brane c is O(2) they are impossible to satisfy simultaneously, because the O(2) component of the vector boson does not couple to any axions, and hence the B -L and Y bosons have the same mass. But even in models with a U(1) group on brane c it happens rather rarely that both constraints are satisfied simultaneously, at least in the searches that have been done so far.\\n\\nWe will make use here of the data presented in [6, 7] , which are available in slightly improved form on the website www.nikhef.nl/ ∼t58/filtersols.php. This database consist of 211634 distinct spectra. Here \\\"distinct\\\" means that they are physically different for a given MIPF 15 if the hidden sector is ignored. Hence the differences can be the number of vector-like states of various kinds or the dilaton couplings of branes a, b, c, d. Geometrically, these spectra may originate from the same moduli space, but then in any case from different points on this moduli space. The improvements in comparison with the data presented in [6] consist of taking into account the full global anomaly conditions from probe branes. In some cases this required nothing more than checking these conditions for an existing solution of the tadpole conditions, but in other cases a new solution had to be found. As a result, a few models disappeared from the original database, but due to improved algorithms a few new ones could be added. The net re-15 Rare cases of identical spectra and couplings originating from different MIPFs are treated as distinct.\\n\\n32 sult is some small but inconsequential changes in the total number of models of various kinds. The numbers we will mention below are based on the improved database.\\n\\nThe total number of models in that database with a Chan-Paton group U(3) × Sp(2) × U(1) × U(1) is 10587. Of these, 391 (about 4%) have a massive B -L vector boson. For U(3) × U(2) × U(1) × U(1) these numbers are, respectively, 51 and 0. Hence no examples of the latter type were found, although they were found with 1,2 and 4 families (in a limited search), in a few percent of the total number of models. It seems therefore reasonable to expect that U(3) × U(2) × U(1) × U(1) with massive B -L do exist, and that their absence is just a matter of statistics. Just for comparison, the total number of U(3) × Sp(2) × O(2) × U(1) models is 56627.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Fermion zero modes for instantons on RCFT's\",\n \"text\": \"In this section we discuss D-brane instantons for general compactifications, including abstract CFT ones. We also provide the spectrum of zero modes on an instanton brane, using the information about their internal structure i.e. in the compactified dimension in geometric models, or of the internal CFT in more abstract setups like in previous section. We will be interested in the latter case.\\n\\nA first question that should be addressed is what this internal structure is. For instance, in type IIA geometric compactifications, it corresponds to a supersymmetric (i.e. special lagrangian) 3-cycle. Notice that these are the same kind of 3-cycles already used to wrap the D6-branes that give rise to the 4d gauge symmetry of such models.\\n\\nFor general CFT's, D-branes are described as boundary states. To describe instantons, one can simply use the same boundary state of the internal CFT to describe the 4d space-filling branes present in the model and the instanton branes. The only difference is that boundaries satisfy Neumann conditions in the 4d space-filling case, and Dirichlet in the instanton case. This exploits the fact that whenever a boundary state of the internal CFT, and with Neumann boundary conditions in the 4d space is an acceptable state of the full CFT, the same boundary state of the internal CFT, combined with Dirichlet boundary conditions in the 4d space also gives an acceptable state of the full CFT. For geometric compactifications this is related to Bott periodicity of the K-theory classes associated to the D-brane charges, but it is possible to show it in general.\\n\\nSince instanton D-branes can thus be naturally associated to the boundary states of 4d space-filling branes, it is convenient to express the spectrum of zero modes of the former in terms of the massless states of the latter. This is particularly useful, since 33 the computation of the spectra on 4d space-filling branes for Gepner model orientifolds has already been described (although the arguments below are valid also for geometric compactifications). Hence, let us denote by M a 4d space-filling brane associated with the same boundary state of the internal CFT as the instanton brane M of interest.\\n\\nNote that the 4d space-filling brane M is an auxiliary tool, and need not be (and, for our instantons of interest, will not be) one of the 4d space-filling branes present in the model.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"'Real' brane instantons\",\n \"text\": \"Let us first consider the case of 'real' brane instantons. Consider a set of m 4d spacefilling branes M, and focus first on the massless spectrum in the MM sector. Before the orientifold projection, it leads to a universal 4d N = 1 U(m) vector multiplet, and a number L MM of adjoint chiral multiplets. The orientifold operation maps this sector to itself, acting on the Chan-Paton with a matrix γ Ω,M . This matrix satisfies\\n\\nγ T Ω,M γ -1 Ω,M = ±1 m (5.1)\\n\\nThe two possibilities can be chosen to correspond to\\n\\nγ Ω,M = 1 m or γ Ω,M = ǫ m , with ǫ m = 0 1 r -1 r 0\\n\\n, and m = 2r hence necessarily even in the latter case. They correspond to the SO and Sp projections, respectively.\\n\\nThe orientifold projection on the N = 1 vector multiplet Chan-Paton matrices is given by\\n\\nλ = -γ Ω,M λ T γ -1 Ω,M (5.2)\\n\\nand leads to SO(m) or Sp(m) vector multiplets for the SO or Sp projection (hence the name). Concerning the N = 1 chiral multiplets, they fall in two classes of p -, p +\\n\\n(with p -+ p + = L MM ) which suffer the projections λ = ±γ Ω,M λ T γ -1 Ω,M (5.3)\\n\\nFor the SO projection, this leads to p + , p -chiral multiplets in the , representation.\\n\\nFor the Sp projection, there are p + , p -chiral multiplets in the , representation.\\n\\nThe sectors Ma (where a is a 4d space-filling branes present in the model) are mapped to sectors Ma ′ , so it is enough to focus on the former. After the orientifold projection one gets L Ma , L Ma ′ chiral multiplets in the ( M , a ), ( M , a ).\\n\\nLet us now obtain the zero modes for a set of m instanton branes M in terms of the above spectrum. The MM sector is closely related to the MM sector, by changing the NN boundary conditions in 4d spacetime to DD boundary conditions 34 (which can be done in a covariant formalism, but not in the light-cone gauge). Before the orientifold projection, one obtains the same set of states (since moddings for NN and DD boundary conditions are identical, both in the NS and R sector), but with different world-volume interpretation. Also, the change in boundary conditions implies that some polarization states which are unphysical for the 4d spacefilling brane are physical in the instanton brane. Hence, the U(m) gauge bosons on the 4d space-filling brane M correspond to four adjoint real scalars in the instanton brane M. Similarly, the 4d spinors in M, correspond to four fermion zero modes on M, transforming as two spinors of opposite chiralities θ α , θ α of the SO(4) rotation group in transverse space. The orientifold projection maps the MM sector to itself, acting on Chan-Paton indices with a matrix γ Ω,M . In close analogy with the argument in [50] for the familiar D5-D9-brane system in type I (see [51, 52] for related derivations), one can show that the condition (5.1) flips sign upon changing four NN boundary conditions to DD, hence\\n\\nγ T Ω,M γ -1 Ω,M = ∓1 m (5.4)\\n\\nNamely, the instanton brane has Sp(m) gauge group when the 4d space-filling brane (with same internal boundary state) has gauge group O(m), and vice-versa. We still refer to these projections as SO and Sp, hoping no confusion arises. Note that, as mentioned in Section 2.2, although there are no gauge bosons in 0 + 0 dimensions, the gauge group is present on the instantons in that it acts on open string endpoints.\\n\\nLet us consider the effect of the orientifold projection on the MM states, as compared with the effect on MM states. Again, following arguments familiar in the D5-D9 brane system in type I, one can show that the signs in conditions like (5.2), (5.3) remain unchanged upon changing four NN dimensions to DD, except for bosonic modes polarized along the directions longitudinal to these four dimensions (and for fermions related to them by the unbroken susy of the total system). To be concrete, considering the four MM adjoint bosons, and two MM adjoint fermions θ α associated to the universal MM vector multiplets, they suffer the projection\\n\\nλ = +γ Ω,M λ T γ -1 M (5.5)\\n\\nHence they transform in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection. On the other hand, for the two fermion zero modes θ α, the projection is\\n\\nλ = -γ Ω,M λ T γ -1 M (5.6) 35\\n\\nand leads to two fermion zero modes in the of Sp(m) for the SO projection, and in the of SO(m) for the Sp projection.\\n\\nThis implies that in order to obtain two fermion zero modes from this universal multiplet, in order to generate a superpotential, one should consider instantons with orthogonal gauge group and multiplicity one (O(1) instantons). For instantons with symplectic gauge group and multiplicity two (Sp(2) instantons), there are two additional fermion zero modes in the triplet representation. As mentioned, we will continue to consider such instantons in our relaxed scan. Multiple instantons, i.e. boundary states with higher multiplicity, lead to a larger amount of additional fermion zero modes (due to the larger gauge representations for the fermions), and do not contribute to superpotentials; we will not consider such cases even in relaxed scans, since they also very often lead to too many charged fermion zero modes and cannot contribute to the operators of interest (except possibly for O(2) and U(2) instantons with low intersections, which are kept in our scan as a curiosity).\\n\\nSimilarly, for the p ± sets of MM scalars and fermions associated to the MM 4d chiral multiplets, the projection is\\n\\nλ = ±γ Ω,M λ T γ -1 Ω,M (5.7)\\n\\nwith the same sign choice as in (5.3). The different structure of γ Ω implies that, for the SO projection we get p + , p -sets of scalars and fermions in the , , while for the Sp projection there are p + , p -sets of scalars and fermions in the , .\\n\\nThis concludes the discussion of the MM sector. Let us not consider the Ma sectors, from the information from the Ma sectors. Notice that this implies changing four NN boundary conditions to DN, which have different moddings. Hence the states are different in both situations, but the information on the multiplicities is preserved.\\n\\nSpecifically, in the NS sector the DN boundary condition introduce an additional vacuum energy which generically makes all states massive. Hence there are no massless scalar zero modes in generic Ma sectors. In the R sector, the change in the moddings reduces the dimension of the massless ground state, leading to a single (chiral) fermionic degree of freedom. Since the orientifold action maps the Ma sector to Ma ′ sectors, there are no subtleties in the orientifold projection. The end result is L Ma , L Ma ′ fermion zero modes in the ( M , a ), ( M , a ). The net number of chiral fermion zero modes in the ( M , a ) is given by I Ma = L Ma ′ -L Ma , i.e. the net number of chiral multiplets in the related Ma sector.\\n\\nThe results for orientifold projections for real branes are shown in table 3. 36 Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) SO N = 1 vect. mult. U(m) O(m) Sp(m)\\n\\n2 f + 2 f + 4 b N = 1 ch. mult. (p + + p -) Ad p + + p - 2p + ( f + b ) + 2p -( f + b ) Sp N = 1 vect. mult. U(m) Sp(m) O(m) 2 f + 2 f + 4 b N = 1 ch. mult. (p + + p -) Ad p + + p - 2p + ( f + b ) + 2p -( f + b ) Any N = 1 ch. mult. L Ma ′ ( M , a )+ L Ma ′ ( M , a )+ L M a ′ ( M , a ) f L Ma ( M , a ) L Ma ( M , a ) L M a ( M , a ) f net I Ma ( M , a ) net I M a ( M , a ) f\\n\\nTable 3: Orientifold projection for real branes: Massless modes of the 4d space-filling branes M (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Complex brane instantons\",\n \"text\": \"We now consider the case of complex brane instantons. The arguments are very similar, hence the discussion is more sketchy. Consider m 4d spacefilling branes M, associated to the internal boundary state of the instanton brane M of interest. The MM leads to a 4d N = 1 U(m) vector multiplet and a number L MM ′ of adjoint chiral multiplets. The orientifold action maps it to the M ′ M ′ sector, hence we may keep just the former and impose no projection. The MM ′ sector is mapped to itself under the orientifold projection. Denoting by γ Ω,M the action on Chan-Paton indices, the MM ′\\n\\nmodes split into sets L ± M M , L ± M ′ M ′ , which suffer a projection λ = ±γ Ω,M λ T γ -1 Ω,M (5.8) leading, for γ Ω,M = 1 m , to L + MM , L - MM chiral multiplets in the , , and L + M ′ M ′ , L - M ′ M ′ chiral\\n\\nmultiplets in the , . The net number of chiral multiplets in the , is\\n\\nI + MM ′ = L + M,M -L + M ′ M ′ , I - MM ′ = L - M,M -L - M ′ M ′ .\\n\\nAnd oppositely for γ Ω,M = ǫ m . Finally, the Ma, Ma ′ and related sectors lead, after the orientifold projection, to\\n\\nL Ma ′ , L Ma , L M ′ a ′ , L M ′ a chiral multiplets in the ( M , a ), ( M , a ), ( M , a ), ( M , a ).\\n\\nIn order to simplify notation, we replace M → M in these expressions in our discussions of instanton zero modes.\\n\\nLet us now consider m brane instantons M and compute their zero mode spectrum in terms of the above. In the MM (and its image M ′ M ′ ) sector there are four scalar modes and four fermions in the adjoint of the U(m) gauge symmetry group; these are related to the 4d vector multiplet in the MM sector. In addition, there are L M M ′ sets of scalars and fermions in the adjoint, related to the L MM ′ non-universal chiral multiplets in the MM sector. The MM ′ sector is mapped to itself, and one has to impose the orientifold projection (recalling that the matrix γ Ω,M differs from γ Ω,M ).\\n\\nFor γ Ω,M = 1, hence γ Ω,M = ǫ, we obtain L + M M , L - M M chiral multiplets in the , , and L + M ′ M ′ , L - M ′ M ′ chiral multiplets in the , . The net number of chiral multiplets in the ,\\n\\nis I + M M ′ = L + M M -L + M ′ M ′ , I - M M ′ = L - M M -L - M ′ M ′ .\\n\\nAnd oppositely for γ Ω,M = ǫ hence γ Ω,M = 1.\\n\\nIn the Ma, Ma ′ and related sectors, there are generically no bosonic zero modes, and there are and ( M , a ) respectively. The net number of chiral fermion zero modes in the ( M , a ) and\\n\\nL Ma , L M ′ a ′ , L Ma ′ , L M ′ a chiral fermion zero modes in the ( M , a ), ( M , a ), ( M , a ),\\n\\n( M , a ) is given by I Ma = L Ma ′ -L M ′ a ′ and I Ma ′ = L Ma -L M ′ a . In\\n\\norder to simplify notation, we replace M → M in these expressions in our discussions of instanton zero modes.\\n\\nThe results for orientifold projections for real branes are shown in table 4.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Search for M instantons\",\n \"text\": \"In this section we perform a search of models which admit an instanton inducing a right-handed neutrino Majorana mass operator. Namely, for each model with the chiral content of the SM in the classification described in Section 4.2, we first scan over boundary states, searching for all instantons with the required uncharged and charged fermion zero mode structure to yield neutrino masses. We then relax our criteria a bit and allow for instantons with correct charged zero mode structure but having extra non-chiral zero modes (both charged and uncharged). The idea is that these non-chiral zero modes could be lifted by diverse effects, as discussed.\\n\\nIt is important to recall that the cubic couplings between instanton zero modes and 4d chiral multiplets are difficult to compute in Gepner model orientifolds. Hence, we will simply assume that such couplings are non-zero if there is no symmetry forbidding them. 38 Proj. Multiplet in M M (before orient.) M (after orient.) M (after orient.) Any N = 1 vect. mult. U(m) × U(m) ′ U(m) U(m) 4 Ad f + 4 Ad b N = 1 ch. mult. p adj Ad + p adj Ad ′ p adj Ad 2p adj ( Ad f + Ad b ) SO N = 1 ch.mult.\\n\\nL MM ( M , M ′ ) L + MM M + L - MM M 2L + M M b,f + 2L - M M b,f L M ′ M ′ ( M , M ′ ) L + M ′ M ′ M + L - M ′ M ′ M 2L + M ′ M ′ b,f + 2L - M ′ M ′ b,f Sp N = 1 ch.mult. L MM ( M , M ′ ) L + MM M + L - MM M 2L + M M b,f + 2L - M M b,f L M ′ M ′ ( M , M ′ ) L + M ′ M ′ M + L - M ′ M ′ M L + M ′ M ′ b,f + L - M ′ M ′ b,f Any N = 1 ch. mult. L Ma ′ ( M , a )+ L Ma ′ ( M , a ) L M a ′ ( M , a ) f . . . L Ma ( M , a ) L M a ( M , a ) f . . . L M ′ a ′ ( M , a ) L M ′ a ′ ( M , a ) f . . . L M ′ a ( M , a ) L M ′ a ( M , a ) f net I Ma ( M , a ) net I M a ( M , a ) f net I Ma ′ ( M , a ) net I M a ′ ( M , a ) f\\n\\nTable 4: Orientifold projection for complex branes: Massless modes of the 4d space-filling branes M (before and after the orientifold projection) and zero modes on the instanton branes M (denoted with sub-indices b, f for bosonic and fermionic modes)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The instanton scan\",\n \"text\": \"Our detailed strategy will become clear along the description of the results. Given a set of a,b,c,d standard model branes, we must look for additional boundary states M that satisfy the requirements of a (B -L)-violating instanton. From the internal CFT point of view this is just another boundary state, differing from 4d spacefilling branes only in the fully localized 4d spacetime structure. The minimal requirement for such a boundary state is B -L violation, which means explicitly\\n\\nI M a -I M a ′ -I M d + I M d ′ = 0 (6.1)\\n\\nIt is easy to see that the existence of such an instanton implies (and hence requires) the existence of a Stückelberg coupling making B -L massive. To see this, consider adding to the Standard Model configuration a 4d spacefilling brane M (in fact used in Section 5) associated to the boundary state M (RR tadpoles can be avoided by simultaneously including M antibranes, which will not change the argument). The new sector in the chiral spectrum charged under the branes M can be obtained by reversing the argument in Section 5, and is controlled by the intersection numbers of M. From 39 the above condition it follows that the complete system has mixed U(1) B-L × (G M ) 2 anomalies, where G M is the Chan-Paton-factor of brane M. These anomalies are cancelled by a Green-Schwarz mechanism involving a (B -L)-axion bilinear coupling, which ends up giving a mass to B -L via the Stückelberg mechanism. This coupling is in fact not sensitive to the presence of the brane M, hence it must have been present already in the initial model (without M).\\n\\nHence the existence of a boundary label M that satisfies (6.1) implies that B -L is massive. Unfortunately the converse is not true: even if B -L has a Stückelberg mass, this still does not imply the existence of suitable instantons satisfying (6.1) 16 Indeed, in several models we found not a single boundary state satisfying (6.1).\\n\\nNote that, since hypercharge must be massless, one can use the reverse argument and obtain that\\n\\nI M a -I M a ′ -I M c + I M c ′ -I M d + I M d ′ = 0 (6.2)\\n\\nin all models. We verified this for all models we considered as a check on the computations.\\n\\nAs already discussed in Section 4.2, in the search for SM constructions in Gepner orientifold, there are 391 models with massless hypercharge and massive B-L. In these models we found a total of 29680 instantons with B -L violation, i.e. with intersection numbers satisfying (6.1). Of course, in order to serve our purpose of generating a Majorana mass superpotential, the instantons have to satisfy some more conditions.\\n\\nLet us consider them in order of importance, and start with the conditions on the net number of chiral fermion zero modes charged under the 4d observable sector. Clearly we need I M a = I M a ′ and I M b = I M b ′ . The latter condition is automatically satisfied in this case, because the b-brane is real in all 391 models. The chiral conditions on the zero modes charged under the branes c and d are as in [3] 17 and are given in equations (3.6), (3.7) (3.8) of the present paper. These are the instantons of most interest, and on which we mainly focus. However, as discussed in Section 3.4, other important Band/or L-violating operators (such as the Weinberg operator or the LH operator) can 16 From intuition in geometric compactifications, one expects that there may always exist a D-brane with the appropriate topological pairings, but there is no guarantee that there is a supersymmetric representative in that topological sector, and even less that it would have no additional fermion zero modes. Note also that even if such D-brane instantons exists, there is no guarantee that it will fall in the scan over RCFT boundary states. 17 Note that there is a sign change in the contribution of the U (1) d generator to Y in comparison to [3] 40 be generated by instantons with similar intersection numbers, up to a factor of 2 and a sign, see table 2 . For this reason we also allow at this stage any instanton which has the correct number of charged zero modes to generate them. Imposing these conditions reduces the number of candidate instantons potentially contributing to neutrino masses in any of the models to 1315.\\n\\nAll instantons satisfying these requirements are summarized in the table 5. In columns 1,2 and 3 we list the tensor combination, MIPF and orientifold choice for which the model occurred. The latter two numbers codify simple current data that describe respectively a MIPF and an orientifold. MIPFs are in general defined by means of a subgroup H of the simple current group G, plus a certain matrix X of rational numbers [55] . Orientifolds are defined by a simple current and a set of signs [43] . In previous work [6] we have enumerated these quantities (up to permutation symmetries) and assigned integer labels to them for future reference. We only refer to these numbers here, but further details are available upon request. Usually for each MIPF and orientifold which contains the standard model there are several choices a,b,c,d for which it is obtained. For a given choice of tensor combination, MIPF and orientifold and SM branes there may be several instantons. For clarity we put all such instantons together in the information in table 5 . In column 4 we indicate which type of instanton branes were found. Five types are distinguished: O1, O2, S2, U1 and U2, corresponding to O(1), O(2), Sp(2), U(1) and U(2) Chan-Paton symmetry on the instanton volume. The number indicates the instanton brane multiplicity that gives the correct number of instanton charged zero modes from the a, b, c, d branes, to lead to right-handed neutrino Majorana masses. The number of zero modes is in general the product of the instanton brane multiplicity and 'intersection number' with the corresponding 4d spacefilling brane. As discussed in Section 5, for symplectic branes the smallest possible brane multiplicity is 2. As we discussed there, only O1 instantons may have the required universal minimal set of two zero modes in the uncharged sector.\\n\\nStill we look for all O(1), Sp(2) and U(1) instantons which may yield a superpotential if the extra uncharged fermion zero modes. In this vein we also include a search for O2 and U2 instantons. Note also that such O2 or U2 instantons imply the existence of other instantons involving the same boundary state, but with multiplicity 1, which may lead to the R-parity violating operator LH. We will discuss the generation of R-parity violating operators at the end of this section. The third character (+ or -)\\n\\nin the instanton in table 5 is the sign of I M c ′ -I M c . For the instantons giving rise to right-handed neutrino Majorana masses this sign should be negative, whereas it should 41 be positive for instantons giving rise to the Weinberg operator (or the LH operator), see table 2 .\\n\\nThe 1315 instantons are divided in the following way over the different types: 3 of types O1+ and O1-, 46 of type U1+, 24 of type U1-, 550 S2+, 627 S2-, 27 of types U2+ and U2-and four of types O2+ and O2-. Notice that the vast majority (97.5%) of the instanton solutions are of type S2+ and S2-. This is encouraging given the nice properties of such instantons, concerning e.g. R-parity conservation. Note also that in almost all cases both S-and S+ are simultaneously present, 18 so both sources of physical neutrino Majorana masses (from the see-saw mechanism or the Weinberg operator) are present. The other instanton classes possibly generating right-handed neutrino masses are O1-and U1-, which are much less abundant. There is just one orientifold with O1-instantons, for which one can obtain cancellation of RR tadpoles, see below. On the other hand we have found no orientifold with U1-instantons and cancellation of tadpoles, see below.\\n\\nTable 5: Summary of instanton branes. (1, 16, 16, 16) (2, 4, 12, 82) 2, 4, 12, 82) (2, 4, 12, 82) (2,4,14,46) 10 0 (2,4,14,46) 16 0 (2,4,16,34) 15 0 (2,4,16,34) 15 1 (2, 4, 16, 34) (2,4,16,34) 16 1 (2,4,16,34) (2, 4, 16, 34) (2, 4, 16, 34)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Tensor MIPF Orientifold Instanton Solution\",\n \"text\": \"12 0 S2 + , S2 - Yes\\n\\n19 0 S2 -! ? (\\n\\n19 0 U2 + !, U2 -! No\\n\\n19 0 U1 + , U1 - No\\n\\n16 0 S2 + , S2 - Yes\\n\\n18 0 S2 - Yes\\n\\n18 0 U1 + , U1 -, U2 + , U2 -No\\n\\n49 0 U2 + , S2 -!, U1 + Yes\\n\\nContinued on next page 18 In some models contributing many instantons there is an exact symmetry between S-and S+.\\n\\nThis explains the approximate symmetry in the full set. In some cases this symmetry can be understood in terms of flipping the degeneracy labels of boundary states. We regard it as accidental, since it is not found in all models.\\n\\nTable 5 -continued from previous page Tensor MIPF Orientifold Instanton Solution (2,4,16,34) 49 0 U1 - No (2,4,18,28) 17 0 2, 4, 22, 22) 2, 4, 22, 22) (2, 4, 22, 22) (2, 4, 22, 22) (2, 4, 22, 22) (2,4,22,22) 20 0 (2,4,22,22) 46 0 (2,4,22,22) (2, 6, 14, 14) 1 1 U1 + No (2,6,14,14) 22 2 (2,6,14,14) 60 2 (2,6,14,14) 64 0 (2,6,14,14) 65 0 (2,6,10,22) 22 2 2,6,8,38) 16 0 (2,8,8,18) (2, 8, 8, 18) (2, 10, 10, 10) (4,6,6,10) 41 0 (4,4,6,22) 43 0 (6,6,6,6) 18 0 Most models have a hidden sector containing extra boundary states beyond the SM ones. In the same spirit of imposing chiral conditions first, we should require\\n\\n(\\n\\n13 3 S2 + !, S2 -! Yes! (\\n\\n13 2 S2 + !, S2 -! Yes\\n\\n13 1 S2 + , S2 - No\\n\\n13 0 S2 + , S2 - Yes\\n\\n31 1 U1 + , U1 - No\\n\\n49 1 O2 + , O2 -, O1 + , O1 -Yes\\n\\n(\\n\\n14 2 S2 + !, S2 -! Yes\\n\\n14 0 S2 + !, S2 -! No\\n\\n52 0 U1 + , U1 - No\\n\\nthat I M h = I M h ′ ,\\n\\nwhere h is a hidden sector brane. This is to guarantee that the generated superpotential does not violate some hidden sector gauge symmetry which would require the presence of hidden sector fields along with the ν R bilinear. The latter condition is not imposed on the previously known hidden sector (i.e. the one in [6, 7] ), but instead a new search for tadpole solutions was performed, for each M, restricting the candidate hidden sector branes to those satisfying I M h = I M h ′ (as discussed in 43 Section 5). This is because in general the known hidden sector in [6, 7] is just a sample out of a huge number of possibilities.\\n\\nIn column 5 we indicate for which instantons it was possible to satisfy the tadpole conditions with this additional constraint. With regard to observable-hidden matter we use the same condition as in [6] , namely that it is allowed only if it is vector-like. Such a solution could be found for 879 of the 1315 instantons, with ten cases inconclusive (i.e it was computationally too difficult to decide if a solution does or does not exist).\\n\\nThe latter are indicated with a question mark in column 5 (for most of the undecidable cases there is a tadpole solution for a different instanton with the same characteristics;\\n\\nfor that reason just one question mark appears).\\n\\nIndependently of the RR tadpole condition (since there may be alternative sources for its cancellation, or hidden sectors which fall beyond the reach of RCFT), we can also consider the further constraint that the number of charged fermion zero modes is exactly right, not just in the chiral sense. This means\\n\\nI M a = I M a ′ = I M b = I M b ′ = 0, I M c = 2, I M c ′ = 0 and I M d = -2, I M d ′ = 0 or vice-versa.\\n\\nFurthermore we require that there are no adjoint or rank-2 tensor zero-modes (note that the latter could be chiral if the instanton brane is complex, and indeed they are in some of the 1315 cases).\\n\\nThis reduces the 1315 instantons to 263. In column 4 we indicate those cases with an exclamation mark. It is noteworthy that the success rate for solving the tadpole conditions is highest for these instantons: 254 of the 263 allow a solution (with 3 undecided). If an exclamation mark appears in column 4, this only indicates that some of the instantons are free of the aforementioned zero modes, not that all of them are. But in all cases, if there are tadpole solutions, they exist in particular for the configurations with an exclamation mark. Finally we may impose the condition that I M h and I M h ′ are separately zero. This is indicated with and exclamation mark in column 5. This turns out to be very restrictive. The only cases where this happens have no hidden sector at all.\\n\\nIt is worth remarking that the only instantons having exactly the correct set of charged zero modes and cancelling tadpoles are of S2 ± type. Also those instantons are the only cases marked with an exclamation mark in column 4 and 5. These examples, which will be discussed below in some detail, also have just the minimal set of fermion zero modes, except for the universal sector (which for Sp(2) instantons contains two extra triplets).\\n\\nThe main conclusion about this scan is that we did not find any instantons with exactly the zero mode fermions to generate the neutrino mass superpotential. However 44 we have found a number of examples which come very close to that, with exactly the required charged zero modes and a very reduced set of extra uncharged zero modes from the universal sector. These extra zero modes are non-chiral and hence one expects that e.g. RR/NS fluxes or other effects may easily lift them, as we discussed in section 2. Concerning O(1) instantons, which have just the two required fermion zero modes in the universal sector, we have found one example, with the appropriate net structure of charged zero modes. However, it has plenty of other extra zero modes. We discuss examples of O(1) and Sp(2) instantons in the following subsections.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"An O1 example\",\n \"text\": \"Let us first discuss the case of O(1) instantons. In principle they would be the more attractive since they have no undesirable universal zero modes at all. Unfortunately this type of instanton is rare within the set we scanned, and we found just one example with a solution to the tadpole equations without any unwanted chiral zero-modes. The instanton however has a very large number of uncharged and charged vector-like zero modes.\\n\\nThe standard model brane configuration occurs for tensor product (2, 4, 22, 22), MIPF 49, orientifold 1, boundaries (a,b,c,d) = (487, 1365, 576, 486) . As usual we only provide this information in order to locate this model in the database. Further details are available on request.\\n\\nThe bi-fundamental fermion spectrum of this model in the (a,b,c,d) sector is fairly close to the MSSM: there is an extra up-quark mirror pair, two mirror pairs of leptoquarks with down quark charges and one with up-quark charges, plus two extra righthanded neutrinos (i.e. a total of five right-handed neutrinos). There are three MSSM Higgs pairs. The tensor spectrum is far less appealing, in particular for brane c: this has 25 adjoints and 7 vector-like pairs of anti-symmetric tensors.\\n\\nAs we said, there is just one instanton brane of type O1-. It has exactly the right number of zero-modes with brane d, but five superfluous pairs of vector-like zeromodes with brane c, plus one vector-like pair with brane a. In addition there are four symmetric tensor zero-modes on the instanton brane (which of course are vector-like, since it is a real brane): the parameter p + in table 3 is equal to 2.\\n\\nThe tadpole solution that is (chirally speaking) compatible with this instanton has a large hidden sector: O(1)×O(2) 4 ×O(3)×U(1) 2 ×Sp(2) 2 ×U(3) (there are other possibilities, but no simple ones). This hidden sector introduces more undesirable features: vector-like observable/hidden matter, vector-like instanton/hidden sector modes, plus 45 chiral and non-chiral matter within the hidden sector. Finally the coupling ratios are as follows: α 3 /α 2 = .54, sin 2 θ w = .094, and the instanton coupling is 3.4 times weaker than the QCD coupling (α 3 /α Instanton = 3.4).\\n\\nDespite these unappealing features this model does demonstrate the existence of this kind of solution.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The S2 models\",\n \"text\": \"As we have mentioned, these are the examples which come closer to the minimal set of fermion zero modes. As we see in Table 5 , all such instantons satisfying the criteria on the zero mode structure (except for the extra universal zero modes) appear for models based on the same CFT orientifold. It is the one obtained from the (2, 4, 22, 22) Gepner model with MIPF 13 and orientifold 3 in the table. The model is obtained as follows.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The closed string sector\",\n \"text\": \"We start with the tensor product (2, 4, 22, 22) . This yields a CFT with 12060 primary fields, 48 of which are simple currents, forming a discrete group G = Z 12 ×Z 2 ×Z 2 . After taking into account the permutation symmetry of the last two factors, we find that this tensor product has 54 symmetric MIPFs, and we choose one of them to build the model of interest. For convenience we specify all quantities in terms of a standard minimal model notation, but also in terms of the labelling of the computer program \\\"kac\\\" that generates the spectrum. This particular MIPF is nr. 13. To build it we choose a subgroup of G, which is isomorphic to H = Z 12 × Z 2 . The generator of the Z 12 factor is primary field nr. 1, (0, 0, 0, {24, -24, 0}, {24, 20, 0}), and the Z 2 factor is generated by primary field nr. 24, (0, 0, 0, 0, {24, 20, 2}). The representations are specified on a basis (NSR, k = 2, k = 4, k = 22, k = 22), i.e. the boundary conditions of the NSRfermions and the four minimal models in the tensor product. Here 0 indicates the CFT vacuum, and for all other states we use the familiar (l, q, s) notation for the N = 2 minimal models. The first generator has conformal weight h = 11 12 and has ground state dimension 1. The second has weight h = 11 2 and has ground state dimension 2: the ground state contains both (0, 0, 0, 0, {24, 20, 2}) and (0, 0, 0, {24, 20, 2}, 0) . The matrix X defining the MIPF according to the prescription given in [53] [54][55]\\n\\nis 1 12 0 0 1 2 (6.3) 46\\n\\nThis simple current modification is applied to the charge conjugation invariant of the tensor product. This defines a MIPF that corresponds to an automorphism of the fusion rules, and that pairs all the primaries in the CFT off-diagonally. The number of Ishibashi states, and hence the number of boundary states is 1080. The MIPF is invariant under exchange of the two k = 22 factors: this maps current 24 to itself, and current 1 to current 11, which is also in H. Hence this symmetry of the tensor product maps H into itself, and it also preserves the matrix X.\\n\\nTo define an orientifold, we must specify a \\\"Klein bottle current\\\" plus two signs defined on the basis of the simple current group. For the current K we use the generator of the second Z 2 in G, primary field nr. 12. This is the representation (0, 0, {4, -4, 0}, {(24, 16, 2)}, {(24, -12, 2)}) which is degenerate with nine other states, all of dimension 1 and conformal weight 7. The crosscap signs are chosen, on the aforementioned basis of H as (+, -). This results in a crosscap coefficient of 0.0464731, and it is orientifold nr. 3 of a total of 8. The orientifold is also invariant under permutation of the identical factors.\\n\\nThe closed string spectrum contains 14 vector multiplets and 60 chiral multiplets.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The standard model branes\",\n \"text\": \"To build a standard model configuration we have to specify the boundary state labels.\\n\\nIt turns out that we have four choices for label a and b, one for c and two for d.\\n\\nThis leads to a total of 32 possibilities. Among these 32 there are 22 have distinct spectra (distinguished by the number of vector-like states), but for all 32 choices one obtains the same set of dilaton couplings. It seems plausible that these choices simply correspond to putting the a, b and d branes in slightly different positions, so that we move the configuration in brane moduli space. The choices are as follows (these are boundary labels assigned by the computer program, and can be decomposed in terms of minimal model representations; this will be explained in table 6 below) a : 10, 22, 130, 142 b : 210, 282, 290, 291 c : 629 d : 712, 797 There are additional possibilities, but they do not give rise to additional distinct spectra. 47 Table 6: Branes appearing in standard model configurations Label Orbit/Deg. Reps Weight Dimension 10 240 (0, 0, 0, 0, {10, 0, 0}) 5/4 1 130 2760 (0, 0, 0, {10, 0, 0}, 0) 5/4 1 22 [528,0] (0, 0, 0, {1, -1, 0}, {11, 1, 0}) 3/2 1 (0, 0, 0, {1, 1, 0}, {11, -1, 0}) 3/2 1 142 [3048,0] (0, 0, 0, {11, -1, 0}, {1, 1, 0}) 3/2 1 (0, 0, 0, {11, 1, 0}, {1, -1, 0}) 3/2 1 210 4248 (0, 0, {3, 3, 0}, {3, -3, 0}, {9, -9, 0}) 1/2 1 282 5760 (0, 0, {3, 3, 0}{9, -9, 0}{3, -3, 0}) 1/2 1 290 [5952,0] (0, 0, {1, 1, 0}{9, 7, 0}{11, -11, 0}) 5/6 1 291 [5952,24] (0, 0, {1,\\n\\n1, 0}{9, 7, 0}{11, -11, 0}) 5/6 1 629 [9348,30] (0, (1, -1, 0), 0, {9, 9, 0}{5, -3, 0} 7/12 1 712 [9852,0] (0, {1, 1, 0}{3, -3, 0}{1, 1, 0}{5, 5, 0}) 1/2 2 (0, {1, 1, 0}{1, -1, 0}{1, 1, 0}{5, -3, 0}) 1/2 2 797 [10356,30] (0, {1, 1, 0}{3, -3, 0}{5, 5, 0}{1, 1, 0}) 1/2 2 (0, {1, 1, 0}{1, -1, 0}{5, -3, 0}{1, 1, 0}) 1/2 2\\n\\nThe second column gives the boundary labels in terms of a primary field label and a degeneracy label (boundaries not indicated by square brackets are not degenerate).\\n\\nThe labels appearing in columns 1 and 2 are assigned by the computer program, and are listed here only for the purpose of reproducing the results using that program.\\n\\nIn column 2, the boundary labels are expressed in terms of primary field labels, as in formula (A.4). If a single number appears, this is a representative of an H-orbit corresponding to the boundary. If square brackets are used, this means that the Horbit has fixed points, and that it corresponds to more than one boundary label. The second entry in the square brackets is the degeneracy label, and refers to a character of the \\\"Central Stabilizer\\\" defined in [43] ; the details of the definition and the labelling will not be important here. In this case the first entry within the square brackets refers to an orbit representative.\\n\\nThese orbit representatives can also be expressed in a standard form for minimal model tensor products. This is done in column 3. This is basically the same expansion shown in (A.4), except that the degeneracy label Ψ I turns out to be trivial in all cases, 48 both for the standard model and for the instanton branes shown below (although the theory does contain primaries with non-trivial Ψ's). In columns 4 and 5 we specify the weight and ground state dimension of the corresponding highest weight representation.\\n\\nThese data are not directly relevant for the boundary state, but helps in identifying it.\\n\\nSince boundaries are specified by orbit representatives, it is not straightforward to compare them, since the standard choice (the one listed in column 2) is arbitrary. For this reason we have used another representative in columns 3, 4 and 5, selected by an objective criterion: we choose the one of minimal dimension and minimal conformal weight (in that order). If there is more than one representative satisfying these criteria we list all.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The open string spectrum\",\n \"text\": \"In Table 7 we summarize the spectra of the 32 models. The first four columns list the a,b,c,d brane labels. The last eight columns specify the total number of multiplets of types Q (quark doublet), U (up quark singlet), D (down quark singlet), L (lepton doublet), E (charged lepton singlet), N (neutrino singlet), Y (lepto-quark) and H (Higgs).\\n\\nThe numbers given are for the total number of lefthanded fermions in the representation, plus their complex conjugates. So for example a 7 in column \\\"Q\\\" means that there are 5 quark doublets in the usual representation (3, 2, 1 6 ), plus two in the complex conjugate representation (3 * , 2, -1 6 ). This yields the required three families of quark doublets, plus two mirror pairs.\\n\\nHence the smallest number that can occur in the six columns QUDLEN is three, if there are no mirrors (note that cubic anomaly cancellation requires three right-handed neutrinos in this class of models). The lepto-quarks Y are all in the same representation as the down-quarks (D), or the conjugate thereof, and they occur only as vector-like mirror pairs. They differ from D-type mirror quarks because they carry lepton number, because they come from open strings ending on the d-brane instead of the c-brane.\\n\\nIn general, there can also exist U-type lepto-quarks, but in these models they do not occur. Finally the numbers 10, 18 and 26 in column 'H' mean that there are 5, 9 or 13 MSSM Higgs pairs H + H. It is worth noticing that right-handed quarks U, D and neutrinos N = ν R do not have vectorlike copies. On the other hand right-handed\\n\\nleptons E always have one and the left-handed fields Q, L may have up to 3 vector-like copies.\\n\\nTable 7: Spectrum all 32 configurations. U(3) Sp(2) U(1) U(1) Q U D L E N Y H 10 210 629 712 7 3 3 9 5 3 6 10 22 210 629 712 7 3 3 9 5 3 6 10 130 210 629 712 3 3 3 9 5 3 2 10 142 210 629 712 3 3 3 9 5 3 2 10 10 282 629 712 3 3 3 5 5 3 6 26 22 282 629 712 3 3 3 5 5 3 6 26 130 282 629 712 7 3 3 5 5 3 2 26 142 282 629 712 7 3 3 5 5 3 2 26 10 290 629 712 3 3 3 3 5 3 6 18 22 290 629 712 3 3 3 3 5 3 6 18 130 290 629 712 3 3 3 3 5 3 2 18 142 290 629 712 3 3 3 3 5 3 2 18 10 291 629 712 3 3 3 3 5 3 6 18 22 291 629 712 5 3 3 3 5 3 6 18 130 291 629 712 3 3 3 3 5 3 2 18 142 291 629 712 3 3 3 3 5 3 2 18 10 210 629 797 7 3 3 5 5 3 2 10 22 210 629 797 7 3 3 5 5 3 2 10 130 210 629 797 3 3 3 5 5 3 6 10 142 210 629 797 3 3 3 5 5 3 6 10 10 282 629 797 3 3 3 9 5 3 2 26 22 282 629 797 3 3 3 9 5 3 2 26 130 282 629 797 7 3 3 9 5 3 6 26 142 282 629 797 7 3 3 9 5 3 6 26 10 290 629 797 3 3 3 3 5 3 2 18 22 290 629 797 3 3 3 3 5 3 2 18 130 290 629 797 3 3 3 3 5 3 6 18 142 290 629 797 3 3 3 3 5 3 6 18 10 291 629 797 3 3 3 3 5 3 2 18 22 291 629 797 5 3 3 3 5 3 2 18 130 291 629 797 3 3 3 3 5 3 6 18 Continued on next page 50 Table 7 -continued from previous page U(3) Sp(2) U(1) U(1) Q U D L E N Y H 142 291 629 797 3 3 3 3 5 3 6 18 In the following table we list the multiplicities L aa and L aa ′ of the branes that occur in these models, leading to vector-like sets of adjoints and rank-2 tensors. Since brane b is symplectic, the number of adjoints is equal to the number of symmetric tensors.\\n\\nTable 8: 4d matter from the aa and aa ′ sectors.\\n\\nBoundary Adjoints Anti-symm. Symm. a(10) 2 2 6 a(22) 2 2 2 a(130) 2 2 6 a(142) 2 2 2 b(210) -14 10 b(282) -14 10 b(290) -14 6 b(291) -14 6 c(629) 9 -14 d(712) 3 -6 d(797) 3 -6 It should be emphasized that CFT constructions generically correspond to particular points in moduli space of CY orientifolds. Due to this, they usually have an 'enhanced' massless particle content with extra vector-like matter and closed string gauge interactions. Thus one would expect that many of the massless vector-like chiral fields present in this class of models could gain masses while moving to a nearby point in moduli space. 51 6.3.4 The instantons Each of these 32 Standard Model compactifications admits 8 instantons. The instanton labels are identical for all the 32 models. They are listed in Table 9 . The first five columns use the same notation as for the standard model boundary labels. In column 6 we list the numerical value of the dilaton coupling to the instanton brane. This quantity is proportional to 1 g 2 . It is instructive to compare these couplings to the gauge couplings, in order to gain intuition on the suppression factor for our instantons. In these models the U(3) dilaton couplings are 0.00622, so that the instantons are more strongly coupled than QCD 19 On the other hand in this particular model the ratio α 3 /α 2 at the string scale is 3.23 (the value of sin 2 θ w at the string scale is 0.527). All of these couplings are subject to renormalization group running, and there are plenty of vector-like states to contribute to this, if one assumes that they acquire masses at a sufficiently low scale. One should perform a detailed renormalization group analysis to check whether one may obtain consistency with the gauge couplings measured at low-energies. Let us emphasize however that one expects that moving in moduli space many of these vector-like states will gain masses and also the values of the different gauge couplings will also generically vary.\\n\\nSince the value of the Type II dilaton is a free parameter at this level, one can get the appropriate (intermediate) mass scale for the right-handed neutrino Majorana masses by choosing an appropriate value for the dilaton. In this context, it is satisfactory to verify that the instanton couplings are unrelated to the gauge couplings, as expected since they do not correspond to gauge instantons [3], and are in fact less suppressed than the latter.\\n\\nNote that the 8 instantons fall into two distinct classes (evidently not related by any discrete symmetry, since the conformal weight on the boundary orbit is distinct, and the coupling is different as well). Within each class, the orbits of the four instanton boundaries appear to be related by the Z 2 symmetries of interchange of the last two tensor factors, and simultaneous inversion of the charge q of the minimal model. However, one has to be very careful in reading off symmetries directly from the labels in columns 3 of Tables ( 6) and (9) for a number of reasons. First of all the entries in column 3 are representatives of boundary orbits, and these representatives themselves are merely representatives of extension orbits. Secondly the action of any discrete 19 Note that the Type II dilaton in this compactifications is an arbitrary parameter which can always be chosen so that we consistently work at weak coupling. It is the relative value of gauge couplings which we are comparing here.\\n\\nTable 9: Instantons for all 32 configurations Lbl. Orbit/Deg. Reps Weight Dim. coupling 414 [8064,0] (0, {1, 1, 0}, 0, {22, -22, 0}, {20, 16, 0}) 5/2 1 0.0016993 417 [8076,30] (0, {1, -1, 0}, 0, {22, 22, 0}, {20, -16, 0}) 5/2 1 0.0016993 456 [8316,0] (0, {1, 1, 0}, 0, {20, 16, 0}, {22, -22, 0}) 5/2 1 0.0016993 459 [8328,30] (0, {1, -1, 0}, 0, {20, -16, 0}, {22, 22, 0}) 5/2 1 0.0016993 418 [8088,0] (0, {1, 1, 0}, 0, {22, -22, 0}, {18, 16, 0}) 5/3 1 0.0027033 420 [8100,0] (0, {1, -1, 0}, 0, {22, 22, 0}, {18, -16, 0}) 5/3 1 0.0027033 502 [8592,0] (0, {1, 1, 0}, 0, {18, 16, 0}, {22, -22, 0}) 5/3 1 0.0027033 505 [8604,30] (0, {1, -1, 0}, 0, {18, -16, 0}, {22, 22, 0}) 5/3 1 0.0027033 symmetry on the degeneracy labels can be non-trivial. In appendix B we discuss these symmetries in more detail.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Other examples\",\n \"text\": \"The Sp(2) instanton examples just discussed are the ones which get closer to the required minimal set of fermion zero modes. Under slightly weaker conditions, we find many more solutions. In all these cases some additional mechanism beyond exact RCFT will be needed to lift some undesirable zero modes.\\n\\nThe simplest such case is the following. The tensor product is (2, 8, 8, 18) , MIPF nr. 14, orientifold 2 (the precise spectra may be found using this information in the database www.nikhef.nl/ ∼t58/filtersols.php). There are three distinct brane configurations for which almost perfect instantons exist, namely (a, b, c, d) = (64, 562, 389, 67) and (64, 577, 389, 67) and (65, 560, 189, 66) . Each has six instantons, three of type S2+ and three of type S2-. As in the foregoing example, the six instantons are identical for the three standard model configurations. In this example, they have three different dilaton coupling strengths: .00254, .00665 and .0108 (each value occurs once for S2+ and once for S2-). By comparison, the U(3)-brane dilaton coupling strength is 0.0119338, so that the instanton brane coupling is quite a bit stronger than the QCD coupling. This is again an interesting point if we want that ν R masses are not too much 53 suppressed. Furthermore in this example there are three distinct instanton couplings, so that one may expect three non-zero eigenvalues (with a hierarchy) in the mass matrix. As in the previous examples there is not gauge coupling unification, one rather has α 3 /α 2 = .4813 and sin 2 (θ w ) = .183 at the string scale. Again a full renormalization group analysis should be performed in order to check consistency with the measured low-energy gauge coupling values.\\n\\nThese models all have a hidden sector consisting of a single Sp(2) factor. They have respectively 3, 1 and 3 susy Higgs pairs, and a spectrum of bi-fundamentals that is closer to that of the standard model than the previously discussed Sp(2) examples: quarks and leptons do not have vector-like copies (there are only some vector-like leptoquarks), and even one of the three models have the minimal set of Higgs fields of the MSSM. The rest of the spectrum is purely vector-like, and contains a number of rank-2 tensors, including eight or six adjoints of U(3). Furthermore there is vectorlike observable-hidden matter. The only undesirable instanton zero-mode is a single bi-fundamental between the hidden sector Sp(2) brane and the instanton brane. Still, these SM brane configurations without the hidden sector, provide interesting and very simple local models of D-brane sectors admitting instantons generating neutrino masses (with the additional ingredients required to eliminate the extra universal triplets of fermion zero modes).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"R-parity violation\",\n \"text\": \"We now turn to the generation of other possible superpotentials violating B -L. An instanton violates R-parity if the amount of B -L violation,\\n\\nI M a -I M a ′ -I M d + I M d ′ (6.4)\\n\\nis odd. Examples of instantons with that property were found in the following tensor product/MIPF/orientifold combinations: [(1, 16, 16, 16) , 12, 0], [(2, 4, 16, 34) , 49, 0], [(2, 4, 12, 82) , 19, 0] [(2, 4, 22, 22), 49, 0] and [(2, 4, 16, 34), 18, 0] . Note that all cases for which O2 or U2 instantons were found necessarily have R-parity violating instantons as well: the corresponding O1 and U1 instantons have I M d or I M d ′ equal to ±1, whereas the intersection with the a is non-chiral. In principle, there are many more ways to obtain R-parity violating instantons (either due to non-vanishing contributions to\\n\\nI M a -I M a ′ or higher values of I M d -I M d ′ )\\n\\n, and indeed, many such instantons turn out to exist. But the number of tensor product/MIPF/orientifold combinations where they occur hardly increases: only in the case [(1, 16, 16, 16), 12, 0] we found R-parity 54 violating instantons, but no U1 or O1 instantons. This suggests that in the other cases R-parity is a true symmetry of the model. Unfortunately we have no way of rigorously ruling out any other non-perturbative effects, but at least the set we can examine respects R-parity. This includes in particular the models without hidden sector (found for [(2, 4, 22, 22), 13, 3] ) discussed above.\\n\\nThe following table list the total number of instantons with the chiral intersections listed in table 2. The total number of instantons (boundaries violating the sum rule, as defined in (6.1)) is 29680, for all standard model configurations combined. The last four columns indicate how many unitary instantons satisfy the sum rule exactly as listed in table (6.1), how many satisfy it with I M x ↔ -I M x ′ (the column U'), and how many O-type and S-type instantons there are. Here 'S' refers to boundaries with a symplectic Chan-Paton group if the boundary is used as an instanton brane. All intersection numbers for type S have been multiplied by 2 before comparing with table 2. For real branes, the relevant quantities used in the comparison are\\n\\nI M a -I M a ′ , I M c -I M c ′ and I M d -I M d ′ , while I M b = 0.\\n\\nThere are fewer unitary instantons possibly generating Majorana masses then the numbers mentioned above because the conditions we use here are stricter: we require here that I M x and I M x ′ match exactly, not just their difference. Note however that this still allows additional vector-like zero-modes. If we only wish to consider cases without any spurious zero-modes, we may limit ourselves to the O-type instantons in the last column. There are very few to inspect, and all of them turn out to have a few non-universal zero modes.\\n\\nD = 4 Operator U U' S O ν R ν R 1 2 627 3 L HL H 0 5 550 3 L H 3 3 0 4 QDL 8 4 0 4 UDD 0 0 0 4 LLE 8 4 0 4 QQQL 0 4 0 3892 UUDE 4 0 0 3880\\n\\nTable 10: Number of instantons in our search which may induce neutrino masses (first 2 rows), R-parity violation (next 4 rows) or proton decay operators (last 2 rows).\\n\\nThe last two cases are B -L preserving dimension five operators, and obviously do not come from the set of 29680 B -L violating instantons. They were searched for 55 separately, but the search was limited to the same 391 models we used in the rest of the paper. Obviously, one could equally well look for such instantons in the full database, since their existence does not require a massive B -L.\\n\\nIt is interesting to note that in the classes of MSSM-like models discussed earlier in this section with the closest to minimal zero mode structure, there are no instantons al all generating either R-parity violating or the B -L dim=5 operators in the table.\\n\\nThis makes them particularly attractive.\\n\\nNote that all numbers in table 10 refer to the occurrence of instantons in the set of 391 tadpole-free models with massive B-L, but without checking the presence of zero-modes between the hidden sector and the instanton. It makes little sense to use the hidden sector in the database for such a check, since this is just one sample from a (usually) large number of possibilities. A meaningful question would be: can one find a hidden sector that has no zero-modes with the instanton. We have done such a search for the B -L violating instantons (see the exclamation marks in the last column of table (5)), but not for the B -L preserving instantons.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions and outlook\",\n \"text\": \"In this paper we have presented a systematic search for MSSM-like Type II Gepner orientifold models allowing for boundary states associated to instantons giving rise to neutrino Majorana masses. This search is very well motivated since neutrino masses are not easily accommodated in the semi-realistic compactifications constructed up to now. String instanton induced Majorana masses provides a novel and promising way to understand the origin of neutrino masses in the string theory context.\\n\\nThe string instantons under discussion are not gauge instantons. Thus, for example, they not only break B + L symmetry (like 't Hooft instantons do) but also B -L, allowing for Majorana neutrino mass generation. The obtained mass terms are of order M s exp(-V /g 2 ) but this suppression is unrelated to the exponential suppression of e.g. electroweak instantons and may be mild. In fact we find in our most interesting examples that the instanton action is typically substantially smaller than that of QCD or electroweak instantons, and hence these effects are much less suppressed than those coming from gauge theory instantons.\\n\\nTo perform our instanton search we have analyzed the structure of the zero modes that these instantons must have in order to induce the required superpotential. This analysis goes beyond the particular context of Gepner orientifolds and has general 56 validity for Type II CY orientifolds. We have found that instantons with O(1) CP symmetry have the required universal sector of just two fermionic zero modes for the superpotential to be generated. Instantons with Sp(2) and U(1) CP symmetries have extra unwanted universal fermionic zero modes, which however may be lifted in a variety of ways in more general setups, as we discuss in the text. In fact we find in our search that around 98 % of the instantons with the correct structure of charged zero modes have Sp(2) CP symmetry. Indeed, from a number of viewpoints the Sp(2)\\n\\ninstantons are specially interesting. The instantons we find with the simplest structure of fermionic zero modes are Sp(2) instantons which are also the ones which are present more frequently in the MSSM-like class of Gepner constructions considered. They have also some interesting features from the phenomenological point of view. Indeed, due to the non-Abelian structure of the CP symmetry, the structure in flavor space of the neutrino Majorana masses factorizes. This makes that, irrespective of what particular compactification is considered, Sp(2) instantons may easily lead to a hierarchical structure of neutrino masses. It would be important to further study the possible phenomenological applications of the present neutrino mass generating mechanism.\\n\\nString instanton effects can also give rise to other B-or L-violating operators. Of particular interest is the dimension 5 Weinberg operator giving direct Majorana masses to the left-handed neutrinos. We find that in the most interesting cases, different instantons giving rise to the Weinberg operator and to ν R Majorana masses are both simultaneously present. Which effect is the dominant one in the generation of the physical light neutrino masses depends on the values of the instanton actions and amplitudes as well as on the value of the string scale. Instantons may also generate dim< 5 operators violating R-parity. We find however that instantons inducing such operators are extremely rare, and in fact are completely absent in the Gepner models with the simplest Sp(2) instantons inducing neutrino masses.\\n\\nThere are many avenues yet to be explored. It would be important to understand better the possible sources (moving in moduli space, addition of RR/NS backgrounds etc.) of uplifting for the extra uncharged fermionic zero modes in the most favoured Sp(2) instantons. A second important question is that we have concentrated on checking the existence of instanton zero modes appropriate to generate neutrino masses; one should further check that the required couplings among the fermionic zero modes and the relevant 4d superfields (i.e. ν R or L H) are indeed present in each particular case. This is in principle possible in models with a known CFT description but could be difficult in practice for the Gepner models here described.\\n\\n57 Instantons can also generate other superpotentials with interesting physical applications. One important example is the generation of a Higgs bilinear (i.e. a µ-term) in MSSM-like models [4, 3]. Thus, e.g., one could perform a systematic search for instantons (boundary states) generating a µ-term in the class of CFT Gepner orientifolds considered in the present article. Other possible application is the search for instantons inducing superpotential couplings involving only closed string moduli. The latter may be useful for the moduli-fixing problem, or for non-perturbative corrections to perturbatively allowed couplings [56].\\n\\nFinally, it would be important to search for analogous instanton effects inducing neutrino masses in other string constructions (heterotic, M-theory etc.). A necessary condition is that the anomaly free U(1) B-L gauge boson should become massive due to a Stückelberg term.\\n\\nThe importance of neutrino masses in physics beyond the Standard Model is unquestionable. We have shown that string theory instantons provide an elegant and simple mechanism to implement them in semi-realistic MSSM-like string vacua, and a powerful constraint in model building. In our opinion, the conditions of the existence of appropriate instantons to generate neutrino masses should be an important guide in a search for a string description of the Standard Model.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"We thank M. Bertolini, R. Blumenhagen, S. Franco, M. Frau, S. Kachru, E. Kiritsis, A. Lerda, D. Lüst, F. Marchesano, T. Weigand for useful discussions. A.M.U. thanks M. González for encouragement and support. The research of A.N. Schellekens was funded in part by program FP 57 of the Foundation for Fundamental Research of Matter (FOM), and Research Project FPA2005-05046 of de Ministerio de Educacion y Ciencia, Spain. The research by L.E. Ibáñez and A.M. Uranga has been supported by the European Commission under RTN European Programs MRTN-CT-2004-503369, MRTN-CT-2004-005105, by the CICYT (Spain), and the Comunidad de Madrid under project HEPHACOS P-ESP-00346.\\n\\nAppendix A CFT Notation\\n\\nHere we summarize the labelling conventions for various CFT quantities. Further details and explanations can be found in [43] .\\n\\nIt is important to keep in mind that there are four steps in the construction, each involving choices of some quantities. The steps are\\n\\n• A CFT tensor product\\n\\n• An extension of the chiral algebra of this tensor product\\n\\n• The choice of a MIPF\\n\\n• The choice of an orientifold The second and third step are easily confused. A MIPF can itself be of extension type (although it can also be of automorphism or mixed type), meaning that it implies an extension of the chiral algebra. The crucial difference between step two and three in that case is that in step 2 all fields that are non-local with respect to the extension are projected out, and the symmetry of the extension is imposed on all states of the CFT,\\n\\ni.e in particular on all boundary states. The extension in step three acts as a bulk invariant, but the boundary states are not required to respect the symmetry implied by the extension.\\n\\nPrimary fields of N = 2 minimal models are labeled in the usual way by three integers (l, q, s). In addition to these minimal models, one building block of our CFT's is of course a set of NSR fermions in four dimensions. They can be represented by the four conjugacy classes (0), (v), (s), (c) analogous to those of a root lattice of type D. Primary fields in a tensor product of M factors are therefore labelled as\\n\\nI = ((x), (l 1 , q 1 , s 1 ), . . . , (l M , q M , s M )) (A.1)\\n\\nwhere x = 0, v, s or c.\\n\\nThis tensor product is extended by the alignment currents and the spin-1 field corresponding to the space-time supersymmetry generator. This organizes the tensor product fields into orbits, which can be labelled by one of the elements of the orbit. We always choose the field of minimal conformal weight (or one of them, in case there 59 are more) as the orbit representative labelling the orbit. The supersymmetry generator may have fixed points, leading to orbits appearing more than once as primary fields of the extended theory. In those cases we need an additional degeneracy label to distinguish them. It is convenient to choose for this label a character of the discrete group that is causing the degeneracy, the \\\"untwisted stabilizer\\\", which depends on I.\\n\\nDenoting this character as Ψ I we get then the following set of labels for the primaries of the extended CFT\\n\\ni = [I, Ψ I ] (A.2)\\n\\nwhere I has the form (A.1). If there are no degeneracies we will leave out the square brackets and the Ψ I .\\n\\nIn boundary CFT's two new labels appear: the labels of Ishibashi-states that propagate in the transverse channel of the annulus, and the boundary labels. In the simplest, \\\"Cardy\\\" case both sets of labels are in one-to-one correspondence with the extended CFT labels i. But if we consider non-trivial MIPFs Z ij both sets of labels are different.\\n\\nThe Ishibashi states are in one-to-one correspondence with the fields i with Z ii c = 0. Degeneracies can occur here if Z ii c > 0. This requires the introduction of a degeneracy label. Such degeneracies may occur if the stabilizer of i (the set of simple currents that fix i) is non-trivial. It is convenient to use elements J of the stabilizer as degeneracy labels, so that the Ishibashi labels get the following form m = (i, J) , (A.3) where i is an extended CFT label, as defined above (to be precise, in some cases a non-trivial degeneracy label is introduced even if Z ii c = 1. The details will not matter here).\\n\\nBoundary states correspond to orbits of the simple current group H that defines a MIPF. To label such orbits we choose a representative. There is no obvious canonical representative (one could use one of minimal conformal weight, but the conformal weight of orbit members of a boundary state does not play any rôle in the formalism, unlike the conformal weight of a primary). So in this case we just make an arbitrary choice. Once again there can be degeneracies. In this case they are due to a subgroup of the stabilizer called the \\\"Central Stabilizer\\\". It is convenient to label the boundary states by an orbit representative i and a character ψ i of the central stabilizer. If we expand the boundary state label in all of its components we get\\n\\na = [i, ψ i ] = [[I, Ψ I ], ψ i ] = [((x), (l 1 , q 1 , s 1 ), . . . , (l M , q M , s M )), Ψ I ], ψ i ] . (A.4) 60\\n\\nNote that i is just a representative of a boundary orbit, and that I is just a representative of an orbit of the extension of the CFT.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B Instanton boundary symmetries\",\n \"text\": \"In the hidden-sector free example discussed in some detail in section 6 we have encountered Sp(2)-type instantons, the most common kind in our scan. This particular model is the one that comes closest to the required zero mode count, although the only superfluous zero modes are rather awkward. Let us assume that the effect of these superfluous universal zero-modes instantons can be avoided. Then there is still another problem we have to face, namely that the two zero modes α i and γ i are related by an Sp(2) transformation of the label i. Then we we need at least three independent instantons (with unrelated couplings) to generate three non-zero neutrino masses, as discussed in Section 3.3. Since the technology to compute the couplings is not yet available, we cannot be completely sure that the relevant couplings are distinct, or indeed that they are non-vanishing, but at least we can inspect if there are obvious symmetries relating them.\\n\\nThe unextended tensor product (2, 4, 22, 22) has 64 discrete symmetries: five separate charge conjugations of the factors (including the NSR space-time factor) and the interchange of the two identical k = 22 minimal models. To get space-time supersymmetry this tensor product is extended with the product of the simple current Ramond ground states of each factor. These Ramond ground states are not invariant under charge conjugation. Therefore this extension breaks the discrete symmetries to Z 2 × Z 2 , the combined charge conjugation of all five factors and the permutation of the two identical factors. The combined charge conjugation also acts non-trivially on the Ramond ground states in each factor, but the result is the charge conjugate of the space-time supersymmetry generator, which is in the chiral algebra. The combined conjugation is in fact the charge conjugation symmetry of the extended CFT. It turns out that only a Z 2 subgroup of Z 2 × Z 2 acts non-trivially on the simple currents of the extended CFT: the permutation of the k = 22 factors acts in the same way as charge conjugation. The action of these symmetries on the complete set of primary fields is more complicated. It is easy to see that the permutation acts differently than charge conjugation. In general the primary fields of the extended CFT are labelled as i = [((x), (i 1 ), . . . , (i 4 )), Ψ] (see Appendix A). The action of the permutation is to interchange i 3 and i 4 , but in cases with a non-trivial degeneracy it is not a priori 61 clear which of the degenerate states is the image of the map. This can be resolved by examining the fusion rules, which should be invariant under the permutation: N ijk = N π(i)π(j)π(k) , (B.1) where N is a fusion coefficient and π a permutation or other automorphism. In general there may be more than one way to resolve these ambiguities, resulting in additional automorphism of the CFT. The standard example of this situation is the extension of the affine algebra A 1 level 4 by the simple current. The resulting CFT has an outer automorphism, non-trivial charge conjugation, that has no counterpart in A 1 level 4.\\n\\nAs mentioned above, the Z 2 permutation symmetry is respected by the MIPF and the orientifold, and since charge conjugation acts in the same way on the simple currents as the permutation, charge conjugation is respected as well.\\n\\nIn this way we end up with (at least) a surviving Z 2 × Z 2 discrete symmetry acting on the boundary labels, or a larger discrete symmetry if that symmetry is extended by the action on the degeneracy labels of the extension. The foregoing story repeats itself for the action on the boundary labels. The boundary labels are given in terms of the CFT labels plus a second degeneracy label, the one indicated by the second entry in the square brackets in column 2 of tables (6) and (9). Once again one has to determine not only how a symmetry acts on the first entry (this is just the action of the symmetry in the extended CFT, respecting its fusion rules), but also how it acts on the degeneracy labels. In this case the precise action can be determined from the invariance of the annulus coefficients A i ab = A π(i) π(a)π(b) , (B.2)\\n\\nwhere π is the action on the primaries of the extended CFT (as determined above) and π is the action on the boundary labels induced by π.\\n\\nSince the orientifold choice is non-trivial, boundary charge conjugation does not coincide with CFT charge conjugation. Indeed, the eight instanton boundary states are invariant under boundary charge conjugation (which they must be in order to produce a \\\"real\\\" Sp(2)-type instanton). However, just as permutations, CFT charge conjugation may induce a non-trivial discrete symmetry on the boundary states.\\n\\nIn addition to these \\\"outer automorphisms\\\" there is the notion of boundary simple currents, introduced in the appendix of [6] . These may be thought of as remnants of the original simple currents, and imply relations between annulus amplitudes of the form A i ab ′ = A i Ja(Jb) ′ (B.3) 62 All of the aforementioned symmetries might relate instanton couplings, and hence threaten their numerical independence. However, in order to do that they have to be symmetries of the full standard model/instanton configuration, not just relate some of the eight instantons to each other. It is easy to see that the permutation of the k = 22 factors changes the standard model brane configuration. Consider brane c: it turns out that under permutation boundary state 629 it is mapped to boundary state 544 or 545 (depending on the action on the degeneracy label), which in any case is distinct. Hence even if the instanton boundaries 414 and 456 resp. 418 and 502 are mapped to each other by boundary permutation, at the same time the standard model configuration is\\n\\nmapped to a distinct one.\\n\\nThis means that we may expect at least four distinct couplings, which should be sufficient. It is of course possible to work out the discrete symmetries exactly, but in view of this argument this would not yield any additional insight.\\n\\nWe do know the exact boundary orbits. The orbit of instanton label 414 is (414, 415, 416, 417) , so that instantons 414 and 417 are related. But the orbit of brane c under the same action is (629, 628, 626, 627) . Hence the action that relates 414 and 417 maps 629 to 627. In fact all four standard model boundaries a,b,c,d are mapped to different ones. This implies that instantons 414 and 417 may produce different couplings as well, so that all eight instantons may contribute in a different way.\\n\\nThese distinctions concern the disk correlators d (r) a in (3.12). The factors exp(-Re U r ) will be related by discrete symmetries, and it seems reasonable to expect them to be identical for instantons 414, 417, 456 and 459, which is indeed correct. However there is no reason to expect the other four instantons to have the same suppression factor, and indeed they do not.\\n\\nNote that these symmetries imply the existence of a much larger set of standard model configurations than the 32 discussed here. However, as mentioned before, the 32 models considered here display all possible distinct spectra.\"\n }\n]","isConversational":false}}},{"rowIdx":190,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1084"},"arxiv_id":{"kind":"string","value":"0704.1084"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1751-8113/40/25/S21"},"content_hash":{"kind":"string","value":"a7acfb95a2edadd76e4c81acce00f735d0f24f7fbf0f6b4f279d6055f94998ba"},"title":{"kind":"string","value":"Vacuum Energy and Renormalization on the Edge"},"abstract":{"kind":"string","value":"The vacuum dependence on boundary conditions in quantum field theories is\nanalysed from a very general viewpoint. From this perspective the\nrenormalization prescriptions not only imply the renormalization of the\ncouplings of the theory in the bulk but also the appearance of a flow in the\nspace of boundary conditions. For regular boundaries this flow has a large\nvariety of fixed points and no cyclic orbit. The family of fixed points\nincludes Neumann and Dirichlet boundary conditions. In one-dimensional field\ntheories pseudoperiodic and quasiperiodic boundary conditions are also RG fixed\npoints. Under these conditions massless bosonic free field theories are\nconformally invariant. Among all fixed points only Neumann boundary conditions\nare infrared stable fixed points. All other conformal invariant boundary\nconditions become unstable under some relevant perturbations. In finite volumes\nwe analyse the dependence of the vacuum energy along the trajectories of the\nrenormalization group flow providing an interesting framework for dark energy\nevolution. On the contrary, the renormalization group flow on the boundary does\nnot affect the leading behaviour of the entanglement entropy of the vacuum in\none-dimensional conformally invariant bosonic theories."},"authors":{"kind":"list like","value":["M. Asorey","D. Garcia-Alvarez and J. M. Munoz-Castaneda"],"string":"[\n \"M. Asorey\",\n \"D. Garcia-Alvarez and J. M. Munoz-Castaneda\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The emergence of the dark energy as one of the basic ingredients of the current standard cosmological scenario, and the absence of an even vague understanding of its possible origin, opens a window to the analysis of all possible mechanisms that generate background energy (see e.g. [1] for a review of recent proposals). The main problem is that the apparent value of the dark energy is very tiny compared with any physical energy scale. A second problem is that in a generic quantum field theory there is generation of vacuum energy and any renormalization prescription requires a fine tuning, which is not very convincing without the quantisation of the gravitational interaction.\n\nThe guess that dark energy might change with the evolution of the Universe can be understood even if dark energy is just vacuum energy. The finite corrections due to finite size of the causal Hubble domain decrease as the Universe continues to expand.\n\nThe aim of this paper is to analyse the variation of these finite size corrections under of renormalization group on the space of boundary conditions for scalar field theories in flat space, although the results are generalisable for more general backgrounds.\n\nThe dependence of the vacuum energy on the boundary conditions [2] is well known since the discovery of Casimir effect [3] (see [4, 5] and [6] and references therein for recent revisions). However, boundaries might also be considered as a source of new, although peculiar, interactions and therefore can undergo renormalization [7, 8] . The renormalization of boundary conditions modifies the critical behaviour of the theory [9, 10, 11] . In systems with boundaries or defects, the boundary RG flow induces a dynamical behaviour on the boundaries. The dynamics of D-branes in string theory emerges in this way [12] .\n\nThe renormalization group flow is analysed from a global viewpoint in the most general framework for boundary conditions of scalar field theories introduced in Ref. [13] . In particular, we consider the possible existence of topological transitions [14] induced by the renormalization of boundary conditions or cyclic orbits in the boundary RG flow [15] . The dependence of the finite size corrections to the vacuum energy and vacuum entanglement entropy [16, 17] under the boundary RG flow is analysed from a very general perspective.\n\nThe action which governs the dynamics of scalar field theory in a bounded domain Ω of flat space consists of two different of terms, S(φ) = S B (φ) + S b (φ). The first one\n\nS B (φ) = 1 2 dt Ω √ g d D x | φ| 2 -|∇φ| 2 -V (|φ| 2 ) ( 1\n\n)\n\nis defined in terms of the values of the fields in the bulk. The second term\n\nS b (ϕ) = 1 2 dt ∂Ω √ g ∂Ω d D-1 x | φ| 2 + 1 2 ϕ * ∂ n ϕ+ 1 2 (∂ n ϕ * )ϕ-|∇ϕ| 2 ( 2\n\n)\n\nVacuum Energy and Renormalization on the Edge 3 depends only on the values of the fields at the boundary ∂Ω † . g ∂Ω denotes the metric induced on the boundary by the bulk flat metric, and ∂ n is the normal derivative at the boundary\n\nϕ = φ| ∂Ω ∂ n ϕ = ∂ n φ| ∂Ω . ( 3\n\n)\n\nThe presence of the boundary term S b allows the generation of local classical equations of motion without requiring any specific type of boundary conditions [19, 20] . Indeed, the gradient term\n\nV = 1 2 Ω |∇φ| 2 ( 4\n\n)\n\ncan be rewritten as\n\nV = 1 2 Ω φ † ∆φ + 1 2 ∂Ω φ † ∂ n φ ( 5\n\n)\n\nwhere ∆ is the Laplace-Beltrami operator ∆ = -∂ µ ∂ µ . In the quantum theory the Laplace-Beltrami operator must have a real spectrum in order to have a selfadjoint Hamiltonian\n\nH = 1 2 √ ∆ + m 2 . ( 6\n\n)\n\nfor the free field theory (The inclusion of interactions does not changes the picture [21] ). This means that the classical fields must satisfy boundary conditions which make the operator ∆ selfadjoint. The complete set of boundary conditions which satisfy this requirement [13] are in one-to-one correspondence with the group of unitary operators of the boundary Hilbert space L 2 (∂Ω, ). For any unitary operator U ∈ L 2 (∂Ω, ), the fields satisfying the boundary condition\n\nϕ -i ∂ n ϕ = U (ϕ + i ∂ n ϕ) ( 7\n\n)\n\ndefine a domain where ∆ is a selfadjoint operator. In the case of open strings, the corresponding conformal 1+1 dimensional scalar field theories is defined on the space interval Ω = [0, 1] ⊂ IR and there is a large variety of admissible boundary conditions described by the unitary group M = U(2). The unitary matrices\n\nU D = -1 0 0 -1 U N = 1 0 0 1 U P = 0 1 1 0 ( 8\n\n)\n\ndefine Dirichlet, Neumann and periodic boundary conditions, which in string theory correspond to a string attached to a D-brane background, free open and closed string theories, respectively. For higher N-dimensional target spaces, or N-component strings, the set of boundary conditions becomes M = U(2N) which includes matrices which interpolate between one single closed string or N disconnected strings [13] . The topology change is described in this picture by a simple change of boundary conditions in L 2 (∂Ω, N ) [14].\n\n† We will assume that the boundary is regular and smooth. See e.g. [18] for the peculiarities associated to the presence of irregular boundaries\n\nVacuum Energy and Renormalization on the Edge 4 If the spectrum of eigenvalues of the unitary operator U does not include the value ±1 (i.e. ±1 / ∈ Sp U) the boundary condition (7) can be rewritten as\n\n∂ n ϕ = -i Á ± U Á ∓ U ϕ ( 9\n\n)\n\nwhich means that only the boundary values of the fields at the boundary can have an arbitrary value ϕ whereas its normal derivative is determined by U and ϕ.\n\nThe corresponding operator mappings from unitary into selfadjoint operators\n\nA ± = -i Á ± U Á ∓ U ( 10\n\n)\n\nare the celebrated Cayley transforms. The inverse Cayley transform\n\nU = Á∓ iA ± Á± iA ± ( 11\n\n)\n\nrecovers the unitary operator U from their selfadjoint Cayley transforms A ± . The condition of ∆ being selfadjoint is necessary but not sufficient to guarantee the unitarity of the corresponding quantum field theory. Indeed, in the case of free field theory the Hamiltonian (6) must be selfadjoint. This requires that the spectrum of ∆ + m 2 must be not only real but also positive which restricts the set of admissible boundary conditions to a subset M of L 2 (∂Ω, ).\n\nBecause of the existence of the boundary term in (5) the Hamiltonian H (6) is not selfadjoint if the spectrum of the unitary operator U intersects the following domain of phase factors\n\nS 1 m = {e 2αi ; -π < α ≤ π, 0 < α < π 2 -arctan m 2 , or π 2 < -α < π -arctan m 2 }.\n\nIn any other case, -m 2 is a lower bound for the spectrum of the operator ∆ and H is selfadjoint. One possible source of unitarity loss is the existence of edge estates with large negative eigenvalues of operator ∆. The consistency of the quantum field theory imposes, thus, a very stringent condition on the type of acceptable boundary conditions, even in the case of massive theories in order to prevent this type of pathological behaviour of vacuum energy.\n\nFor real scalar fields there is a further condition. U has to satisfy a CP symmetry preserving condition\n\nU † = U * , U = U T . ( 12\n\n)\n\nThe usual Neumann and Dirichlet boundary conditions U = ±Á satisfy this condition.\n\nIn general, for\n\nU = A 1 B B T A 2 ( 13\n\n)\n\nthe condition requires that\n\nA 1 = A T 1 , A 2 = A T 2 , A 1 B * + BA † 2 = 0 ( 14\n\n) BB † + A 1 A † 1 = Á, A 2 A † 2 + B T B * = Á ( 15\n\n)\n\nVacuum Energy and Renormalization on the Edge 5 In particular, the quasi-periodic condition ϕ(L) = M -1 ϕ(0), ∂ n ϕ(L) = M∂ n ϕ(0) is also compatible if M = M t = M * .\n\nIn the case of one single real massless scalar the set of compatible boundary conditions has two connected components: M 0 given by the operators of the form\n\nU β = cos β Á + i sin β σ y , and M 1 given by U α = cos α σ z + sin α σ x . ( 16\n\n)\n\nM 0 includes Neumann (β = 0) and Dirichlet (β = π) conditions; and M 1 contains the quasi-periodic boundary conditions\n\nϕ(L) = tan α 2 ϕ(0); ∂ n ϕ(L) = tan α 2 -1 ∂ n ϕ(0) ( 17\n\n)\n\nwhich include periodic (α = π 2 ) and antiperiodic (α = -π 2 ) boundary conditions.\n\nSince boundary conditions appear more naturally in the Schrödinger picture of field theory and the theory is plagued of ultraviolet singularities some doubts were raised about their relevance for the quantum field theory. The pioneer work of Symanzik [21] confirmed the consistence of the standard picture even in presence of bulk renormalizable interactions (see [22] for an explanation of a recent controversy [23] ).\n\nMoreover, there is a renormalization of the very boundary conditions because the boundary terms are the source of new interactions.\n\nThe renormalization group can be defined in the continuum approach by\n\nφ Λ x Λ = Λ [φ] [φ(x) -ξ Λ (x)] ( 18\n\n)\n\nby means of a fluctuating field ξ Λ with short range fluctuations of order 1 Λ . This implies that the boundary condition\n\n∂ n ϕ = A ϕ ( 19\n\n)\n\nis renormalised to\n\n∂ n ϕ Λ = A Λ ϕ Λ , ( 20\n\n) since ∂ n φ Λ x b Λ = Λ [φ]+1 [∂ n φ(x b ) -∂ n ξ Λ (x b )] = A Λ [φ]+1 φ(x b ) = A Λ φ Λ x b Λ ( 21\n\n)\n\nwith A Λ = ΛA. For more general boundary conditions the continuum renormalization group is given by\n\nΛ U † Λ ∂ Λ U Λ = 1 2 U † Λ -U Λ ( 22\n\n) or U † t ∂ t U t = 1 2 U † t -U t ( 23\n\n)\n\nVacuum Energy and Renormalization on the Edge 6 for Λ = Λ 0 e t . Fixed points correspond, therefore, to self-adjoint boundary conditions U † = U. In particular, Dirichlet and Neumann (U = ∓ Á) are renormalization group fixed points. For mixed boundary conditions the RG flows from Dirichlet (UV) toward Neumann (IR) conditions. U = e 2i arctan e -t Á.\n\n(24) Critical exponents can be identified with the eigenvalues of the matrix U c at the fixed points. Since U c is also hermitian all critical exponents are either 1 or -1 and there is no room for cyclic orbits. It is well known, however, that some quantum systems with singular boundaries and singular interactions [15, 24] exhibit cyclic renormalization group flows. Moreover, some topological field theories (e.g. Russian doll models) present a similar behaviour [25]. In scalar field theories, this phenomenon simply does not occur for regular boundaries. For the same reasons topological transitions do not occur for finite scale transformations since the flip of eigenvalues from -1 to +1 requires a change in the parameter t of the flow from -∞ to ∞ as in (24)).\n\nIn 1+1 dimensions the theory of massless scalar fields is formally conformal invariant. However, boundary conditions might break this symmetry [9, 10, 11] .\n\nConformal invariance is only preserved if the boundary conditions are stable under the boundary renormalization group flow. The fixed points can easily be identified. For a complex scalar field, besides the above mentioned fixed points, which correspond to Dirichlet, Neumann and pseudo-periodic boundary conditions and obviously are conformal invariant, there are fixed points corresponding to quasi-periodic boundary conditions (17). They also preserve the conformal symmetry.\n\nIn 1+1 dimensions this exhausts the whole set of conformal invariant boundary conditions. Any other boundary condition flows toward one of these fixed points. The most stable fixed point corresponds to Neumann conditions because all its critical exponents are +1. The most unstable is that of Dirichlet conditions since all critical exponents -1. This is compatible with the fact that the neighbourhood of Dirichlet boundary conditions is plagued of singularities Periodic, quasi-periodic and pseudo-periodic fixed points present relevant and irrelevant perturbations with critical exponents ±1, respectively. Negative values label the possible instabilities. Implications of these results for string theory are well known. Periodic boundary conditions, appear as attractors of systems with quasi-periodic and pseudo-periodic conditions which stresses the stability of closed string theory vacuum. For open strings the (stable) attractor points are standard free strings (Neumann). Any other boundary condition flow toward one of those fixed points.\n\nNotice that the absence of topological transitions in the boundary renormalization group flow is a consequence of the fact that all relevant perturbations are always associated with -1 critical exponents.\n\nVacuum Energy and Renormalization on the Edge 7 In higher dimensions (D > 1) conformal invariance requires, even in the massless case m = 0, that Neumann boundary conditions have to be modified in order to preserve conformal invariance with a term\n\n∂ n ϕ = D -1 4 D K ϕ, ( 25\n\n)\n\nproportional to the extrinsic curvature K of the boundary. In the case of singular boundaries some more interesting boundary renormalization group flows arise (see e.g. [18] for a review): fixed points and cyclic orbits of the boundary renormalization group flow can appear [15, 25, 24] and conformal invariance can be partially broken to a discrete subgroup [24].\n\nThe infrared properties of quantum field theory are very sensitive to boundary conditions [26] . In particular, physical properties of the quantum vacuum state like the vacuum energy may exhibit a very strong dependence on the type of boundary conditions. This can be explicitly shown in the simple case of a massless field defined on a finite one-dimensional interval [0, L].\n\nFor pseudoperiodic boundary conditions defined by the unitary operator\n\nU θ = cos θ σ x -sin θ σ y : ϕ(L) = e iθ ϕ(0) ( 26\n\n)\n\nthe Casimir vacuum energy (see e.g. Ref. [5] and references therein) is given by\n\nE 0 = π L 1 12 -min n∈Z θ 2π +n - 1 2 2 ( 27\n\n)\n\nThe vacuum energy dependence on θ is in this case relatively smooth. The only cuspidal point at θ = 0 corresponds to periodic boundary conditions. A completely regular behaviour is obtained for Robin boundary conditions\n\nU = e 2αi Á : ∂ n ϕ(0) = tan α ϕ(0), ∂ n ϕ(L) = tan α ϕ(L) ( 28\n\n)\n\nwhich smoothly interpolate between Dirichlet (α = π 2 ) and Neumann (α = π) conditions when α is restricted to the interval α ∈ [ π 2 , π] [27, 28, 29] . Finally, the Casimir energy for quasi-periodic boundary conditions [30]\n\nE 0 = π L 1 12 -min n∈Z α 2π + n + 1 4 2 ( 29\n\n)\n\nis also dependent on the choice of the parameter α. Two particularly interesting cases are given by\n\nα = 0, U Z = σ z ; ϕ(L) = 0, ∂ n ϕ(0) = 0 and α = π, U ′ Z = σ z ; ϕ(0) = 0, ∂ n ϕ(L) = 0\n\nwhich correspond to a Zaremba (mixed) boundary conditions: one boundary is Dirichlet and the other Neumann.\n\nVacuum Energy and Renormalization on the Edge 8 6. Vacuum entanglement entropy\n\nThe dependence of vacuum energy on boundary conditions seems to suggest that many other observables may suffer the same effect. In particular, one may wonder whether or not the entropy of the system is dependent on the type of conditions that constrain the values of the fields at the boundary. The entropy of the field theory at finite temperature scales with the volume of physical space. Only in quantum gravity or string theory the entropy can scale with the area of black hole horizon. However, in field theory it is possible ω Ω Figure 1. Information loss by integration over the fluctuations of the fields inside the domain ω to generate a mixed state from the pure vacuum state Ψ 0 by integrating out the fluctuating modes in a bounded domain ω of the physical space Ω (see Figure 1 )\n\nρ ω = ω Ψ * 0 Ψ 0 . ( 30\n\n)\n\nThe entropy of this state S ω = -T r ρ ω log ρ ω , although ultravioletly divergent, provides a measure of the degree of entanglement of the vacuum state. In the case of a free massless real scalar field theory in one-dimensional spaces (D = 1) this entropy scales logarithmically with the size l ω of ω and the ultraviolet cut-off ǫ introduced to split apart the domain ω and its complement Ω\\ω\n\nS ω = 1 3 log l ω ǫ , ( 31\n\n)\n\nand in D = 2 dimensions it scales linearly with the perimeter R ω of ω\n\nS ω = c 2 R ω ǫ -γ ( 32\n\n)\n\nand in D > 2 dimensions as the volume of the boundary of ω\n\nS ω = c D V ω ǫ 1-D . ( 33\n\n)\n\nIn particular in three-dimensional spaces it scales with the area of the boundary of ω like in the presence of a blackhole [16, 17] . Although the coefficients of the leading terms c 2 , c D in (32) and (33) have been explicitly computed, they are not universal because they obviously depend on the choice of the UV cutoff ǫ. On the contrary, the coefficient\n\nVacuum Energy and Renormalization on the Edge 9 c 1 = 1/3 of the logarithmic term in (31) is universal and does coincide with one third of the central charge of the corresponding conformal field theory. Similarly, the finite γ term in (32) is also universal in D = 2 dimension and is related to a degree of topological entanglement [31] . It is remarkable that in D = 1 the coefficient c 1 = 1/3 is also independent of the choice of boundary condition in Ω. This in contrast with what happens for the finite size corrections to vacuum energy. The coefficient of the 1/L term is also proportional to the central charge but in that case the corresponding factor is very sensitive to the type of boundary conditions imposed at the boundary of Ω. The above results indicate that whereas the Casimir energy is closely related with the infrared properties of the conformal theory which are sensitive to the boundary conditions, the entanglement entropy is rather associated to the behaviour at the interface between ω and its complement Ω\\ω which do not depend on the choice of boundary conditions at the edge of the physical space.\n\nThe description of regular boundary conditions in terms of unitary matrices provides a very useful framework for the description of the boundary renormalization group flow and the breaking of conformal invariance due to boundary effects. Neumann conditions turn out to be the only boundary conditions which are absolutely stable under RG flow. All other boundary conditions may have some relevant perturbations which are the source of RG instabilities. However, the global structure of the flow does not permit topological transitions.\n\nThe finite size corrections to vacuum energy are very sensitive to the choice of boundary conditions which discriminate between the different fixed points of the renormalization group flow. On the contrary, the leading contribution to entanglement entropy of the vacuum is insensitive, for one-dimensional massless scalar field theories, to the change of boundary conditions. In D=2 dimensions the same property holds for the finite correction to the entanglement entropy of massless scalar theories. This fact, is very relevant for the implementation of quantum codes with topological stability [31] . However, these properties do not hold for the leading terms contributing to the entanglement entropy.\n\nWe thank E. Elizalde, J.G. Esteve, S. Odintsov and G. Sierra for interesting discussions on closely related subjects. This work is partially supported by CICYT (grant FPA2004-02948) and DGIID-DGA (grant2006-E24/2).\n\nVacuum Energy and Renormalization on the Edge 10 References [1] T. Padmanabhan, Phys. Rept. 380 (2003) 235 [2] D. Deutsch, P. Candelas, Phys. Rev. D20 (1979) 3063 [3] H. B. G. Casimir, Proc. K. ned. Akad. Wet. 51(1948) 793 [4] K. A. Milton, The Casimir Effect: Physical Manifestations of the Zero Point Energy, World Sci., Singapore (2001) [5] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. 353 (2002) 1 [6] M. Asorey, D. García-Alvarez and J.M. Muñoz-Castañeda, J. Phys. A 39 (2006) 6127 [7] I. G. Moss, Class. Quant. Grav. 6 (1989) 759 [8] S. D. Odintsov, Class. Quant. Grav., 7(1990)445 [9] I. Affleck, Nucl. Phys. (Proc. Suppl.) B58 (1997) 35 [10] J. B. Zuber and V. B. Petkova, Arxive preprint [hep-th/0103007] [11] J. Cardy, In Encyclopedia of Mathematical Physics, Eds. J.-P. Françoise, G. L. Naber and T. S.\n\nTsun, Academic Press (2006) [12] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724 [13] M. Asorey, A. Ibort and G. Marmo, Int. J. Mod. Phys. A 20 (2005) 1001 [14] A. P. Balachandran, G. Bimonte, G. Marmo and A. Simoni, Nucl. Phys. B 446 (1995) 299 [15] S. D. Glazek and K. G. Wilson, Phys. Rev. Lett. 89 (2002) 230401 [16] L. Bombelli, R.K. Koul, J. Lee and R. Sorkin, Phys. Rev. D 34 (1986) 373 [17] M. Srednicki, Phys. Rev. Lett. 71 (1993) 666 [18] I. Tsutsui and T. Fülöp, Arxive preprint [quant-ph/0312028] [19] M. Bordag, H. Falomir, E. M. Santangelo and D. V. Vassilevich, Phys.Rev. D65 (2002) 064032 [20] A. A. Saharian, Phys. Rev. D69 (2004) 085005 [21] K. Symanzik, Nucl. Phys. B190 (1981) 1 [22] E. Elizalde, J. Phys. A 36 (2003) L567 [23] N.Graham, R.L.Jaffe, V.Khemani, M.Quandt, O. Schroeder and H.Weigel, Nucl.Phys. B677 (2004) 379-404 [24] M. Asorey, J.G. Esteve and G. Sierra, In preparation (2006) [25] A. LeClair, J. M. Roman and G. Sierra, Phys. Rev. B69 (2004) 20505 [26] M.Asorey, J. Geom. Phys. 11(1993)94 [27] A. Romeo and A. A. Saharian, J. Phys. A35 (2002) 1297 [28] L. C. Albuquerque, R. M. Cavalcanti, J.Phys. A37 (2004) 7039 [29] B. Mintz, C. Farina, P. A. Maia Neto and R. B. Rodrigues, J. Phys. A39 (2006) 11325 [30] M. Asorey, unpublished (1994) [31] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404"},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction","text":"The emergence of the dark energy as one of the basic ingredients of the current standard cosmological scenario, and the absence of an even vague understanding of its possible origin, opens a window to the analysis of all possible mechanisms that generate background energy (see e.g. [1] for a review of recent proposals). The main problem is that the apparent value of the dark energy is very tiny compared with any physical energy scale. A second problem is that in a generic quantum field theory there is generation of vacuum energy and any renormalization prescription requires a fine tuning, which is not very convincing without the quantisation of the gravitational interaction.\n\nThe guess that dark energy might change with the evolution of the Universe can be understood even if dark energy is just vacuum energy. The finite corrections due to finite size of the causal Hubble domain decrease as the Universe continues to expand.\n\nThe aim of this paper is to analyse the variation of these finite size corrections under of renormalization group on the space of boundary conditions for scalar field theories in flat space, although the results are generalisable for more general backgrounds.\n\nThe dependence of the vacuum energy on the boundary conditions [2] is well known since the discovery of Casimir effect [3] (see [4, 5] and [6] and references therein for recent revisions). However, boundaries might also be considered as a source of new, although peculiar, interactions and therefore can undergo renormalization [7, 8] . The renormalization of boundary conditions modifies the critical behaviour of the theory [9, 10, 11] . In systems with boundaries or defects, the boundary RG flow induces a dynamical behaviour on the boundaries. The dynamics of D-branes in string theory emerges in this way [12] .\n\nThe renormalization group flow is analysed from a global viewpoint in the most general framework for boundary conditions of scalar field theories introduced in Ref. [13] . In particular, we consider the possible existence of topological transitions [14] induced by the renormalization of boundary conditions or cyclic orbits in the boundary RG flow [15] . The dependence of the finite size corrections to the vacuum energy and vacuum entanglement entropy [16, 17] under the boundary RG flow is analysed from a very general perspective."},{"section_type":"OTHER","section_title":"Boundary conditions in Field Theory","text":"The action which governs the dynamics of scalar field theory in a bounded domain Ω of flat space consists of two different of terms, S(φ) = S B (φ) + S b (φ). The first one\n\nS B (φ) = 1 2 dt Ω √ g d D x | φ| 2 -|∇φ| 2 -V (|φ| 2 ) ( 1\n\n)\n\nis defined in terms of the values of the fields in the bulk. The second term\n\nS b (ϕ) = 1 2 dt ∂Ω √ g ∂Ω d D-1 x | φ| 2 + 1 2 ϕ * ∂ n ϕ+ 1 2 (∂ n ϕ * )ϕ-|∇ϕ| 2 ( 2\n\n)\n\nVacuum Energy and Renormalization on the Edge 3 depends only on the values of the fields at the boundary ∂Ω † . g ∂Ω denotes the metric induced on the boundary by the bulk flat metric, and ∂ n is the normal derivative at the boundary\n\nϕ = φ| ∂Ω ∂ n ϕ = ∂ n φ| ∂Ω . ( 3\n\n)\n\nThe presence of the boundary term S b allows the generation of local classical equations of motion without requiring any specific type of boundary conditions [19, 20] . Indeed, the gradient term\n\nV = 1 2 Ω |∇φ| 2 ( 4\n\n)\n\ncan be rewritten as\n\nV = 1 2 Ω φ † ∆φ + 1 2 ∂Ω φ † ∂ n φ ( 5\n\n)\n\nwhere ∆ is the Laplace-Beltrami operator ∆ = -∂ µ ∂ µ . In the quantum theory the Laplace-Beltrami operator must have a real spectrum in order to have a selfadjoint Hamiltonian\n\nH = 1 2 √ ∆ + m 2 . ( 6\n\n)\n\nfor the free field theory (The inclusion of interactions does not changes the picture [21] ). This means that the classical fields must satisfy boundary conditions which make the operator ∆ selfadjoint. The complete set of boundary conditions which satisfy this requirement [13] are in one-to-one correspondence with the group of unitary operators of the boundary Hilbert space L 2 (∂Ω, ). For any unitary operator U ∈ L 2 (∂Ω, ), the fields satisfying the boundary condition\n\nϕ -i ∂ n ϕ = U (ϕ + i ∂ n ϕ) ( 7\n\n)\n\ndefine a domain where ∆ is a selfadjoint operator. In the case of open strings, the corresponding conformal 1+1 dimensional scalar field theories is defined on the space interval Ω = [0, 1] ⊂ IR and there is a large variety of admissible boundary conditions described by the unitary group M = U(2). The unitary matrices\n\nU D = -1 0 0 -1 U N = 1 0 0 1 U P = 0 1 1 0 ( 8\n\n)\n\ndefine Dirichlet, Neumann and periodic boundary conditions, which in string theory correspond to a string attached to a D-brane background, free open and closed string theories, respectively. For higher N-dimensional target spaces, or N-component strings, the set of boundary conditions becomes M = U(2N) which includes matrices which interpolate between one single closed string or N disconnected strings [13] . The topology change is described in this picture by a simple change of boundary conditions in L 2 (∂Ω, N ) [14].\n\n† We will assume that the boundary is regular and smooth. See e.g. [18] for the peculiarities associated to the presence of irregular boundaries\n\nVacuum Energy and Renormalization on the Edge 4 If the spectrum of eigenvalues of the unitary operator U does not include the value ±1 (i.e. ±1 / ∈ Sp U) the boundary condition (7) can be rewritten as\n\n∂ n ϕ = -i Á ± U Á ∓ U ϕ ( 9\n\n)\n\nwhich means that only the boundary values of the fields at the boundary can have an arbitrary value ϕ whereas its normal derivative is determined by U and ϕ.\n\nThe corresponding operator mappings from unitary into selfadjoint operators\n\nA ± = -i Á ± U Á ∓ U ( 10\n\n)\n\nare the celebrated Cayley transforms. The inverse Cayley transform\n\nU = Á∓ iA ± Á± iA ± ( 11\n\n)\n\nrecovers the unitary operator U from their selfadjoint Cayley transforms A ± . The condition of ∆ being selfadjoint is necessary but not sufficient to guarantee the unitarity of the corresponding quantum field theory. Indeed, in the case of free field theory the Hamiltonian (6) must be selfadjoint. This requires that the spectrum of ∆ + m 2 must be not only real but also positive which restricts the set of admissible boundary conditions to a subset M of L 2 (∂Ω, ).\n\nBecause of the existence of the boundary term in (5) the Hamiltonian H (6) is not selfadjoint if the spectrum of the unitary operator U intersects the following domain of phase factors\n\nS 1 m = {e 2αi ; -π < α ≤ π, 0 < α < π 2 -arctan m 2 , or π 2 < -α < π -arctan m 2 }.\n\nIn any other case, -m 2 is a lower bound for the spectrum of the operator ∆ and H is selfadjoint. One possible source of unitarity loss is the existence of edge estates with large negative eigenvalues of operator ∆. The consistency of the quantum field theory imposes, thus, a very stringent condition on the type of acceptable boundary conditions, even in the case of massive theories in order to prevent this type of pathological behaviour of vacuum energy.\n\nFor real scalar fields there is a further condition. U has to satisfy a CP symmetry preserving condition\n\nU † = U * , U = U T . ( 12\n\n)\n\nThe usual Neumann and Dirichlet boundary conditions U = ±Á satisfy this condition.\n\nIn general, for\n\nU = A 1 B B T A 2 ( 13\n\n)\n\nthe condition requires that\n\nA 1 = A T 1 , A 2 = A T 2 , A 1 B * + BA † 2 = 0 ( 14\n\n) BB † + A 1 A † 1 = Á, A 2 A † 2 + B T B * = Á ( 15\n\n)\n\nVacuum Energy and Renormalization on the Edge 5 In particular, the quasi-periodic condition ϕ(L) = M -1 ϕ(0), ∂ n ϕ(L) = M∂ n ϕ(0) is also compatible if M = M t = M * .\n\nIn the case of one single real massless scalar the set of compatible boundary conditions has two connected components: M 0 given by the operators of the form\n\nU β = cos β Á + i sin β σ y , and M 1 given by U α = cos α σ z + sin α σ x . ( 16\n\n)\n\nM 0 includes Neumann (β = 0) and Dirichlet (β = π) conditions; and M 1 contains the quasi-periodic boundary conditions\n\nϕ(L) = tan α 2 ϕ(0); ∂ n ϕ(L) = tan α 2 -1 ∂ n ϕ(0) ( 17\n\n)\n\nwhich include periodic (α = π 2 ) and antiperiodic (α = -π 2 ) boundary conditions."},{"section_type":"OTHER","section_title":"Boundary Conditions and Renormalization Group","text":"Since boundary conditions appear more naturally in the Schrödinger picture of field theory and the theory is plagued of ultraviolet singularities some doubts were raised about their relevance for the quantum field theory. The pioneer work of Symanzik [21] confirmed the consistence of the standard picture even in presence of bulk renormalizable interactions (see [22] for an explanation of a recent controversy [23] ).\n\nMoreover, there is a renormalization of the very boundary conditions because the boundary terms are the source of new interactions.\n\nThe renormalization group can be defined in the continuum approach by\n\nφ Λ x Λ = Λ [φ] [φ(x) -ξ Λ (x)] ( 18\n\n)\n\nby means of a fluctuating field ξ Λ with short range fluctuations of order 1 Λ . This implies that the boundary condition\n\n∂ n ϕ = A ϕ ( 19\n\n)\n\nis renormalised to\n\n∂ n ϕ Λ = A Λ ϕ Λ , ( 20\n\n) since ∂ n φ Λ x b Λ = Λ [φ]+1 [∂ n φ(x b ) -∂ n ξ Λ (x b )] = A Λ [φ]+1 φ(x b ) = A Λ φ Λ x b Λ ( 21\n\n)\n\nwith A Λ = ΛA. For more general boundary conditions the continuum renormalization group is given by\n\nΛ U † Λ ∂ Λ U Λ = 1 2 U † Λ -U Λ ( 22\n\n) or U † t ∂ t U t = 1 2 U † t -U t ( 23\n\n)\n\nVacuum Energy and Renormalization on the Edge 6 for Λ = Λ 0 e t . Fixed points correspond, therefore, to self-adjoint boundary conditions U † = U. In particular, Dirichlet and Neumann (U = ∓ Á) are renormalization group fixed points. For mixed boundary conditions the RG flows from Dirichlet (UV) toward Neumann (IR) conditions. U = e 2i arctan e -t Á.\n\n(24) Critical exponents can be identified with the eigenvalues of the matrix U c at the fixed points. Since U c is also hermitian all critical exponents are either 1 or -1 and there is no room for cyclic orbits. It is well known, however, that some quantum systems with singular boundaries and singular interactions [15, 24] exhibit cyclic renormalization group flows. Moreover, some topological field theories (e.g. Russian doll models) present a similar behaviour [25]. In scalar field theories, this phenomenon simply does not occur for regular boundaries. For the same reasons topological transitions do not occur for finite scale transformations since the flip of eigenvalues from -1 to +1 requires a change in the parameter t of the flow from -∞ to ∞ as in (24))."},{"section_type":"OTHER","section_title":"Conformal Invariance and boundary conditions","text":"In 1+1 dimensions the theory of massless scalar fields is formally conformal invariant. However, boundary conditions might break this symmetry [9, 10, 11] .\n\nConformal invariance is only preserved if the boundary conditions are stable under the boundary renormalization group flow. The fixed points can easily be identified. For a complex scalar field, besides the above mentioned fixed points, which correspond to Dirichlet, Neumann and pseudo-periodic boundary conditions and obviously are conformal invariant, there are fixed points corresponding to quasi-periodic boundary conditions (17). They also preserve the conformal symmetry.\n\nIn 1+1 dimensions this exhausts the whole set of conformal invariant boundary conditions. Any other boundary condition flows toward one of these fixed points. The most stable fixed point corresponds to Neumann conditions because all its critical exponents are +1. The most unstable is that of Dirichlet conditions since all critical exponents -1. This is compatible with the fact that the neighbourhood of Dirichlet boundary conditions is plagued of singularities Periodic, quasi-periodic and pseudo-periodic fixed points present relevant and irrelevant perturbations with critical exponents ±1, respectively. Negative values label the possible instabilities. Implications of these results for string theory are well known. Periodic boundary conditions, appear as attractors of systems with quasi-periodic and pseudo-periodic conditions which stresses the stability of closed string theory vacuum. For open strings the (stable) attractor points are standard free strings (Neumann). Any other boundary condition flow toward one of those fixed points.\n\nNotice that the absence of topological transitions in the boundary renormalization group flow is a consequence of the fact that all relevant perturbations are always associated with -1 critical exponents.\n\nVacuum Energy and Renormalization on the Edge 7 In higher dimensions (D > 1) conformal invariance requires, even in the massless case m = 0, that Neumann boundary conditions have to be modified in order to preserve conformal invariance with a term\n\n∂ n ϕ = D -1 4 D K ϕ, ( 25\n\n)\n\nproportional to the extrinsic curvature K of the boundary. In the case of singular boundaries some more interesting boundary renormalization group flows arise (see e.g. [18] for a review): fixed points and cyclic orbits of the boundary renormalization group flow can appear [15, 25, 24] and conformal invariance can be partially broken to a discrete subgroup [24]."},{"section_type":"OTHER","section_title":"Vacuum energy and boundary conditions","text":"The infrared properties of quantum field theory are very sensitive to boundary conditions [26] . In particular, physical properties of the quantum vacuum state like the vacuum energy may exhibit a very strong dependence on the type of boundary conditions. This can be explicitly shown in the simple case of a massless field defined on a finite one-dimensional interval [0, L].\n\nFor pseudoperiodic boundary conditions defined by the unitary operator\n\nU θ = cos θ σ x -sin θ σ y : ϕ(L) = e iθ ϕ(0) ( 26\n\n)\n\nthe Casimir vacuum energy (see e.g. Ref. [5] and references therein) is given by\n\nE 0 = π L 1 12 -min n∈Z θ 2π +n - 1 2 2 ( 27\n\n)\n\nThe vacuum energy dependence on θ is in this case relatively smooth. The only cuspidal point at θ = 0 corresponds to periodic boundary conditions. A completely regular behaviour is obtained for Robin boundary conditions\n\nU = e 2αi Á : ∂ n ϕ(0) = tan α ϕ(0), ∂ n ϕ(L) = tan α ϕ(L) ( 28\n\n)\n\nwhich smoothly interpolate between Dirichlet (α = π 2 ) and Neumann (α = π) conditions when α is restricted to the interval α ∈ [ π 2 , π] [27, 28, 29] . Finally, the Casimir energy for quasi-periodic boundary conditions [30]\n\nE 0 = π L 1 12 -min n∈Z α 2π + n + 1 4 2 ( 29\n\n)\n\nis also dependent on the choice of the parameter α. Two particularly interesting cases are given by\n\nα = 0, U Z = σ z ; ϕ(L) = 0, ∂ n ϕ(0) = 0 and α = π, U ′ Z = σ z ; ϕ(0) = 0, ∂ n ϕ(L) = 0\n\nwhich correspond to a Zaremba (mixed) boundary conditions: one boundary is Dirichlet and the other Neumann.\n\nVacuum Energy and Renormalization on the Edge 8 6. Vacuum entanglement entropy\n\nThe dependence of vacuum energy on boundary conditions seems to suggest that many other observables may suffer the same effect. In particular, one may wonder whether or not the entropy of the system is dependent on the type of conditions that constrain the values of the fields at the boundary. The entropy of the field theory at finite temperature scales with the volume of physical space. Only in quantum gravity or string theory the entropy can scale with the area of black hole horizon. However, in field theory it is possible ω Ω Figure 1. Information loss by integration over the fluctuations of the fields inside the domain ω to generate a mixed state from the pure vacuum state Ψ 0 by integrating out the fluctuating modes in a bounded domain ω of the physical space Ω (see Figure 1 )\n\nρ ω = ω Ψ * 0 Ψ 0 . ( 30\n\n)\n\nThe entropy of this state S ω = -T r ρ ω log ρ ω , although ultravioletly divergent, provides a measure of the degree of entanglement of the vacuum state. In the case of a free massless real scalar field theory in one-dimensional spaces (D = 1) this entropy scales logarithmically with the size l ω of ω and the ultraviolet cut-off ǫ introduced to split apart the domain ω and its complement Ω\\ω\n\nS ω = 1 3 log l ω ǫ , ( 31\n\n)\n\nand in D = 2 dimensions it scales linearly with the perimeter R ω of ω\n\nS ω = c 2 R ω ǫ -γ ( 32\n\n)\n\nand in D > 2 dimensions as the volume of the boundary of ω\n\nS ω = c D V ω ǫ 1-D . ( 33\n\n)\n\nIn particular in three-dimensional spaces it scales with the area of the boundary of ω like in the presence of a blackhole [16, 17] . Although the coefficients of the leading terms c 2 , c D in (32) and (33) have been explicitly computed, they are not universal because they obviously depend on the choice of the UV cutoff ǫ. On the contrary, the coefficient\n\nVacuum Energy and Renormalization on the Edge 9 c 1 = 1/3 of the logarithmic term in (31) is universal and does coincide with one third of the central charge of the corresponding conformal field theory. Similarly, the finite γ term in (32) is also universal in D = 2 dimension and is related to a degree of topological entanglement [31] . It is remarkable that in D = 1 the coefficient c 1 = 1/3 is also independent of the choice of boundary condition in Ω. This in contrast with what happens for the finite size corrections to vacuum energy. The coefficient of the 1/L term is also proportional to the central charge but in that case the corresponding factor is very sensitive to the type of boundary conditions imposed at the boundary of Ω. The above results indicate that whereas the Casimir energy is closely related with the infrared properties of the conformal theory which are sensitive to the boundary conditions, the entanglement entropy is rather associated to the behaviour at the interface between ω and its complement Ω\\ω which do not depend on the choice of boundary conditions at the edge of the physical space."},{"section_type":"CONCLUSION","section_title":"Conclusions","text":"The description of regular boundary conditions in terms of unitary matrices provides a very useful framework for the description of the boundary renormalization group flow and the breaking of conformal invariance due to boundary effects. Neumann conditions turn out to be the only boundary conditions which are absolutely stable under RG flow. All other boundary conditions may have some relevant perturbations which are the source of RG instabilities. However, the global structure of the flow does not permit topological transitions.\n\nThe finite size corrections to vacuum energy are very sensitive to the choice of boundary conditions which discriminate between the different fixed points of the renormalization group flow. On the contrary, the leading contribution to entanglement entropy of the vacuum is insensitive, for one-dimensional massless scalar field theories, to the change of boundary conditions. In D=2 dimensions the same property holds for the finite correction to the entanglement entropy of massless scalar theories. This fact, is very relevant for the implementation of quantum codes with topological stability [31] . However, these properties do not hold for the leading terms contributing to the entanglement entropy."},{"section_type":"OTHER","section_title":"Acknowledgements","text":"We thank E. Elizalde, J.G. Esteve, S. Odintsov and G. Sierra for interesting discussions on closely related subjects. This work is partially supported by CICYT (grant FPA2004-02948) and DGIID-DGA (grant2006-E24/2).\n\nVacuum Energy and Renormalization on the Edge 10 References [1] T. Padmanabhan, Phys. Rept. 380 (2003) 235 [2] D. Deutsch, P. Candelas, Phys. Rev. D20 (1979) 3063 [3] H. B. G. Casimir, Proc. K. ned. Akad. Wet. 51(1948) 793 [4] K. A. Milton, The Casimir Effect: Physical Manifestations of the Zero Point Energy, World Sci., Singapore (2001) [5] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. 353 (2002) 1 [6] M. Asorey, D. García-Alvarez and J.M. Muñoz-Castañeda, J. Phys. A 39 (2006) 6127 [7] I. G. Moss, Class. Quant. Grav. 6 (1989) 759 [8] S. D. Odintsov, Class. Quant. Grav., 7(1990)445 [9] I. Affleck, Nucl. Phys. (Proc. Suppl.) B58 (1997) 35 [10] J. B. Zuber and V. B. Petkova, Arxive preprint [hep-th/0103007] [11] J. Cardy, In Encyclopedia of Mathematical Physics, Eds. J.-P. Françoise, G. L. Naber and T. S.\n\nTsun, Academic Press (2006) [12] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724 [13] M. Asorey, A. Ibort and G. Marmo, Int. J. Mod. Phys. A 20 (2005) 1001 [14] A. P. Balachandran, G. Bimonte, G. Marmo and A. Simoni, Nucl. Phys. B 446 (1995) 299 [15] S. D. Glazek and K. G. Wilson, Phys. Rev. Lett. 89 (2002) 230401 [16] L. Bombelli, R.K. Koul, J. Lee and R. Sorkin, Phys. Rev. D 34 (1986) 373 [17] M. Srednicki, Phys. Rev. Lett. 71 (1993) 666 [18] I. Tsutsui and T. Fülöp, Arxive preprint [quant-ph/0312028] [19] M. Bordag, H. Falomir, E. M. Santangelo and D. V. Vassilevich, Phys.Rev. D65 (2002) 064032 [20] A. A. Saharian, Phys. Rev. D69 (2004) 085005 [21] K. Symanzik, Nucl. Phys. B190 (1981) 1 [22] E. Elizalde, J. Phys. A 36 (2003) L567 [23] N.Graham, R.L.Jaffe, V.Khemani, M.Quandt, O. Schroeder and H.Weigel, Nucl.Phys. B677 (2004) 379-404 [24] M. Asorey, J.G. Esteve and G. Sierra, In preparation (2006) [25] A. LeClair, J. M. Roman and G. Sierra, Phys. Rev. B69 (2004) 20505 [26] M.Asorey, J. Geom. Phys. 11(1993)94 [27] A. Romeo and A. A. Saharian, J. Phys. A35 (2002) 1297 [28] L. C. Albuquerque, R. M. Cavalcanti, J.Phys. A37 (2004) 7039 [29] B. Mintz, C. Farina, P. A. Maia Neto and R. B. Rodrigues, J. Phys. A39 (2006) 11325 [30] M. Asorey, unpublished (1994) [31] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404"}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"The emergence of the dark energy as one of the basic ingredients of the current standard cosmological scenario, and the absence of an even vague understanding of its possible origin, opens a window to the analysis of all possible mechanisms that generate background energy (see e.g. [1] for a review of recent proposals). The main problem is that the apparent value of the dark energy is very tiny compared with any physical energy scale. A second problem is that in a generic quantum field theory there is generation of vacuum energy and any renormalization prescription requires a fine tuning, which is not very convincing without the quantisation of the gravitational interaction.\\n\\nThe guess that dark energy might change with the evolution of the Universe can be understood even if dark energy is just vacuum energy. The finite corrections due to finite size of the causal Hubble domain decrease as the Universe continues to expand.\\n\\nThe aim of this paper is to analyse the variation of these finite size corrections under of renormalization group on the space of boundary conditions for scalar field theories in flat space, although the results are generalisable for more general backgrounds.\\n\\nThe dependence of the vacuum energy on the boundary conditions [2] is well known since the discovery of Casimir effect [3] (see [4, 5] and [6] and references therein for recent revisions). However, boundaries might also be considered as a source of new, although peculiar, interactions and therefore can undergo renormalization [7, 8] . The renormalization of boundary conditions modifies the critical behaviour of the theory [9, 10, 11] . In systems with boundaries or defects, the boundary RG flow induces a dynamical behaviour on the boundaries. The dynamics of D-branes in string theory emerges in this way [12] .\\n\\nThe renormalization group flow is analysed from a global viewpoint in the most general framework for boundary conditions of scalar field theories introduced in Ref. [13] . In particular, we consider the possible existence of topological transitions [14] induced by the renormalization of boundary conditions or cyclic orbits in the boundary RG flow [15] . The dependence of the finite size corrections to the vacuum energy and vacuum entanglement entropy [16, 17] under the boundary RG flow is analysed from a very general perspective.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Boundary conditions in Field Theory\",\n \"text\": \"The action which governs the dynamics of scalar field theory in a bounded domain Ω of flat space consists of two different of terms, S(φ) = S B (φ) + S b (φ). The first one\\n\\nS B (φ) = 1 2 dt Ω √ g d D x | φ| 2 -|∇φ| 2 -V (|φ| 2 ) ( 1\\n\\n)\\n\\nis defined in terms of the values of the fields in the bulk. The second term\\n\\nS b (ϕ) = 1 2 dt ∂Ω √ g ∂Ω d D-1 x | φ| 2 + 1 2 ϕ * ∂ n ϕ+ 1 2 (∂ n ϕ * )ϕ-|∇ϕ| 2 ( 2\\n\\n)\\n\\nVacuum Energy and Renormalization on the Edge 3 depends only on the values of the fields at the boundary ∂Ω † . g ∂Ω denotes the metric induced on the boundary by the bulk flat metric, and ∂ n is the normal derivative at the boundary\\n\\nϕ = φ| ∂Ω ∂ n ϕ = ∂ n φ| ∂Ω . ( 3\\n\\n)\\n\\nThe presence of the boundary term S b allows the generation of local classical equations of motion without requiring any specific type of boundary conditions [19, 20] . Indeed, the gradient term\\n\\nV = 1 2 Ω |∇φ| 2 ( 4\\n\\n)\\n\\ncan be rewritten as\\n\\nV = 1 2 Ω φ † ∆φ + 1 2 ∂Ω φ † ∂ n φ ( 5\\n\\n)\\n\\nwhere ∆ is the Laplace-Beltrami operator ∆ = -∂ µ ∂ µ . In the quantum theory the Laplace-Beltrami operator must have a real spectrum in order to have a selfadjoint Hamiltonian\\n\\nH = 1 2 √ ∆ + m 2 . ( 6\\n\\n)\\n\\nfor the free field theory (The inclusion of interactions does not changes the picture [21] ). This means that the classical fields must satisfy boundary conditions which make the operator ∆ selfadjoint. The complete set of boundary conditions which satisfy this requirement [13] are in one-to-one correspondence with the group of unitary operators of the boundary Hilbert space L 2 (∂Ω, ). For any unitary operator U ∈ L 2 (∂Ω, ), the fields satisfying the boundary condition\\n\\nϕ -i ∂ n ϕ = U (ϕ + i ∂ n ϕ) ( 7\\n\\n)\\n\\ndefine a domain where ∆ is a selfadjoint operator. In the case of open strings, the corresponding conformal 1+1 dimensional scalar field theories is defined on the space interval Ω = [0, 1] ⊂ IR and there is a large variety of admissible boundary conditions described by the unitary group M = U(2). The unitary matrices\\n\\nU D = -1 0 0 -1 U N = 1 0 0 1 U P = 0 1 1 0 ( 8\\n\\n)\\n\\ndefine Dirichlet, Neumann and periodic boundary conditions, which in string theory correspond to a string attached to a D-brane background, free open and closed string theories, respectively. For higher N-dimensional target spaces, or N-component strings, the set of boundary conditions becomes M = U(2N) which includes matrices which interpolate between one single closed string or N disconnected strings [13] . The topology change is described in this picture by a simple change of boundary conditions in L 2 (∂Ω, N ) [14].\\n\\n† We will assume that the boundary is regular and smooth. See e.g. [18] for the peculiarities associated to the presence of irregular boundaries\\n\\nVacuum Energy and Renormalization on the Edge 4 If the spectrum of eigenvalues of the unitary operator U does not include the value ±1 (i.e. ±1 / ∈ Sp U) the boundary condition (7) can be rewritten as\\n\\n∂ n ϕ = -i Á ± U Á ∓ U ϕ ( 9\\n\\n)\\n\\nwhich means that only the boundary values of the fields at the boundary can have an arbitrary value ϕ whereas its normal derivative is determined by U and ϕ.\\n\\nThe corresponding operator mappings from unitary into selfadjoint operators\\n\\nA ± = -i Á ± U Á ∓ U ( 10\\n\\n)\\n\\nare the celebrated Cayley transforms. The inverse Cayley transform\\n\\nU = Á∓ iA ± Á± iA ± ( 11\\n\\n)\\n\\nrecovers the unitary operator U from their selfadjoint Cayley transforms A ± . The condition of ∆ being selfadjoint is necessary but not sufficient to guarantee the unitarity of the corresponding quantum field theory. Indeed, in the case of free field theory the Hamiltonian (6) must be selfadjoint. This requires that the spectrum of ∆ + m 2 must be not only real but also positive which restricts the set of admissible boundary conditions to a subset M of L 2 (∂Ω, ).\\n\\nBecause of the existence of the boundary term in (5) the Hamiltonian H (6) is not selfadjoint if the spectrum of the unitary operator U intersects the following domain of phase factors\\n\\nS 1 m = {e 2αi ; -π < α ≤ π, 0 < α < π 2 -arctan m 2 , or π 2 < -α < π -arctan m 2 }.\\n\\nIn any other case, -m 2 is a lower bound for the spectrum of the operator ∆ and H is selfadjoint. One possible source of unitarity loss is the existence of edge estates with large negative eigenvalues of operator ∆. The consistency of the quantum field theory imposes, thus, a very stringent condition on the type of acceptable boundary conditions, even in the case of massive theories in order to prevent this type of pathological behaviour of vacuum energy.\\n\\nFor real scalar fields there is a further condition. U has to satisfy a CP symmetry preserving condition\\n\\nU † = U * , U = U T . ( 12\\n\\n)\\n\\nThe usual Neumann and Dirichlet boundary conditions U = ±Á satisfy this condition.\\n\\nIn general, for\\n\\nU = A 1 B B T A 2 ( 13\\n\\n)\\n\\nthe condition requires that\\n\\nA 1 = A T 1 , A 2 = A T 2 , A 1 B * + BA † 2 = 0 ( 14\\n\\n) BB † + A 1 A † 1 = Á, A 2 A † 2 + B T B * = Á ( 15\\n\\n)\\n\\nVacuum Energy and Renormalization on the Edge 5 In particular, the quasi-periodic condition ϕ(L) = M -1 ϕ(0), ∂ n ϕ(L) = M∂ n ϕ(0) is also compatible if M = M t = M * .\\n\\nIn the case of one single real massless scalar the set of compatible boundary conditions has two connected components: M 0 given by the operators of the form\\n\\nU β = cos β Á + i sin β σ y , and M 1 given by U α = cos α σ z + sin α σ x . ( 16\\n\\n)\\n\\nM 0 includes Neumann (β = 0) and Dirichlet (β = π) conditions; and M 1 contains the quasi-periodic boundary conditions\\n\\nϕ(L) = tan α 2 ϕ(0); ∂ n ϕ(L) = tan α 2 -1 ∂ n ϕ(0) ( 17\\n\\n)\\n\\nwhich include periodic (α = π 2 ) and antiperiodic (α = -π 2 ) boundary conditions.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Boundary Conditions and Renormalization Group\",\n \"text\": \"Since boundary conditions appear more naturally in the Schrödinger picture of field theory and the theory is plagued of ultraviolet singularities some doubts were raised about their relevance for the quantum field theory. The pioneer work of Symanzik [21] confirmed the consistence of the standard picture even in presence of bulk renormalizable interactions (see [22] for an explanation of a recent controversy [23] ).\\n\\nMoreover, there is a renormalization of the very boundary conditions because the boundary terms are the source of new interactions.\\n\\nThe renormalization group can be defined in the continuum approach by\\n\\nφ Λ x Λ = Λ [φ] [φ(x) -ξ Λ (x)] ( 18\\n\\n)\\n\\nby means of a fluctuating field ξ Λ with short range fluctuations of order 1 Λ . This implies that the boundary condition\\n\\n∂ n ϕ = A ϕ ( 19\\n\\n)\\n\\nis renormalised to\\n\\n∂ n ϕ Λ = A Λ ϕ Λ , ( 20\\n\\n) since ∂ n φ Λ x b Λ = Λ [φ]+1 [∂ n φ(x b ) -∂ n ξ Λ (x b )] = A Λ [φ]+1 φ(x b ) = A Λ φ Λ x b Λ ( 21\\n\\n)\\n\\nwith A Λ = ΛA. For more general boundary conditions the continuum renormalization group is given by\\n\\nΛ U † Λ ∂ Λ U Λ = 1 2 U † Λ -U Λ ( 22\\n\\n) or U † t ∂ t U t = 1 2 U † t -U t ( 23\\n\\n)\\n\\nVacuum Energy and Renormalization on the Edge 6 for Λ = Λ 0 e t . Fixed points correspond, therefore, to self-adjoint boundary conditions U † = U. In particular, Dirichlet and Neumann (U = ∓ Á) are renormalization group fixed points. For mixed boundary conditions the RG flows from Dirichlet (UV) toward Neumann (IR) conditions. U = e 2i arctan e -t Á.\\n\\n(24) Critical exponents can be identified with the eigenvalues of the matrix U c at the fixed points. Since U c is also hermitian all critical exponents are either 1 or -1 and there is no room for cyclic orbits. It is well known, however, that some quantum systems with singular boundaries and singular interactions [15, 24] exhibit cyclic renormalization group flows. Moreover, some topological field theories (e.g. Russian doll models) present a similar behaviour [25]. In scalar field theories, this phenomenon simply does not occur for regular boundaries. For the same reasons topological transitions do not occur for finite scale transformations since the flip of eigenvalues from -1 to +1 requires a change in the parameter t of the flow from -∞ to ∞ as in (24)).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Conformal Invariance and boundary conditions\",\n \"text\": \"In 1+1 dimensions the theory of massless scalar fields is formally conformal invariant. However, boundary conditions might break this symmetry [9, 10, 11] .\\n\\nConformal invariance is only preserved if the boundary conditions are stable under the boundary renormalization group flow. The fixed points can easily be identified. For a complex scalar field, besides the above mentioned fixed points, which correspond to Dirichlet, Neumann and pseudo-periodic boundary conditions and obviously are conformal invariant, there are fixed points corresponding to quasi-periodic boundary conditions (17). They also preserve the conformal symmetry.\\n\\nIn 1+1 dimensions this exhausts the whole set of conformal invariant boundary conditions. Any other boundary condition flows toward one of these fixed points. The most stable fixed point corresponds to Neumann conditions because all its critical exponents are +1. The most unstable is that of Dirichlet conditions since all critical exponents -1. This is compatible with the fact that the neighbourhood of Dirichlet boundary conditions is plagued of singularities Periodic, quasi-periodic and pseudo-periodic fixed points present relevant and irrelevant perturbations with critical exponents ±1, respectively. Negative values label the possible instabilities. Implications of these results for string theory are well known. Periodic boundary conditions, appear as attractors of systems with quasi-periodic and pseudo-periodic conditions which stresses the stability of closed string theory vacuum. For open strings the (stable) attractor points are standard free strings (Neumann). Any other boundary condition flow toward one of those fixed points.\\n\\nNotice that the absence of topological transitions in the boundary renormalization group flow is a consequence of the fact that all relevant perturbations are always associated with -1 critical exponents.\\n\\nVacuum Energy and Renormalization on the Edge 7 In higher dimensions (D > 1) conformal invariance requires, even in the massless case m = 0, that Neumann boundary conditions have to be modified in order to preserve conformal invariance with a term\\n\\n∂ n ϕ = D -1 4 D K ϕ, ( 25\\n\\n)\\n\\nproportional to the extrinsic curvature K of the boundary. In the case of singular boundaries some more interesting boundary renormalization group flows arise (see e.g. [18] for a review): fixed points and cyclic orbits of the boundary renormalization group flow can appear [15, 25, 24] and conformal invariance can be partially broken to a discrete subgroup [24].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Vacuum energy and boundary conditions\",\n \"text\": \"The infrared properties of quantum field theory are very sensitive to boundary conditions [26] . In particular, physical properties of the quantum vacuum state like the vacuum energy may exhibit a very strong dependence on the type of boundary conditions. This can be explicitly shown in the simple case of a massless field defined on a finite one-dimensional interval [0, L].\\n\\nFor pseudoperiodic boundary conditions defined by the unitary operator\\n\\nU θ = cos θ σ x -sin θ σ y : ϕ(L) = e iθ ϕ(0) ( 26\\n\\n)\\n\\nthe Casimir vacuum energy (see e.g. Ref. [5] and references therein) is given by\\n\\nE 0 = π L 1 12 -min n∈Z θ 2π +n - 1 2 2 ( 27\\n\\n)\\n\\nThe vacuum energy dependence on θ is in this case relatively smooth. The only cuspidal point at θ = 0 corresponds to periodic boundary conditions. A completely regular behaviour is obtained for Robin boundary conditions\\n\\nU = e 2αi Á : ∂ n ϕ(0) = tan α ϕ(0), ∂ n ϕ(L) = tan α ϕ(L) ( 28\\n\\n)\\n\\nwhich smoothly interpolate between Dirichlet (α = π 2 ) and Neumann (α = π) conditions when α is restricted to the interval α ∈ [ π 2 , π] [27, 28, 29] . Finally, the Casimir energy for quasi-periodic boundary conditions [30]\\n\\nE 0 = π L 1 12 -min n∈Z α 2π + n + 1 4 2 ( 29\\n\\n)\\n\\nis also dependent on the choice of the parameter α. Two particularly interesting cases are given by\\n\\nα = 0, U Z = σ z ; ϕ(L) = 0, ∂ n ϕ(0) = 0 and α = π, U ′ Z = σ z ; ϕ(0) = 0, ∂ n ϕ(L) = 0\\n\\nwhich correspond to a Zaremba (mixed) boundary conditions: one boundary is Dirichlet and the other Neumann.\\n\\nVacuum Energy and Renormalization on the Edge 8 6. Vacuum entanglement entropy\\n\\nThe dependence of vacuum energy on boundary conditions seems to suggest that many other observables may suffer the same effect. In particular, one may wonder whether or not the entropy of the system is dependent on the type of conditions that constrain the values of the fields at the boundary. The entropy of the field theory at finite temperature scales with the volume of physical space. Only in quantum gravity or string theory the entropy can scale with the area of black hole horizon. However, in field theory it is possible ω Ω Figure 1. Information loss by integration over the fluctuations of the fields inside the domain ω to generate a mixed state from the pure vacuum state Ψ 0 by integrating out the fluctuating modes in a bounded domain ω of the physical space Ω (see Figure 1 )\\n\\nρ ω = ω Ψ * 0 Ψ 0 . ( 30\\n\\n)\\n\\nThe entropy of this state S ω = -T r ρ ω log ρ ω , although ultravioletly divergent, provides a measure of the degree of entanglement of the vacuum state. In the case of a free massless real scalar field theory in one-dimensional spaces (D = 1) this entropy scales logarithmically with the size l ω of ω and the ultraviolet cut-off ǫ introduced to split apart the domain ω and its complement Ω\\\\ω\\n\\nS ω = 1 3 log l ω ǫ , ( 31\\n\\n)\\n\\nand in D = 2 dimensions it scales linearly with the perimeter R ω of ω\\n\\nS ω = c 2 R ω ǫ -γ ( 32\\n\\n)\\n\\nand in D > 2 dimensions as the volume of the boundary of ω\\n\\nS ω = c D V ω ǫ 1-D . ( 33\\n\\n)\\n\\nIn particular in three-dimensional spaces it scales with the area of the boundary of ω like in the presence of a blackhole [16, 17] . Although the coefficients of the leading terms c 2 , c D in (32) and (33) have been explicitly computed, they are not universal because they obviously depend on the choice of the UV cutoff ǫ. On the contrary, the coefficient\\n\\nVacuum Energy and Renormalization on the Edge 9 c 1 = 1/3 of the logarithmic term in (31) is universal and does coincide with one third of the central charge of the corresponding conformal field theory. Similarly, the finite γ term in (32) is also universal in D = 2 dimension and is related to a degree of topological entanglement [31] . It is remarkable that in D = 1 the coefficient c 1 = 1/3 is also independent of the choice of boundary condition in Ω. This in contrast with what happens for the finite size corrections to vacuum energy. The coefficient of the 1/L term is also proportional to the central charge but in that case the corresponding factor is very sensitive to the type of boundary conditions imposed at the boundary of Ω. The above results indicate that whereas the Casimir energy is closely related with the infrared properties of the conformal theory which are sensitive to the boundary conditions, the entanglement entropy is rather associated to the behaviour at the interface between ω and its complement Ω\\\\ω which do not depend on the choice of boundary conditions at the edge of the physical space.\"\n },\n {\n \"section_type\": \"CONCLUSION\",\n \"section_title\": \"Conclusions\",\n \"text\": \"The description of regular boundary conditions in terms of unitary matrices provides a very useful framework for the description of the boundary renormalization group flow and the breaking of conformal invariance due to boundary effects. Neumann conditions turn out to be the only boundary conditions which are absolutely stable under RG flow. All other boundary conditions may have some relevant perturbations which are the source of RG instabilities. However, the global structure of the flow does not permit topological transitions.\\n\\nThe finite size corrections to vacuum energy are very sensitive to the choice of boundary conditions which discriminate between the different fixed points of the renormalization group flow. On the contrary, the leading contribution to entanglement entropy of the vacuum is insensitive, for one-dimensional massless scalar field theories, to the change of boundary conditions. In D=2 dimensions the same property holds for the finite correction to the entanglement entropy of massless scalar theories. This fact, is very relevant for the implementation of quantum codes with topological stability [31] . However, these properties do not hold for the leading terms contributing to the entanglement entropy.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgements\",\n \"text\": \"We thank E. Elizalde, J.G. Esteve, S. Odintsov and G. Sierra for interesting discussions on closely related subjects. This work is partially supported by CICYT (grant FPA2004-02948) and DGIID-DGA (grant2006-E24/2).\\n\\nVacuum Energy and Renormalization on the Edge 10 References [1] T. Padmanabhan, Phys. Rept. 380 (2003) 235 [2] D. Deutsch, P. Candelas, Phys. Rev. D20 (1979) 3063 [3] H. B. G. Casimir, Proc. K. ned. Akad. Wet. 51(1948) 793 [4] K. A. Milton, The Casimir Effect: Physical Manifestations of the Zero Point Energy, World Sci., Singapore (2001) [5] M. Bordag, U. Mohideen, V. M. Mostepanenko, Phys. Rep. 353 (2002) 1 [6] M. Asorey, D. García-Alvarez and J.M. Muñoz-Castañeda, J. Phys. A 39 (2006) 6127 [7] I. G. Moss, Class. Quant. Grav. 6 (1989) 759 [8] S. D. Odintsov, Class. Quant. Grav., 7(1990)445 [9] I. Affleck, Nucl. Phys. (Proc. Suppl.) B58 (1997) 35 [10] J. B. Zuber and V. B. Petkova, Arxive preprint [hep-th/0103007] [11] J. Cardy, In Encyclopedia of Mathematical Physics, Eds. J.-P. Françoise, G. L. Naber and T. S.\\n\\nTsun, Academic Press (2006) [12] J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724 [13] M. Asorey, A. Ibort and G. Marmo, Int. J. Mod. Phys. A 20 (2005) 1001 [14] A. P. Balachandran, G. Bimonte, G. Marmo and A. Simoni, Nucl. Phys. B 446 (1995) 299 [15] S. D. Glazek and K. G. Wilson, Phys. Rev. Lett. 89 (2002) 230401 [16] L. Bombelli, R.K. Koul, J. Lee and R. Sorkin, Phys. Rev. D 34 (1986) 373 [17] M. Srednicki, Phys. Rev. Lett. 71 (1993) 666 [18] I. Tsutsui and T. Fülöp, Arxive preprint [quant-ph/0312028] [19] M. Bordag, H. Falomir, E. M. Santangelo and D. V. Vassilevich, Phys.Rev. D65 (2002) 064032 [20] A. A. Saharian, Phys. Rev. D69 (2004) 085005 [21] K. Symanzik, Nucl. Phys. B190 (1981) 1 [22] E. Elizalde, J. Phys. A 36 (2003) L567 [23] N.Graham, R.L.Jaffe, V.Khemani, M.Quandt, O. Schroeder and H.Weigel, Nucl.Phys. B677 (2004) 379-404 [24] M. Asorey, J.G. Esteve and G. Sierra, In preparation (2006) [25] A. LeClair, J. M. Roman and G. Sierra, Phys. Rev. B69 (2004) 20505 [26] M.Asorey, J. Geom. Phys. 11(1993)94 [27] A. Romeo and A. A. Saharian, J. Phys. A35 (2002) 1297 [28] L. C. Albuquerque, R. M. Cavalcanti, J.Phys. A37 (2004) 7039 [29] B. Mintz, C. Farina, P. A. Maia Neto and R. B. Rodrigues, J. Phys. A39 (2006) 11325 [30] M. Asorey, unpublished (1994) [31] A. Kitaev and J. Preskill, Phys. Rev. Lett. 96 (2006) 110404\"\n }\n]","isConversational":false}}},{"rowIdx":191,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1085"},"arxiv_id":{"kind":"string","value":"0704.1085"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1016/j.physletb.2007.06.056"},"content_hash":{"kind":"string","value":"8bd7acc2c1e6a1228e15ea29ec0203416a3e3b53fc075678499862e4db6f75e6"},"title":{"kind":"string","value":"Gauge Mediation in String Theory"},"abstract":{"kind":"string","value":"We show that a large class of phenomenologically viable models for gauge\nmediation of supersymmetry breaking based on meta-stable vacua can be realized\nin local Calabi-Yau compactifications of string theory."},"authors":{"kind":"list like","value":["Teruhiko Kawano","Hirosi Ooguri","Yutaka Ookouchi"],"string":"[\n \"Teruhiko Kawano\",\n \"Hirosi Ooguri\",\n \"Yutaka Ookouchi\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th","hep-ph"],"string":"[\n \"hep-th\",\n \"hep-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The use of meta-stable vacua in supersymmetric model building has attracted much attention lately, especially after the discovery [1] that generic supersymmetric field theories in four dimensions such as the supersymmetric QCD with massive flavors have metastable vacua with broken supersymmetry. In [2] , realistic models of direct mediation were constructed using superpotentials without U (1) R symmetry. Though explicit breaking of the U (1) R symmetry generates meta-stable vacua, there is a range of parameters where one can make them sufficiently long lived, while satisfying the phenomenological constraints on the masses of the gauginos, the gravitino, and the scalars without artificially elaborate constructions. The models can also avoid producing Landau poles in standard model gauge interactions below the unification scale. Recently beautiful realizations of these models in string theory, including a natural mechanism to generate small parameters of these models, were found in [3] .\n\nGauge mediation models were also constructed using meta-stable vacua with similar phenomenological benefits [4, 5] . Related ideas have been explored in [6] [7] [8] [9] [10] . Accepting the possibility that our universe may be in a meta-stable state allows us to circumvent the theoretical constraints due to the Nelson-Seiberg theorem on R-symmetry [11] and the Witten index [12] and gives us greater flexibility in model building, as emphasized in [13] among other recent papers.\n\nAmong the models constructed recently based on meta-stable vacua, the ones discussed in [4] are particularly simple. In this paper, we will show that they have ultra-violet completions in supersymmetric quiver gauge theories which can be realized in string compactifications. Moreover, our construction can be naturally generalized to a large class of quiver gauge theories, providing a basis for the speculation in [4] that \"gauge mediation may be a rather generic phenomenon in the landscape of possible supersymmetric theories.\" In this paper, we will demonstrate the idea by explicitly working out one example: a model based on type IIB superstrings compactified on the A 4 -fibered geometry [14] . We will also give an outline of generalizations of this construction to a large class of quiver gauge theories. Detailed analysis of meta-stable vacua in these models will be given in a separate paper [15]. 1\n\nThe model we will consider in this paper is realized in string theory compactified on the local Calabi-Yau manifold described by the equation,\n\nx 2 + y 2 + 5 i=1 (z + t i (w)) = 0, 5 i=1 t i (w) = 0, t i (w) -t i+1 (w) = µ i (w -x i ), (x, y, z, w) ∈ C 4 .\n\n(2.1) Since t i 's are functions of w, this gives the A 4 singularity fibered over w ∈ C. In particular, there exist four two-cycles S 2 on which D branes can be wrapped. The low energy limit of D5 branes wrapping the two-cycles S 2 and extending along the four uncompactified dimensions is the A 4 quiver gauge theory with the gauge group 2, 3, 4 for the four gauge group factors and the bi-fundamental chiral multiplets (Q 12 , Q 21 ), (Q 23 , Q 32 ), and (Q 34 , Q 43 ). This quiver gauge theory can also be realized on intersecting brane configuration with NS5 and D4 branes, as expected from the T-duality between the A n singularity and NS5 branes [16] .\n\nU (N 1 ) × U (N 2 ) × U (N 3 ) × U (N 4 ) with the adjoint chiral multiplets X i=1,\n\nN 1 N 2 N 3 N 4 Q 12,21 Q 23,32 Q 34,43 X 1 X 2 X3 X 4 Fig. 1: A 4 quiver diagram\n\nFrom the Calabi-Yau singularity (2.1), one can read off the superpotential of the quiver theory as [17,18]\n\nW A 4 = 3 i=1 tr (Q i+1,i X i Q i,i+1 -Q i,i+1 X i+1 Q i+1,i ) + 4 i=1 tr µ i 2 (X i -x i ) 2 . (2.2)\n\nNote that the dimensionful parameters µ i and x i are the moduli of the Calabi-Yau manifold given by (2.1), namely they are closed string moduli. The dynamical scales Λ i=1,•••,4 of the four gauge group factors are also closed string moduli, related to the sizes of the S 2 's.\n\nThese closed string moduli are frozen and can be regarded as parameters of the low energy theory. Let us suppose that µ i are sufficiently larger than Λ i so that we can integrate out all the adjoints X i to obtain the effective superpotential\n\nW eff = 3 i=1 m i tr Q i i+1 Q i+1 i - 3 i=1 1 μi tr (Q i i+1 Q i+1 i ) 2 + 1 µ 2 tr Q 21 Q 12 Q 23 Q 32 + 1 µ 3 tr Q 32 Q 23 Q 34 Q 43 , (2.3) 2\n\nwhere\n\nm i = c i -c i+1 , μi = 2µ i µ i+1 µ i + µ i+1 (i = 1, 2, 3).\n\nThis quiver gauge theory can be used as a gauge mediation model as follows. We identify the bi-fundamentals (Q 34 , Q 43 ) as messenger fields. One way to incorporate the standard model sector would be to identify a subgroup of the U (N 4 ) gauge group with the standard model gauge group or a GUT gauge group. Alternatively, we can replace the 4th node of the quiver diagram of Fig. 1 carrying the U (N 4 ) gauge group with a string theory construction of the standard model. For example, if the standard model is realized on intersecting branes, messengers can be open strings connecting the 3rd node carrying the U (N 3 ) gauge group to the standard model branes. 1 In the following, we will denote the bi-fundamental fields (Q 34 , Q 43 ) as (f, f ) to distinguish them from the rest of the quiver gauge theory and to emphasize their role as the messengers.\n\nThe rest of the quiver gauge theory is treated as a hidden sector, where supersymmetry is broken dynamically. To use the result of [1], let us assume that the ranks of the gauge group factors satisfy\n\nN 2 + 1 ≤ N 1 + N 3 < 3 2 N 2 (2.4) and that Λ 1 , Λ 3 , Λ 4 ≪ Λ 2 ≪ µ i .\n\nIn this case, one can identify the gauge group SU (N c ) of the model of [1] with SU (N 2 ) ⊂ U (N 2 ) of the quiver theory. Since the metastable vacuum can be found near the origin of\n\nthe meson fields M 11 ∼ Q 12 Q 21 , M 33 ∼ Q 32 Q 23 , the terms tr (Q 12 Q 21 ) 2 , tr (Q 32 Q 23 ) 2 and tr (Q 12 Q 21 Q 32 Q 23 ) in the superpotential (2.\n\n3) are irrelevant in our discussion below, if the masses µ i of the adjoints satisfy the following bounds [4,5],\n\nΛ 2 2 μ1,2 , Λ 2 2 µ 2 ≤ min 1 4π m 1,2 Λ 2 , 1 16π 2 m 3 µ 3 Λ 2 . (2.5)\n\nIn this range of the parameters, the hidden sector and its interaction with the messenger sector is described by the superpotential,\n\nW = mtr Q 12 Q 21 + mtr Q 32 Q 23 + 1 µ 3 tr Q 32 Q 23 f f + m 3 tr f f - 1 μ3 tr f f 2 . (2.6)\n\n1 In this case, we can still use the effective potential (2.3) to describe the interaction of the messengers and the hidden sector, but we should set 1/μ 4 = 1/(2µ 3 ) since we do not have the adjoint field X 4 .\n\nHere, we set the mass parameters m 1 = m 2 = m, for simplicity. Consider the case when N 1 = N 2 = 3 and N 3 = 1 so that the Landau pole problem can easily be avoided. The resulting model is a variant of the models proposed in [4] . The model [4] has the global symmetry U (4) × U (1) mess , where U (4) is the flavor symmetry of the ISS model and U (1) mess acts on the messengers (f, f). The meta-stable vacuum spontaneously breaks the U (4) symmetry, giving rise to Nambu-Goldstone bosons, when m 1 ≃ m 2 . In our model, the would-be Nambu-Goldstone bosons are eaten by the gauge symmetry. This difference is not important in the low energy analysis of supersymmetry breaking effects.\n\nLet us discuss phenomenological constraints on the parameters in (2.6). We will focus on the following part of the superpotential (2.6),\n\nW mess = Λ 2 µ 3 M 33 f f + m 3 f f , (2.7)\n\nwhere M 33 = Q 32 Q 23 /Λ 2 is neutral under the U (N 3 ) = U (1) gauge group. We have dropped the irrelevant quartic term (f f ) 2 because the messengers (f, f) are weakly interacting at energies above the electroweak scale, if the mass parameter μ3 is large enough.\n\nThe F -component of the meson superfield M 33 develops the vacuum expectation value and breaks supersymmetry [1]. The supersymmetric mass and the soft supersymmetry breaking mass of the messenger fields (f, f) are then given by\n\nW mess ≃ m 3 + θ 2 mΛ 2 2 µ 3 f f . (2.8)\n\nFollowing the analysis in [4, 5] , we find that all the phenomenological requirements for the messenger sector can be satisfied, for example, in the following range of parameters,\n\nΛ 2 ≃ 10 11 GeV, m ≃ 10 8 GeV, m 3 ≃ 10 7 GeV, µ 1 ≥ µ 2 ≥ 10 13 GeV, µ 3 ≃ 10 18 GeV. (2.9)\n\nWe found that both the messenger sector and the hidden sector of the models proposed in [4] can be realized in the A 4 quiver gauge theory. This construction naturally suggests the following generalization. Consider a quiver diagram which can be separated into two disjoint diagrams Γ 1 and Γ 2 by cutting at one node, which we denote by a. If the scale Λ a associated to the gauge group on the a-node is sufficiently low, and if superpotential 4 interactions between them are small, we have effectively two separate quiver gauge theories for phenomena much above the scale Λ a , one associated to Γ 1 and another associated to Γ 2 , which are weakly interacting with each other through the a-node. If supersymmetry is broken in the sector Γ 1 , it can be communicated to the sector Γ 2 by the gauge mediation mechanism. The beauty of the quiver gauge theory construction is that, because of the presence of bi-fundamental and adjoint fields on links and nodes, an effective superpotential of the form (2.8) is naturally generated when supersymmetry is broken in a part of the diagram connected to the a-node.\n\nIt follows trivially that any quiver theory that is vector like with adjustable mass terms has meta-stable supersymmetry breaking vacua in some range of its parameter. All one has to do is to identify a part of the diagram where supersymmetry can be broken using a known mechanism, for example as in [1] or its variant [19], and to have its effect communicated to the rest of the diagram by messengers. One can also consider the scenario where the quiver theory associated to a sub-diagram Γ 2 has a supersymmetric vacuum with dynamically generated small scales, which can be used to set parameters of the theory associated to another sub-diagram Γ 1 , where supersymmetry is broken. The supersymmetry breaking effect can then be communicated back to the sub-diagram Γ 2 . This would give a string theory realization of the idea of [20] . These and other mechanisms of supersymmetry breaking will be explored further in [15] .\n\nThese supersymmetry breaking quiver gauge theories can be coupled to the messenger sector. In fact, as in the case of the A 4 model discussed in the previous section, the messenger sector itself can be included in quiver theories. If the messenger sector is attached at the end of the quiver diagram, the effective low energy superpotential always takes the form (2.3). Thus, one can see that the models in [4] and their generalizations are robust and naturally appear in this large class of string compactifications.\n\nWe thank D. Berenstein, M. Dine, R. Kitano, J. Marsano, C. S. Park, N. Seiberg, M. Shigemori, and T. Watari for discussions. H.O. thanks the hospitality of the high energy theory group at the University of Tokyo at Hongo. 5 H.O. and Y.O. are supported in part by the DOE grant DE-FG03-92-ER40701. The research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st Century COE Program at the University of Tokyo. Y.O. is also supported in part by the JSPS Fellowship for Research Abroad. The research of T.K. was supported in part by the Grants-in-Aid (#16740133) and (#16081206) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. 6 References [1] K. Intriligator, N. Seiberg and D. Shih, \"Dynamical SUSY breaking in meta-stable vacua,\" JHEP 0604, 021 (2006), arXiv:hep-th/0602239. [2] R. Kitano, H. Ooguri and Y. Ookouchi, \"Direct mediation of meta-stable supersymmetry breaking,\" Phys. Rev. D 75, 045022 (2007), arXiv:hep-ph/0612139. [3] R. Argurio, M. Bertolini, S. Franco and S. Kachru, \"Metastable vacua and D-branes at the conifold,\" arXiv:hep-th/0703236. [4] H. Murayama and Y. Nomura, \"Gauge mediation simplified,\" arXiv:hep-ph/0612186. [5] H. Murayama and Y. Nomura, \"Simple scheme for gauge mediation,\" arXiv:hepph/0701231. [6] A. Amariti, L. Girardello and A. Mariotti, \"On meta-stable SQCD with adjoint matter and gauge mediation,\" arXiv:hep-th/0701121. [7] M. Dine and J. Mason, \"Gauge mediation in metastable vacua,\" arXiv:hep-ph/0611312. [8] C. Csaki, Y. Shirman and J. Terning, \"A simple model of low-scale direct gauge mediation,\" arXiv:hep-ph/0612241. [9] O. Aharony and N. Seiberg, \"Naturalized and simplified gauge mediation,\" JHEP 0702, 054 (2007), arXiv:hep-ph/0612308. [10] D. Shih, \"Spontaneous R-symmetry breaking in O'Raifeartaigh models,\" arXiv:hepth/0703196. [11] A. E. Nelson and N. Seiberg, \"R symmetry breaking versus supersymmetry breaking,\" Nucl. Phys. B 416, 46 (1994), arXiv:hep-ph/9309299. [12] E. Witten, \"Dynamical Breaking Of Supersymmetry,\" Nucl. Phys. B 188, 513 (1981). [13] K. Intriligator, N. Seiberg and D. Shih, \"Supersymmetry breaking, R-symmetry breaking and metastable vacua,\" arXiv:hep-th/0703281. [14] F. Cachazo, S. Katz and C. Vafa, \"Geometric transitions and N = 1 quiver theories,\" arXiv:hep-th/0108120. [15] T. Kawano, H. Ooguri, Y. Ookouchi and C.S. Park, in preparation [16] H. Ooguri and C. Vafa, \"Two-dimensional black hole and singularities of CY manifolds,\" Nucl. Phys. B 463, 55 (1996), arXiv:hep-th/9511164. [17] C. Vafa, \"Superstrings and topological strings at large N ,\" J. Math. Phys. 42, 2798 (2001), arXiv:hep-th/0008142. [18] F. Cachazo, B. Fiol, K. A. Intriligator, S. Katz and C. Vafa, \"A geometric unification of dualities,\" Nucl. Phys. B 628, 3 (2002), arXiv:hep-th/0110028. [19] H. Ooguri and Y. Ookouchi, 'Landscape of supersymmetry breaking vacua in geometrically realized gauge theories,\" Nucl. Phys. B 755, 239 (2006), arXiv:hep-th/0606061. [20] M. Dine, J. L. Feng and E. Silverstein, \"Retrofitting O'Raifeartaigh models with dynamical scales,\" Phys. Rev. D 74, 095012 (2006), arXiv:hep-th/0608159. 7"},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"Introduction","text":"The use of meta-stable vacua in supersymmetric model building has attracted much attention lately, especially after the discovery [1] that generic supersymmetric field theories in four dimensions such as the supersymmetric QCD with massive flavors have metastable vacua with broken supersymmetry. In [2] , realistic models of direct mediation were constructed using superpotentials without U (1) R symmetry. Though explicit breaking of the U (1) R symmetry generates meta-stable vacua, there is a range of parameters where one can make them sufficiently long lived, while satisfying the phenomenological constraints on the masses of the gauginos, the gravitino, and the scalars without artificially elaborate constructions. The models can also avoid producing Landau poles in standard model gauge interactions below the unification scale. Recently beautiful realizations of these models in string theory, including a natural mechanism to generate small parameters of these models, were found in [3] .\n\nGauge mediation models were also constructed using meta-stable vacua with similar phenomenological benefits [4, 5] . Related ideas have been explored in [6] [7] [8] [9] [10] . Accepting the possibility that our universe may be in a meta-stable state allows us to circumvent the theoretical constraints due to the Nelson-Seiberg theorem on R-symmetry [11] and the Witten index [12] and gives us greater flexibility in model building, as emphasized in [13] among other recent papers.\n\nAmong the models constructed recently based on meta-stable vacua, the ones discussed in [4] are particularly simple. In this paper, we will show that they have ultra-violet completions in supersymmetric quiver gauge theories which can be realized in string compactifications. Moreover, our construction can be naturally generalized to a large class of quiver gauge theories, providing a basis for the speculation in [4] that \"gauge mediation may be a rather generic phenomenon in the landscape of possible supersymmetric theories.\" In this paper, we will demonstrate the idea by explicitly working out one example: a model based on type IIB superstrings compactified on the A 4 -fibered geometry [14] . We will also give an outline of generalizations of this construction to a large class of quiver gauge theories. Detailed analysis of meta-stable vacua in these models will be given in a separate paper [15]. 1"},{"section_type":"OTHER","section_title":"The Model","text":"The model we will consider in this paper is realized in string theory compactified on the local Calabi-Yau manifold described by the equation,\n\nx 2 + y 2 + 5 i=1 (z + t i (w)) = 0, 5 i=1 t i (w) = 0, t i (w) -t i+1 (w) = µ i (w -x i ), (x, y, z, w) ∈ C 4 .\n\n(2.1) Since t i 's are functions of w, this gives the A 4 singularity fibered over w ∈ C. In particular, there exist four two-cycles S 2 on which D branes can be wrapped. The low energy limit of D5 branes wrapping the two-cycles S 2 and extending along the four uncompactified dimensions is the A 4 quiver gauge theory with the gauge group 2, 3, 4 for the four gauge group factors and the bi-fundamental chiral multiplets (Q 12 , Q 21 ), (Q 23 , Q 32 ), and (Q 34 , Q 43 ). This quiver gauge theory can also be realized on intersecting brane configuration with NS5 and D4 branes, as expected from the T-duality between the A n singularity and NS5 branes [16] .\n\nU (N 1 ) × U (N 2 ) × U (N 3 ) × U (N 4 ) with the adjoint chiral multiplets X i=1,\n\nN 1 N 2 N 3 N 4 Q 12,21 Q 23,32 Q 34,43 X 1 X 2 X3 X 4 Fig. 1: A 4 quiver diagram\n\nFrom the Calabi-Yau singularity (2.1), one can read off the superpotential of the quiver theory as [17,18]\n\nW A 4 = 3 i=1 tr (Q i+1,i X i Q i,i+1 -Q i,i+1 X i+1 Q i+1,i ) + 4 i=1 tr µ i 2 (X i -x i ) 2 . (2.2)\n\nNote that the dimensionful parameters µ i and x i are the moduli of the Calabi-Yau manifold given by (2.1), namely they are closed string moduli. The dynamical scales Λ i=1,•••,4 of the four gauge group factors are also closed string moduli, related to the sizes of the S 2 's.\n\nThese closed string moduli are frozen and can be regarded as parameters of the low energy theory. Let us suppose that µ i are sufficiently larger than Λ i so that we can integrate out all the adjoints X i to obtain the effective superpotential\n\nW eff = 3 i=1 m i tr Q i i+1 Q i+1 i - 3 i=1 1 μi tr (Q i i+1 Q i+1 i ) 2 + 1 µ 2 tr Q 21 Q 12 Q 23 Q 32 + 1 µ 3 tr Q 32 Q 23 Q 34 Q 43 , (2.3) 2\n\nwhere\n\nm i = c i -c i+1 , μi = 2µ i µ i+1 µ i + µ i+1 (i = 1, 2, 3).\n\nThis quiver gauge theory can be used as a gauge mediation model as follows. We identify the bi-fundamentals (Q 34 , Q 43 ) as messenger fields. One way to incorporate the standard model sector would be to identify a subgroup of the U (N 4 ) gauge group with the standard model gauge group or a GUT gauge group. Alternatively, we can replace the 4th node of the quiver diagram of Fig. 1 carrying the U (N 4 ) gauge group with a string theory construction of the standard model. For example, if the standard model is realized on intersecting branes, messengers can be open strings connecting the 3rd node carrying the U (N 3 ) gauge group to the standard model branes. 1 In the following, we will denote the bi-fundamental fields (Q 34 , Q 43 ) as (f, f ) to distinguish them from the rest of the quiver gauge theory and to emphasize their role as the messengers.\n\nThe rest of the quiver gauge theory is treated as a hidden sector, where supersymmetry is broken dynamically. To use the result of [1], let us assume that the ranks of the gauge group factors satisfy\n\nN 2 + 1 ≤ N 1 + N 3 < 3 2 N 2 (2.4) and that Λ 1 , Λ 3 , Λ 4 ≪ Λ 2 ≪ µ i .\n\nIn this case, one can identify the gauge group SU (N c ) of the model of [1] with SU (N 2 ) ⊂ U (N 2 ) of the quiver theory. Since the metastable vacuum can be found near the origin of\n\nthe meson fields M 11 ∼ Q 12 Q 21 , M 33 ∼ Q 32 Q 23 , the terms tr (Q 12 Q 21 ) 2 , tr (Q 32 Q 23 ) 2 and tr (Q 12 Q 21 Q 32 Q 23 ) in the superpotential (2.\n\n3) are irrelevant in our discussion below, if the masses µ i of the adjoints satisfy the following bounds [4,5],\n\nΛ 2 2 μ1,2 , Λ 2 2 µ 2 ≤ min 1 4π m 1,2 Λ 2 , 1 16π 2 m 3 µ 3 Λ 2 . (2.5)\n\nIn this range of the parameters, the hidden sector and its interaction with the messenger sector is described by the superpotential,\n\nW = mtr Q 12 Q 21 + mtr Q 32 Q 23 + 1 µ 3 tr Q 32 Q 23 f f + m 3 tr f f - 1 μ3 tr f f 2 . (2.6)\n\n1 In this case, we can still use the effective potential (2.3) to describe the interaction of the messengers and the hidden sector, but we should set 1/μ 4 = 1/(2µ 3 ) since we do not have the adjoint field X 4 .\n\nHere, we set the mass parameters m 1 = m 2 = m, for simplicity. Consider the case when N 1 = N 2 = 3 and N 3 = 1 so that the Landau pole problem can easily be avoided. The resulting model is a variant of the models proposed in [4] . The model [4] has the global symmetry U (4) × U (1) mess , where U (4) is the flavor symmetry of the ISS model and U (1) mess acts on the messengers (f, f). The meta-stable vacuum spontaneously breaks the U (4) symmetry, giving rise to Nambu-Goldstone bosons, when m 1 ≃ m 2 . In our model, the would-be Nambu-Goldstone bosons are eaten by the gauge symmetry. This difference is not important in the low energy analysis of supersymmetry breaking effects.\n\nLet us discuss phenomenological constraints on the parameters in (2.6). We will focus on the following part of the superpotential (2.6),\n\nW mess = Λ 2 µ 3 M 33 f f + m 3 f f , (2.7)\n\nwhere M 33 = Q 32 Q 23 /Λ 2 is neutral under the U (N 3 ) = U (1) gauge group. We have dropped the irrelevant quartic term (f f ) 2 because the messengers (f, f) are weakly interacting at energies above the electroweak scale, if the mass parameter μ3 is large enough.\n\nThe F -component of the meson superfield M 33 develops the vacuum expectation value and breaks supersymmetry [1]. The supersymmetric mass and the soft supersymmetry breaking mass of the messenger fields (f, f) are then given by\n\nW mess ≃ m 3 + θ 2 mΛ 2 2 µ 3 f f . (2.8)\n\nFollowing the analysis in [4, 5] , we find that all the phenomenological requirements for the messenger sector can be satisfied, for example, in the following range of parameters,\n\nΛ 2 ≃ 10 11 GeV, m ≃ 10 8 GeV, m 3 ≃ 10 7 GeV, µ 1 ≥ µ 2 ≥ 10 13 GeV, µ 3 ≃ 10 18 GeV. (2.9)"},{"section_type":"OTHER","section_title":"Generalization","text":"We found that both the messenger sector and the hidden sector of the models proposed in [4] can be realized in the A 4 quiver gauge theory. This construction naturally suggests the following generalization. Consider a quiver diagram which can be separated into two disjoint diagrams Γ 1 and Γ 2 by cutting at one node, which we denote by a. If the scale Λ a associated to the gauge group on the a-node is sufficiently low, and if superpotential 4 interactions between them are small, we have effectively two separate quiver gauge theories for phenomena much above the scale Λ a , one associated to Γ 1 and another associated to Γ 2 , which are weakly interacting with each other through the a-node. If supersymmetry is broken in the sector Γ 1 , it can be communicated to the sector Γ 2 by the gauge mediation mechanism. The beauty of the quiver gauge theory construction is that, because of the presence of bi-fundamental and adjoint fields on links and nodes, an effective superpotential of the form (2.8) is naturally generated when supersymmetry is broken in a part of the diagram connected to the a-node.\n\nIt follows trivially that any quiver theory that is vector like with adjustable mass terms has meta-stable supersymmetry breaking vacua in some range of its parameter. All one has to do is to identify a part of the diagram where supersymmetry can be broken using a known mechanism, for example as in [1] or its variant [19], and to have its effect communicated to the rest of the diagram by messengers. One can also consider the scenario where the quiver theory associated to a sub-diagram Γ 2 has a supersymmetric vacuum with dynamically generated small scales, which can be used to set parameters of the theory associated to another sub-diagram Γ 1 , where supersymmetry is broken. The supersymmetry breaking effect can then be communicated back to the sub-diagram Γ 2 . This would give a string theory realization of the idea of [20] . These and other mechanisms of supersymmetry breaking will be explored further in [15] .\n\nThese supersymmetry breaking quiver gauge theories can be coupled to the messenger sector. In fact, as in the case of the A 4 model discussed in the previous section, the messenger sector itself can be included in quiver theories. If the messenger sector is attached at the end of the quiver diagram, the effective low energy superpotential always takes the form (2.3). Thus, one can see that the models in [4] and their generalizations are robust and naturally appear in this large class of string compactifications."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"We thank D. Berenstein, M. Dine, R. Kitano, J. Marsano, C. S. Park, N. Seiberg, M. Shigemori, and T. Watari for discussions. H.O. thanks the hospitality of the high energy theory group at the University of Tokyo at Hongo. 5 H.O. and Y.O. are supported in part by the DOE grant DE-FG03-92-ER40701. The research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st Century COE Program at the University of Tokyo. Y.O. is also supported in part by the JSPS Fellowship for Research Abroad. The research of T.K. was supported in part by the Grants-in-Aid (#16740133) and (#16081206) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. 6 References [1] K. Intriligator, N. Seiberg and D. Shih, \"Dynamical SUSY breaking in meta-stable vacua,\" JHEP 0604, 021 (2006), arXiv:hep-th/0602239. [2] R. Kitano, H. Ooguri and Y. Ookouchi, \"Direct mediation of meta-stable supersymmetry breaking,\" Phys. Rev. D 75, 045022 (2007), arXiv:hep-ph/0612139. [3] R. Argurio, M. Bertolini, S. Franco and S. Kachru, \"Metastable vacua and D-branes at the conifold,\" arXiv:hep-th/0703236. [4] H. Murayama and Y. Nomura, \"Gauge mediation simplified,\" arXiv:hep-ph/0612186. [5] H. Murayama and Y. Nomura, \"Simple scheme for gauge mediation,\" arXiv:hepph/0701231. [6] A. Amariti, L. Girardello and A. Mariotti, \"On meta-stable SQCD with adjoint matter and gauge mediation,\" arXiv:hep-th/0701121. [7] M. Dine and J. Mason, \"Gauge mediation in metastable vacua,\" arXiv:hep-ph/0611312. [8] C. Csaki, Y. Shirman and J. Terning, \"A simple model of low-scale direct gauge mediation,\" arXiv:hep-ph/0612241. [9] O. Aharony and N. Seiberg, \"Naturalized and simplified gauge mediation,\" JHEP 0702, 054 (2007), arXiv:hep-ph/0612308. [10] D. Shih, \"Spontaneous R-symmetry breaking in O'Raifeartaigh models,\" arXiv:hepth/0703196. [11] A. E. Nelson and N. Seiberg, \"R symmetry breaking versus supersymmetry breaking,\" Nucl. Phys. B 416, 46 (1994), arXiv:hep-ph/9309299. [12] E. Witten, \"Dynamical Breaking Of Supersymmetry,\" Nucl. Phys. B 188, 513 (1981). [13] K. Intriligator, N. Seiberg and D. Shih, \"Supersymmetry breaking, R-symmetry breaking and metastable vacua,\" arXiv:hep-th/0703281. [14] F. Cachazo, S. Katz and C. Vafa, \"Geometric transitions and N = 1 quiver theories,\" arXiv:hep-th/0108120. [15] T. Kawano, H. Ooguri, Y. Ookouchi and C.S. Park, in preparation [16] H. Ooguri and C. Vafa, \"Two-dimensional black hole and singularities of CY manifolds,\" Nucl. Phys. B 463, 55 (1996), arXiv:hep-th/9511164. [17] C. Vafa, \"Superstrings and topological strings at large N ,\" J. Math. Phys. 42, 2798 (2001), arXiv:hep-th/0008142. [18] F. Cachazo, B. Fiol, K. A. Intriligator, S. Katz and C. Vafa, \"A geometric unification of dualities,\" Nucl. Phys. B 628, 3 (2002), arXiv:hep-th/0110028. [19] H. Ooguri and Y. Ookouchi, 'Landscape of supersymmetry breaking vacua in geometrically realized gauge theories,\" Nucl. Phys. B 755, 239 (2006), arXiv:hep-th/0606061. [20] M. Dine, J. L. Feng and E. Silverstein, \"Retrofitting O'Raifeartaigh models with dynamical scales,\" Phys. Rev. D 74, 095012 (2006), arXiv:hep-th/0608159. 7"}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"Introduction\",\n \"text\": \"The use of meta-stable vacua in supersymmetric model building has attracted much attention lately, especially after the discovery [1] that generic supersymmetric field theories in four dimensions such as the supersymmetric QCD with massive flavors have metastable vacua with broken supersymmetry. In [2] , realistic models of direct mediation were constructed using superpotentials without U (1) R symmetry. Though explicit breaking of the U (1) R symmetry generates meta-stable vacua, there is a range of parameters where one can make them sufficiently long lived, while satisfying the phenomenological constraints on the masses of the gauginos, the gravitino, and the scalars without artificially elaborate constructions. The models can also avoid producing Landau poles in standard model gauge interactions below the unification scale. Recently beautiful realizations of these models in string theory, including a natural mechanism to generate small parameters of these models, were found in [3] .\\n\\nGauge mediation models were also constructed using meta-stable vacua with similar phenomenological benefits [4, 5] . Related ideas have been explored in [6] [7] [8] [9] [10] . Accepting the possibility that our universe may be in a meta-stable state allows us to circumvent the theoretical constraints due to the Nelson-Seiberg theorem on R-symmetry [11] and the Witten index [12] and gives us greater flexibility in model building, as emphasized in [13] among other recent papers.\\n\\nAmong the models constructed recently based on meta-stable vacua, the ones discussed in [4] are particularly simple. In this paper, we will show that they have ultra-violet completions in supersymmetric quiver gauge theories which can be realized in string compactifications. Moreover, our construction can be naturally generalized to a large class of quiver gauge theories, providing a basis for the speculation in [4] that \\\"gauge mediation may be a rather generic phenomenon in the landscape of possible supersymmetric theories.\\\" In this paper, we will demonstrate the idea by explicitly working out one example: a model based on type IIB superstrings compactified on the A 4 -fibered geometry [14] . We will also give an outline of generalizations of this construction to a large class of quiver gauge theories. Detailed analysis of meta-stable vacua in these models will be given in a separate paper [15]. 1\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"The Model\",\n \"text\": \"The model we will consider in this paper is realized in string theory compactified on the local Calabi-Yau manifold described by the equation,\\n\\nx 2 + y 2 + 5 i=1 (z + t i (w)) = 0, 5 i=1 t i (w) = 0, t i (w) -t i+1 (w) = µ i (w -x i ), (x, y, z, w) ∈ C 4 .\\n\\n(2.1) Since t i 's are functions of w, this gives the A 4 singularity fibered over w ∈ C. In particular, there exist four two-cycles S 2 on which D branes can be wrapped. The low energy limit of D5 branes wrapping the two-cycles S 2 and extending along the four uncompactified dimensions is the A 4 quiver gauge theory with the gauge group 2, 3, 4 for the four gauge group factors and the bi-fundamental chiral multiplets (Q 12 , Q 21 ), (Q 23 , Q 32 ), and (Q 34 , Q 43 ). This quiver gauge theory can also be realized on intersecting brane configuration with NS5 and D4 branes, as expected from the T-duality between the A n singularity and NS5 branes [16] .\\n\\nU (N 1 ) × U (N 2 ) × U (N 3 ) × U (N 4 ) with the adjoint chiral multiplets X i=1,\\n\\nN 1 N 2 N 3 N 4 Q 12,21 Q 23,32 Q 34,43 X 1 X 2 X3 X 4 Fig. 1: A 4 quiver diagram\\n\\nFrom the Calabi-Yau singularity (2.1), one can read off the superpotential of the quiver theory as [17,18]\\n\\nW A 4 = 3 i=1 tr (Q i+1,i X i Q i,i+1 -Q i,i+1 X i+1 Q i+1,i ) + 4 i=1 tr µ i 2 (X i -x i ) 2 . (2.2)\\n\\nNote that the dimensionful parameters µ i and x i are the moduli of the Calabi-Yau manifold given by (2.1), namely they are closed string moduli. The dynamical scales Λ i=1,•••,4 of the four gauge group factors are also closed string moduli, related to the sizes of the S 2 's.\\n\\nThese closed string moduli are frozen and can be regarded as parameters of the low energy theory. Let us suppose that µ i are sufficiently larger than Λ i so that we can integrate out all the adjoints X i to obtain the effective superpotential\\n\\nW eff = 3 i=1 m i tr Q i i+1 Q i+1 i - 3 i=1 1 μi tr (Q i i+1 Q i+1 i ) 2 + 1 µ 2 tr Q 21 Q 12 Q 23 Q 32 + 1 µ 3 tr Q 32 Q 23 Q 34 Q 43 , (2.3) 2\\n\\nwhere\\n\\nm i = c i -c i+1 , μi = 2µ i µ i+1 µ i + µ i+1 (i = 1, 2, 3).\\n\\nThis quiver gauge theory can be used as a gauge mediation model as follows. We identify the bi-fundamentals (Q 34 , Q 43 ) as messenger fields. One way to incorporate the standard model sector would be to identify a subgroup of the U (N 4 ) gauge group with the standard model gauge group or a GUT gauge group. Alternatively, we can replace the 4th node of the quiver diagram of Fig. 1 carrying the U (N 4 ) gauge group with a string theory construction of the standard model. For example, if the standard model is realized on intersecting branes, messengers can be open strings connecting the 3rd node carrying the U (N 3 ) gauge group to the standard model branes. 1 In the following, we will denote the bi-fundamental fields (Q 34 , Q 43 ) as (f, f ) to distinguish them from the rest of the quiver gauge theory and to emphasize their role as the messengers.\\n\\nThe rest of the quiver gauge theory is treated as a hidden sector, where supersymmetry is broken dynamically. To use the result of [1], let us assume that the ranks of the gauge group factors satisfy\\n\\nN 2 + 1 ≤ N 1 + N 3 < 3 2 N 2 (2.4) and that Λ 1 , Λ 3 , Λ 4 ≪ Λ 2 ≪ µ i .\\n\\nIn this case, one can identify the gauge group SU (N c ) of the model of [1] with SU (N 2 ) ⊂ U (N 2 ) of the quiver theory. Since the metastable vacuum can be found near the origin of\\n\\nthe meson fields M 11 ∼ Q 12 Q 21 , M 33 ∼ Q 32 Q 23 , the terms tr (Q 12 Q 21 ) 2 , tr (Q 32 Q 23 ) 2 and tr (Q 12 Q 21 Q 32 Q 23 ) in the superpotential (2.\\n\\n3) are irrelevant in our discussion below, if the masses µ i of the adjoints satisfy the following bounds [4,5],\\n\\nΛ 2 2 μ1,2 , Λ 2 2 µ 2 ≤ min 1 4π m 1,2 Λ 2 , 1 16π 2 m 3 µ 3 Λ 2 . (2.5)\\n\\nIn this range of the parameters, the hidden sector and its interaction with the messenger sector is described by the superpotential,\\n\\nW = mtr Q 12 Q 21 + mtr Q 32 Q 23 + 1 µ 3 tr Q 32 Q 23 f f + m 3 tr f f - 1 μ3 tr f f 2 . (2.6)\\n\\n1 In this case, we can still use the effective potential (2.3) to describe the interaction of the messengers and the hidden sector, but we should set 1/μ 4 = 1/(2µ 3 ) since we do not have the adjoint field X 4 .\\n\\nHere, we set the mass parameters m 1 = m 2 = m, for simplicity. Consider the case when N 1 = N 2 = 3 and N 3 = 1 so that the Landau pole problem can easily be avoided. The resulting model is a variant of the models proposed in [4] . The model [4] has the global symmetry U (4) × U (1) mess , where U (4) is the flavor symmetry of the ISS model and U (1) mess acts on the messengers (f, f). The meta-stable vacuum spontaneously breaks the U (4) symmetry, giving rise to Nambu-Goldstone bosons, when m 1 ≃ m 2 . In our model, the would-be Nambu-Goldstone bosons are eaten by the gauge symmetry. This difference is not important in the low energy analysis of supersymmetry breaking effects.\\n\\nLet us discuss phenomenological constraints on the parameters in (2.6). We will focus on the following part of the superpotential (2.6),\\n\\nW mess = Λ 2 µ 3 M 33 f f + m 3 f f , (2.7)\\n\\nwhere M 33 = Q 32 Q 23 /Λ 2 is neutral under the U (N 3 ) = U (1) gauge group. We have dropped the irrelevant quartic term (f f ) 2 because the messengers (f, f) are weakly interacting at energies above the electroweak scale, if the mass parameter μ3 is large enough.\\n\\nThe F -component of the meson superfield M 33 develops the vacuum expectation value and breaks supersymmetry [1]. The supersymmetric mass and the soft supersymmetry breaking mass of the messenger fields (f, f) are then given by\\n\\nW mess ≃ m 3 + θ 2 mΛ 2 2 µ 3 f f . (2.8)\\n\\nFollowing the analysis in [4, 5] , we find that all the phenomenological requirements for the messenger sector can be satisfied, for example, in the following range of parameters,\\n\\nΛ 2 ≃ 10 11 GeV, m ≃ 10 8 GeV, m 3 ≃ 10 7 GeV, µ 1 ≥ µ 2 ≥ 10 13 GeV, µ 3 ≃ 10 18 GeV. (2.9)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Generalization\",\n \"text\": \"We found that both the messenger sector and the hidden sector of the models proposed in [4] can be realized in the A 4 quiver gauge theory. This construction naturally suggests the following generalization. Consider a quiver diagram which can be separated into two disjoint diagrams Γ 1 and Γ 2 by cutting at one node, which we denote by a. If the scale Λ a associated to the gauge group on the a-node is sufficiently low, and if superpotential 4 interactions between them are small, we have effectively two separate quiver gauge theories for phenomena much above the scale Λ a , one associated to Γ 1 and another associated to Γ 2 , which are weakly interacting with each other through the a-node. If supersymmetry is broken in the sector Γ 1 , it can be communicated to the sector Γ 2 by the gauge mediation mechanism. The beauty of the quiver gauge theory construction is that, because of the presence of bi-fundamental and adjoint fields on links and nodes, an effective superpotential of the form (2.8) is naturally generated when supersymmetry is broken in a part of the diagram connected to the a-node.\\n\\nIt follows trivially that any quiver theory that is vector like with adjustable mass terms has meta-stable supersymmetry breaking vacua in some range of its parameter. All one has to do is to identify a part of the diagram where supersymmetry can be broken using a known mechanism, for example as in [1] or its variant [19], and to have its effect communicated to the rest of the diagram by messengers. One can also consider the scenario where the quiver theory associated to a sub-diagram Γ 2 has a supersymmetric vacuum with dynamically generated small scales, which can be used to set parameters of the theory associated to another sub-diagram Γ 1 , where supersymmetry is broken. The supersymmetry breaking effect can then be communicated back to the sub-diagram Γ 2 . This would give a string theory realization of the idea of [20] . These and other mechanisms of supersymmetry breaking will be explored further in [15] .\\n\\nThese supersymmetry breaking quiver gauge theories can be coupled to the messenger sector. In fact, as in the case of the A 4 model discussed in the previous section, the messenger sector itself can be included in quiver theories. If the messenger sector is attached at the end of the quiver diagram, the effective low energy superpotential always takes the form (2.3). Thus, one can see that the models in [4] and their generalizations are robust and naturally appear in this large class of string compactifications.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"We thank D. Berenstein, M. Dine, R. Kitano, J. Marsano, C. S. Park, N. Seiberg, M. Shigemori, and T. Watari for discussions. H.O. thanks the hospitality of the high energy theory group at the University of Tokyo at Hongo. 5 H.O. and Y.O. are supported in part by the DOE grant DE-FG03-92-ER40701. The research of H.O. is also supported in part by the NSF grant OISE-0403366 and by the 21st Century COE Program at the University of Tokyo. Y.O. is also supported in part by the JSPS Fellowship for Research Abroad. The research of T.K. was supported in part by the Grants-in-Aid (#16740133) and (#16081206) from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. 6 References [1] K. Intriligator, N. Seiberg and D. Shih, \\\"Dynamical SUSY breaking in meta-stable vacua,\\\" JHEP 0604, 021 (2006), arXiv:hep-th/0602239. [2] R. Kitano, H. Ooguri and Y. Ookouchi, \\\"Direct mediation of meta-stable supersymmetry breaking,\\\" Phys. Rev. D 75, 045022 (2007), arXiv:hep-ph/0612139. [3] R. Argurio, M. Bertolini, S. Franco and S. Kachru, \\\"Metastable vacua and D-branes at the conifold,\\\" arXiv:hep-th/0703236. [4] H. Murayama and Y. Nomura, \\\"Gauge mediation simplified,\\\" arXiv:hep-ph/0612186. [5] H. Murayama and Y. Nomura, \\\"Simple scheme for gauge mediation,\\\" arXiv:hepph/0701231. [6] A. Amariti, L. Girardello and A. Mariotti, \\\"On meta-stable SQCD with adjoint matter and gauge mediation,\\\" arXiv:hep-th/0701121. [7] M. Dine and J. Mason, \\\"Gauge mediation in metastable vacua,\\\" arXiv:hep-ph/0611312. [8] C. Csaki, Y. Shirman and J. Terning, \\\"A simple model of low-scale direct gauge mediation,\\\" arXiv:hep-ph/0612241. [9] O. Aharony and N. Seiberg, \\\"Naturalized and simplified gauge mediation,\\\" JHEP 0702, 054 (2007), arXiv:hep-ph/0612308. [10] D. Shih, \\\"Spontaneous R-symmetry breaking in O'Raifeartaigh models,\\\" arXiv:hepth/0703196. [11] A. E. Nelson and N. Seiberg, \\\"R symmetry breaking versus supersymmetry breaking,\\\" Nucl. Phys. B 416, 46 (1994), arXiv:hep-ph/9309299. [12] E. Witten, \\\"Dynamical Breaking Of Supersymmetry,\\\" Nucl. Phys. B 188, 513 (1981). [13] K. Intriligator, N. Seiberg and D. Shih, \\\"Supersymmetry breaking, R-symmetry breaking and metastable vacua,\\\" arXiv:hep-th/0703281. [14] F. Cachazo, S. Katz and C. Vafa, \\\"Geometric transitions and N = 1 quiver theories,\\\" arXiv:hep-th/0108120. [15] T. Kawano, H. Ooguri, Y. Ookouchi and C.S. Park, in preparation [16] H. Ooguri and C. Vafa, \\\"Two-dimensional black hole and singularities of CY manifolds,\\\" Nucl. Phys. B 463, 55 (1996), arXiv:hep-th/9511164. [17] C. Vafa, \\\"Superstrings and topological strings at large N ,\\\" J. Math. Phys. 42, 2798 (2001), arXiv:hep-th/0008142. [18] F. Cachazo, B. Fiol, K. A. Intriligator, S. Katz and C. Vafa, \\\"A geometric unification of dualities,\\\" Nucl. Phys. B 628, 3 (2002), arXiv:hep-th/0110028. [19] H. Ooguri and Y. Ookouchi, 'Landscape of supersymmetry breaking vacua in geometrically realized gauge theories,\\\" Nucl. Phys. B 755, 239 (2006), arXiv:hep-th/0606061. [20] M. Dine, J. L. Feng and E. Silverstein, \\\"Retrofitting O'Raifeartaigh models with dynamical scales,\\\" Phys. Rev. D 74, 095012 (2006), arXiv:hep-th/0608159. 7\"\n }\n]","isConversational":false}}},{"rowIdx":192,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1086"},"arxiv_id":{"kind":"string","value":"0704.1086"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.75.124002"},"content_hash":{"kind":"string","value":"dcf0d7701a7b1489c0a47a9a712c25a75b13e093f228ff30b9d6ef70503ef573"},"title":{"kind":"string","value":"Higher harmonics increase LISA's mass reach for supermassive black holes"},"abstract":{"kind":"string","value":"Current expectations on the signal to noise ratios and masses of supermassive\nblack holes which the Laser Interferometer Space Antenna (LISA) can observe are\nbased on using in matched filtering only the dominant harmonic of the inspiral\nwaveform at twice the orbital frequency. Other harmonics will affect the\nsignal-to-noise ratio of systems currently believed to be observable by LISA.\nMore significantly, inclusion of other harmonics in our matched filters would\nmean that more massive systems that were previously thought to be {\\it not}\nvisible in LISA should be detectable with reasonable SNRs. Our estimates show\nthat we should be able to significantly increase the mass reach of LISA and\nobserve the more commonly occurring supermassive black holes of masses $\\sim\n10^8M_\\odot.$ More specifically, with the inclusion of all known harmonics LISA\nwill be able to observe even supermassive black hole coalescences with total\nmass $\\sim 10^8 M_\\odot (10^9M_\\odot)$ (and mass-ratio 0.1) for a low frequency\ncut-off of $10^{-4}{\\rm Hz}$ $(10^{-5}{\\rm Hz})$ with an SNR up to $\\sim 60$\n$(\\sim 30)$ at a distance of 3 Gpc. This is important from the astrophysical\nviewpoint since observational evidence for the existence of black holes in this\nmass range is quite strong and binaries containing such supermassive black\nholes will be inaccessible to LISA if one uses as detection templates only the\ndominant harmonic."},"authors":{"kind":"list like","value":["K.G. Arun","Bala R. Iyer","B. S. Sathyaprakash","Siddhartha Sinha"],"string":"[\n \"K.G. Arun\",\n \"Bala R. Iyer\",\n \"B. S. Sathyaprakash\",\n \"Siddhartha Sinha\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["gr-qc","astro-ph"],"string":"[\n \"gr-qc\",\n \"astro-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"gr-qc"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"There is strong observational evidence for the existence of supermassive black holes (SMBHs) with masses in the range of 10 6 M ⊙ -10 9 M ⊙ (see e.g. Ref. [1] and references therein) in most galactic nuclei [2] . Therefore, mergers of galaxies, as evidenced by high-redshift surveys, should give rise to binaries containing SMBHs. Late stage evolution of a SMBH binary is dictated by the emission of gravitational radiation. The resulting loss of orbital energy and angular momentum would lead to the coalescence of the two holes. Indeed, X-ray observations have revealed the existence of at least one such system that would coalesce within the Hubble time [3] . Gravitational waves (GW) emitted in the process could be detected by the planned Laser Interferometer Space Antenna (LISA) [4] .\n\nObservation of SMBH binaries at high redshifts is one of the major science goals of LISA. These observations will allow us to probe the evolution of SMBHs and structure formation [5] and provide an unique opportunity to test General Relativity (and its alternatives) in the strong field regime of the theory [6, 7, 8, 9, 10] . Observing SMBH coalescences with high (100-1000) SNR [8, 9] is crucial for performing all the aforementioned tests.\n\nMotivated by the fact that matched filtering is more sensitive to the phase of the signal than its amplitude [11] , search algorithms so far have deployed a waveform model involving only the dominant harmonic (at twice the orbital frequency), although the phase evolution itself is included to the maximum available post-Newtonian (PN) order 2 (currently 3.5PN, for non-spinning systems [12, 13] ). Waveforms in which all amplitude corrections are neglected, but the phase is treated to the maximum available order, are called restricted waveforms (RWF) and these are what are used so far in the analysis of data from ground-based detectors [14, 15, 16, 17] . This paper will consider the advantage of using the full wave forms (FWF) in the context of LISA.\n\nLISA is designed to detect gravitational waves in the frequency-band 0.1-100 mHz. This frequency range determines the range of masses accessible to LISA because the inspiral signal would end when the system's orbital frequency reaches the mass-dependent last stable orbit (LSO). In the test-mass approximation, the angular velocity ω LSO at LSO is given by ω LSO = 6 -3/2 M -1 , where M is the total mass of the binary. Search templates that contain only the dominant harmonic cannot extract power in the signal beyond f LSO = ω LSO /π ≃ 4.39(M/10 6 M ⊙ ) -1 mHz. This further implies that the frequency range [0.1, 100] mHz corresponds to the range ∼ 4.39 × [10 4 , 10 7 ]M ⊙ for the total mass of binary black holes that would be accessible to LISA 1 . However, as Table 1 of Ref. [1] would reveal, there is observational evidence for the existence of many SMBHs whose masses are of the order of 10 8 -10 9 M ⊙ . LISA will be unable to observe binaries containing SMBHs in this mass range if it used as search templates waveforms containing only the dominant harmonic.\n\nInclusion of higher-order amplitude terms in the waveform introduces the following two new features: (i) appearance of higher harmonics of the orbital phase and (ii) PN amplitude corrections to the leading as well as higher harmonics of the orbital frequency. For example, at 0.5PN order, which is the first-order correction, there are two new harmonics Ψ and 3Ψ, where Ψ is related to the orbital phase of the binary as in Refs. [18, 19] . More interestingly, in the expressions for the 'plus' and 'cross' polarizations, all odd harmonics of the orbital frequency are proportional to δm M , where δm is the difference in the masses of the binary components (see Eq. (5.7)-(5.10) of Refs. [19] ). Another important feature of the full waveform is that the (2n + 2) th harmonic first appears at the n th PN order in amplitude 2 . For example, the fourth harmonic first appears at 1PN, and has PN amplitude corrections to its dominant term at 2PN and 2.5PN (see Refs. [18, 19] for details).\n\nEarly investigations on the importance of amplitude-corrections to search templates were carried out by Sintes and Vecchio [20, 21] . Their study used only the first-order correction at 0.5 PN order. They concluded that the addition of the amplitude terms in the waveform did not improve the accuracy in the estimation of source's angular position and the distance, whereas the estimation of the chirp and reduced masses could be 10 times better when compared to the RWF. Recently, in the context of ground-based detectors, Van Den Broeck and Sengupta [22, 23, 24] examined the implications of going beyond the restricted PN approximation and employing instead the full waveform [18, 19] . The two main implications of the comprehensive analysis in Refs. [22, 23] for terrestrial GW detectors may be summarized as follows:\n\n1. For binary neutron stars and stellar mass black holes, restricted waveforms over-estimate the SNR as compared the full waveform.\n\n2. The use of the full waveforms significantly increases the mass-reach of second and third generation detectors, advanced LIGO and EGO being able to observe systems with total mass ∼ 400M ⊙ and a third generation detector as high as 10 3 M ⊙ .\n\nIn the present paper, we study in the context of LISA the implication of using templates based on the FWF (i.e. including all known harmonics of the orbital phase and all known amplitude corrections in the GW polarisations). Coalescences of SMBH binaries with masses ∼ 10 8-9 M ⊙ will not be observable by LISA if one uses only templates based on the RWF. Using templates based on amplitude corrected full waveforms, instead of the usual restricted waveforms, will enable LISA to observe coalescences of SMBH binaries with total mass ∼ 10 8 M ⊙ (10 9 M ⊙ ) if the lower frequency cut-off LISA can achieve is ∼ 10 -4 Hz (10 -5 Hz).\n\nThe rest of this paper is organized as follows: In Section II, we give the FWF in the frequency domain, by taking into account the orbital motion of LISA around the sun and its changing orientation. Section III discusses the results of our investigations where we compare the performances of the amplitude-corrected waveforms at different PN orders in terms of their mass-reach and distance-reach and correlate it to the 'observed' spectrum in LISA. Section V concludes with a brief summary of the main results and assumptions underlying their derivation.\n\n1 Although, binaries lighter than 10 4 M ⊙ would, in principle, evolve through the LISA band they would not be luminous enough to be visible in LISA unless they are close-by. 2 The 0.5PN term is an exception to this and also introduces a harmonic at the orbital frequency apart from the one at thrice the orbital frequency.\n\nII. TEMPLATE WAVEFORMS FOR LISA A. Amplitude corrected waveform\n\nFor non-spinning binaries in quasi-circular orbits inspiralling due to radiation-reaction, waveforms were computed in Refs. [18, 19] up to 2.5PN order in amplitude and 3.5PN in phase [12, 13] . This waveform h(t) is a linear combination of sine and cosine functions of multiples of the orbital phase Ψ(t). The expression for the 2.5PN polarization contains the first seven harmonics of the orbital phase, the dominant harmonic being the one at twice the orbital phase. The signal depends on the following parameters: D L , the luminosity distance to the binary, m the total (red-shifted) mass, ν the symmetric mass-ratio (reduced mass divided by total mass), the spherical polar angles (θ, φ) determining the direction of the \"line-of-sight\", the inclination angle ι of the angular momentum L of the binary with respect to the direction opposite to the line-of-sight, and the polarization angle ψ which determines the orientation of the projection of L in the plane normal to the line-of-sight.\n\nWe rewrite the waveform in terms of only cosines in a form similar to [23]: h(t) = 2M ν D L 7 k=1 5 n=0 A (k,n/2) cos[kΨ(t) + φ (k,n/2) ] x n 2 +1 (t), (2.1)\n\nwhere the coefficients A (k,n/2) and φ (k,n/2) are functions of (ν, θ, φ, ψ, ı), and x(t) = (2πM F (t)) 2/3 is the post-Newtonian parameter with F (t) the instantaneous orbital frequency. Terms 2Mν DL x n/2+1 (t) A (k,n/2) and φ (k,n/2) are the wave amplitude and polarization phase, respectively, corresponding to the k th harmonic and (n/2) th PN order. We call the coefficients A (k,n/2) the polarization amplitudes. The orbital phase Ψ(t) is a PN series in x, which, in the case of non-spinning binaries, is known to 3.5PN order [12, 13] . For a non-spinning source and a detector whose position and orientation are almost constant during the time of observation of the signal, all the above mentioned angles are constants. For ground-based GW detectors dealt with in Ref. [22, 23] , one is in this situation.\n\nB. Amplitude corrected waveform including modulations due to LISA's orbital motion -Time Domain\n\nLISA will be able to observe many sources from their early stages of inspiral and most would last for a pretty long time. We shall only consider binary sources that last for a year or less before merger. Since the LISA plane is tilted by 60 • with respect to the plane of the ecliptic, during the course of its heliocentric orbit its orientation and position varies periodically, with a period of one year and the signal in Eq. (2.1) will suffer additional amplitude and phase modulations. Thus in the case of LISA the angles θ, φ, and ψ (but not ı) appearing in Eq. 2.1 are functions of time.\n\nTo proceed further, in the frame of a non-rotating observer fixed to the solar-system barycentre, we denote by the location of the source on the sky by the spherical polar angles θ S and φ S and the orientation of the source by the spherical polar angles θ L and φ L determining the direction of the orbital angular momentum L of the binary. The transformation between the fixed set of angles 3 (θ S , φ S , θ L , φ L ) and the time-dependent angular coordinates of the source (θ(t), φ(t), ψ(t), ı) as measured by LISA are given in Ref. [25] .\n\nGeneralizing, Ref. [25] from the RWF to the FWF, the signal as seen in LISA is of the form,\n\nh(t) = √ 3 2 2M ν D L 7 k=1 5 n=0 A (k,n/2) (t) cos[kΨ(t) + φ (k,n/2) (t) + kφ D (t)] x n 2 +1 (t) . (2.2)\n\nThe PN parameter x(t) appearing in Eq. (2.2) is still equal to (2πM F (t)) 2/3 , where F (t), however, is the orbital frequency as measured by a non-rotating observer located at the solar-system barycentre. The term φ D (t) is the Doppler phase [25] , accounting for the phase difference of the gravitational wave-front between LISA and the solarsystem barycentre. The time-dependence of φ D (t) is due to the orbital motion of LISA about the barycentre. It is given by\n\nφ D (t) = 2 π F (t) R sin θ S cos[φ(t) -φ S ], (2.3)\n\nwhere R = 1 AU and φ(t) is the angular position of LISA with respect to the barycentre given by φ(t) = 2 π t T , T being equal to one year.\n\n3 This is a different notation from [25], where the source angles measured in the fixed barycentre frame are denoted by ( θS , φS , θL , φL ) 4 C. Amplitude corrected waveform including modulations due to LISA's orbit -Frequency Domain\n\nThe above waveform is valid in the adiabatic regime, where the radiation-reaction time-scale is much larger than the orbital time-scale. We also note that the additional amplitude and Doppler modulations in the waveform for LISA vary on time-scales of 1 yr (i.e. ∼ 3 × 10 7 s), while LISA can observe orbital periods at most up to 2 × 10 5 s, (i.e. gravitational wave frequencies of order 10 -5 Hz.). Consequently, the Doppler modulations change much more slowly (a hundredth) than the orbital phase. This permits the use of the stationary phase approximation (SPA) to obtain an analytical form for the Fourier transform (FT) h(f ) of the signal:\n\nh(f ) ≃ √ 3 2 2M ν D L 7 k=1 5 n=0\n\nA (k,n/2) (t(f /k)) x n 2 +1 (t(f /k)) e -iφ (k,n/2) (t(f /k))\n\n2 k Ḟ (t(f /k)) exp [i ψ f (t(f /k))] , (2.4)\n\nwhere an over dot denotes derivative with respect to time and ψ f (t(f /k)) is given by ψ f (t(f /k)) = 2πf t(f /k) -k Ψ(t(f /k)) -k φ D (t(f /k)) -π/4. (2.5)\n\nThe PN expansions for t(F ), Ψ(F ), Ḟ (F ) are given in Ref. [12] . The expression in Eq. (2.4) within the summation over k represents the FT due to the k th harmonic. It should be noted that the term Ḟ may be treated in different ways that could lead to numerically different results. In a numerical treatment, for instance, one could avoid performing a further re-expansion. Alternatively, one could re-expand the denominator in the amplitude and truncate the resulting expression at the n th PN order, to obtain the nPN amplitude-corrected waveform. Ref. [22, 23] choose the latter and we follow them in this work. Radiation reaction results in an increase in the orbital frequency F (t) which will ultimately drive the system beyond the adiabatic inspiral phase and the inspiral waveform given above will no longer be valid. In the first approximation this is expected to occur when the orbital frequency F (t) reaches F LSO -the orbital frequency of the LSO of a Schwarzschild solution with the same mass as the binary's total mass M ,\n\nF LSO = (2 π 6 3 2 M ) -1 .\n\n(2.6) Thus, we truncate the signal in the time domain at a time t LSO , given implicitly by F (t LSO ) = F LSO . In the SPA, the main contribution to the FT of the k th harmonic at a given Fourier frequency f , comes from the neighbourhood of the time when the instantaneous value of the k th harmonic sweeps past f . Thus the k th harmonic is not expected to contribute significant power to the FT for frequencies above k F LSO , if the signal is truncated in the time domain beyond t LSO . This motivates the truncation of the FT due to the k th harmonic at frequencies above k F LSO by a step function θ(k F LSO -f ) [θ(x) = 1, for x ≥ 0, and 0 for negative x].\n\nIn this Section we investigate the effect of the higher harmonics in LISA observations of supermassive black hole binaries. The LISA waveform discussed in the previous Section will be used for the analysis.\n\nGiven a waveform h, the best signal-to-noise ratio (SNR) achieved using an optimal filter is given by ρ[h] ≡ (h|h) 1/2 , where ( . | . ) is the usual inner product in terms of the one-sided noise power spectral density S h (f ) of the detector. With the convention for Fourier transforms, x(f ) = ∞ -∞ x(t) exp(-2πif t) dt, the inner product is given by:\n\n(x|y) ≡ 4 f end fs Re[x * (f )ỹ(f )] S h (f ) df. (3.1)\n\nFor an optimal filter, which maximises the overlap of the signal with template, one can write\n\nρ 2 = 4 f end fs | h(f )| 2 S h (f ) df. (3.2)\n\nWe use the non-sky-averaged noise-spectral-density as given in Eqs. (2.28)-(2.32) of Ref. [6]. 5 10 -4 10 -3 10 -2 frequency(in Hz) 0.0 2.0×10 6 4.0×10 6 6.0×10 6 8.0×10 6 Pf(f) RWF FWF 10 -4 frequency(in Hz) 0.0 5.0×10 5 1.0×10 6 1.5×10 6 2.0×10 6 Pf(f) RWF FWF FIG. 1: The observed spectrum, P(f f ) , in LISA using the full (solid, red) and restricted (dashed, blue) waveforms, for two archetypal binaries, (10 5 , 10 6 )M⊙ (left) and 2 × (10 6 , 10 7 )M⊙ (right). The sources are assumed to be at 3 Gpc and their orientation with respect to the solar-system barycentre is chosen to be θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3. The spectrum is much more complicated and highly oscillatory for the FWF than for the RWF, because of interference between various harmonics. The higher frequency reach of the FWF is due to presence of higher harmonics as apparent in the figure . The spectrum for the system in the left panel sharply rises at a frequency ∼ 2 × 10 -3 Hz. Beyond this frequency, the effective LISA noise decreases sharply with increasing frequency (as there are fewer galactic binaries per frequency bin) leading to the observed increase in the spectrum.\n\n) ≡ d(ρ 2 ) d(ln f ) = f | h(f )| 2 S h (\n\nThe upper limit of integration f end is taken to be the minimum of 7 F LSO and 1 Hz, the latter being a conventional upper cut-off for the LISA noise curve. The lower limit f s is chosen assuming LISA observes the inspiral for a duration ∆t obs before it reaches the LSO. Using the quadrupole formula, we find that the orbital frequency at the epoch t LSO -∆t obs is given by\n\nF (t LSO -∆t obs ) = F LSO (1 + 256 ν 5 M ∆t obs v 8 LSO ) 3 8 , (3.3)\n\nwhere v LSO is the orbital velocity and t LSO the epoch at which the orbital frequency reaches the value F LSO . We take v LSO = 1/ √ 6, the orbital velocity at the LSO in the case of a test mass orbiting a Schwarzschild black hole. We designate F (t LSO -∆t obs ) as F in . Thus the k th harmonic will have a frequency k F in , ∆t obs before t LSO . The above formula reduces to the simpler form in Ref. [6] as v LSO → ∞. For the mass values explored in this work there is no significant dependence of the results on this choice. In all our calculations we take ∆t obs to be one year.\n\nThe lower cut-off for the k th harmonic should be the maximum of the lower cut-off of LISA (10 -4 Hz) and k F in and simply implemented by truncating the waveform due to the kth harmonic by another step-function θ(f -k F in ) and choosing f s to be 10 -4 Hz. It is worth noting that the k th harmonic probes a larger interval of the frequency domain i.e. k(F LSO -F in ) relative to the fundamental harmonic. For brevity, we refer to this as the span of the k th harmonic.\n\nThere is a caveat with regard to the use of higher harmonics that is worth mentioning: In the time-domain the waveform should begin when the highest harmonic reaches the lower cutoff. This has an implication on data analysis as the templates will be an order-of-magnitude longer than before. Thus, it might be sensible to use higher harmonics only in the case of higher masses.\n\nTo get some insight into the effect of higher harmonics via amplitude corrections let us first look at the SNR integrand, i.e., the \"noise-weighted signal power\" per unit logarithmic frequency interval [26] . Rewriting the expression for the SNR as\n\nρ 2 = 4 f end fs f | h(f )| 2 S h (f ) d ln(f ), (3.4) 6\n\nTABLE I: SNRs due to successive PN amplitude-corrected waveforms, with phase corrections to 3.5 PN order in all cases. The orientation of the source with respect to the solar-system barycentre is chosen to be θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3. For the (10 6 -10 7 )M⊙ binary system, all harmonics enter deep into the sensitive part of the LISA bandwidth. Apart from an increase at 0.5PN, we see a consistent reduction in the SNR on inclusion of higher PN order amplitude corrections. For the (5.5 × 10 6 , 5.5 × 10 7 )M⊙ binary system, the second harmonic fails to enter the LISA bandwidth, while the third harmonic spans a small insensitive region. Thus the SNR due to the RWF is zero, while the SNR due to the 0.5PN waveform is smaller than the SNRs due to higher order PN terms. Both sources are at a distance of 3 Gpc. PN SNR order (10 6 -10 7 )M⊙ 5.5 × (10 6 -10 7 )M⊙ 0 924.48 0 0.5 1025.8 211.98 1 928.48 343.17 1.5 869.78 319.34 2 824.65 266.65 2.5 809.51 277.34 the quantity of our interest is\n\nP(f ) ≡ d(ρ 2 ) d(ln f ) = f | h(f )| 2 S h (f ) , (3.5)\n\nwhich is designated as the \"observed spectrum\", following [24] . The observed spectrum is plotted versus frequency for given masses in Fig. 1 . As is the case for ground-based detectors [24], the spectrum due to the FWF has a lot more structure and is highly oscillatory because of interference between various harmonics. For the (10 5 , 10 6 )M ⊙ system, the mass being low, the second harmonic and hence the RWF extends up to frequencies ∼ 2 × 10 -3 Hz, where LISA is most sensitive. This leads to a rapid increase in the observed spectrum in this frequency region. The spectrum due to the FWF, containing higher harmonics continue beyond the RWF into the most sensitive part of the LISA band. For the 2(10 6 , 10 7 )M ⊙ system, the frequency span of the second harmonic is small and the sensitive region of the LISA band lies beyond its maximum reach.\n\nFollowing the analysis of Ref. [22, 23] , we classify the sources into two types: In the first category are sources for which the dominant (second) harmonic has a large frequency span in the LISA band. The second category on the other hand comprise sources whose dominant harmonic fails to enter the LISA bandwidth but the higher harmonics do. Since the upper cut-off frequency for each harmonic is inversely proportional to the total mass (from the expression for F LSO ), we note that the sources of the first type will have total mass less than some value which we call the RWF mass-reach, the maximum mass detectable by the RWF, while the second type will have masses greater than this value. The condition that the upper cut-off of the dominant harmonic is less than or equal to the lower cut-off of LISA (i.e., by the inequality 2 F LSO ≤ f s ) determines the RWF mass-reach. The choice of f s for the LISA mission is still not clear and theoretical implications of this choice are explored in e.g. Ref. [27] . For f s in the range [10 -5 , 10 -4 ] Hz the RWF mass-reach varies over the range [4.39 , 43.9] × 10 7 M ⊙ , the lower end of the mass range corresponding to the higher end of the frequency range.\n\nIn Fig. 2 we plot the SNRs computed using the restricted (RWF) and full (FWF) waveforms as a function of the binary's total mass for two values of the mass ratio 4 . We first consider systems whose total mass is less than 4 Our codes are calibrated by reproducing the results of [22, 23] , which considers ground-based detectors, and also of [28] , which computes SNRs in LISA using RWF.\n\n7 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Total Mass (in Msun) 10 0 10 1 10 2 10 3 SNR RWF; m 1 /m 2 =0.1 FWF; m 1 /m 2 =0.1 RWF; m 1 /m 2 =0.01 FWF; m 1 /m 2 =0.01 FIG. 2: SNR versus total mass for mass ratios of 0.1 and 0.01. The figure shows that apart from the dips due to white-dwarf confusion noise, for mass values where the RWF enters the LISA band, the corresponding SNR is consistently more than the SNR produced by the FWF. However, for mass values where the second harmonic terminates before it reaches the LISA bandwidth, the FWF which has higher harmonics that enter the LISA band produces significant SNRs. The frequency reach of a harmonic depends only on the total mass, and thus the mass reach of the FWF is independent of the mass-ratio. For more asymmetric systems, the magnitude of the SNR is low for all masses both for the RWF and the FWF. Sources are at a luminosity distance of 3 Gpc with fixed angles given by θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3.\n\n4 × 10 7 M ⊙ . For these systems, the SNRs computed using the two different approximations agree with each other to within 10%, with the RWF over-estimating the SNR, when compared to the FWF, in most of the range. This is explicitly shown for a (10 6 , 10 7 )M ⊙ binary in the first column of Table I . Indeed, but for the slight increase in SNR as we go from 0PN to 0.5PN, we find a steady decrease as one increases the PN order of the amplitude correction. The reduction in SNR at higher PN orders can be understood by studying the structure of | h(f )| 2 , the numerator in the integrand of the SNR in Eq. (3.2). There are basically three types of terms: 1. direct terms in which the phases in Eq. (2.4) cancel 2. interference terms between different PN corrections of the same harmonic,\n\nA 2 (k,n/2) (t(f /k)) f -7 3 (M f ) 2n 3 ,\n\nA (k,m/2) (t(f /k)) A (k,n/2) (t(f /k)) f -7 3 (M f ) m+n 3 cos[φ (k,m/2) (t(f /k)) -φ (k,n/2) (t(f /k))]\n\n3. harmonic mixtures 5 which are terms containing the interference between different PN corrections of different harmonics, e.g. the m/2 th PN correction of the k th harmonic and n/2 th PN correction of the l th harmonic.\n\nA (k,m/2) (t(f /k)) A (l,n/2) (t(f /l)) f -7 3 (M f ) m+n 3 cos[ψ f (t(f /k)) -φ (k,m/2) (t(f /k)) -ψ f (t(f /l)) + φ (l,n/2) (t(f /l))]\n\nwhere ψ f (t(f /k)) is given by Eq. (2.5), All these terms are scaled by M 5/3 , where M = M ν 3/5 is the chirp-mass. (Additional multiplicative factors have been omitted in the above expressions, among which are the step-functions mentioned earlier and PN expansion coefficients of the denominator of the Fourier amplitude in Eq. (2.4), the latter being time-independent.)\n\nBefore we explain the SNR trends in the context of LISA, we mention that for ground-based detectors a similar effect was found in Ref. [23] for a different but corresponding mass region. The lower cut-off for a typical ground-based 5 We use the term 'harmonic mixtures' at the risk of being mistaken to the well-known 'harmonic mixing' in music. Our use of the phrase 'harmonic mixtures' is simply to convey the physical effect of the interference between different harmonics 8 1×10 -4 1×10 -3 frequency(Hz) 0 0.5 1 polarisation amplitude A (2,0) 0.00 3.15×10 7 6.31×10 7 time (in sec) 0 0.5 1 polarisation amplitude A (2,0) polarisation amplitude A (2,0) time ( in years ) 0 1 2 0.5 1 FIG. 3: Variation of polarisation amplitude of the RWF with frequency and time (inset). The inset, plotted over a duration of two years clearly shows periodicity due to LISA's orbital motion around the Sun. The binary mass, (10 6 -10 4 )M⊙, has been chosen such that it can, in principle, be observed for two years. The plot in the frequency domain shows that the variation of the polarisation amplitude is confined to a very small part of the frequency span of the dominant harmonic, and essentially behaves as a constant in the frequency domain.\n\ndetector, say Advanced LIGO is 20Hz, and the effect of higher harmonics is seen for masses less than ∼ 220M ⊙ . In that case, as mentioned earlier, the polarisation amplitudes and phases are constants. The RWF contains only the Newtonian term of the second harmonic and thus | h(f )| 2 consists of a single direct term with n = 0 and k = 2. With the inclusion of higher-order amplitude terms in the waveform, PN corrections to the dominant harmonic, and higher harmonics and their PN corrections, also contribute to the SNR. In other words, the signal power spectrum | h(f )| 2 will contain all three types of terms discussed before. From the form of the direct terms, it is evident that their contribution to the SNR will be positive definite. We also note that, for ground-based detectors, the frequency dependence of the direct and interference terms will just be a power law. However, the sign of the interference terms (and consequently their contribution to the SNR) depends on the difference between the polarisation phases of different PN corrections for the same harmonic. Van Den Broeck and Sengupta showed that for a given harmonic, for all allowed values of the parameters (ν, θ, φ, ψ, ı), each PN correction is almost \"out of phase\" with both the PN correction preceding and succeeding it 6 . The resulting negative terms (representing destructive interferences) reduce the SNR as one includes higher PN amplitude corrections in the waveform.\n\nThe third type of terms, harmonic mixtures, however, are highly oscillatory functions of the frequency, as the phase difference ψ f (t(f /k)) -ψ f (t(f /l)) between the k th and the l th harmonic become even or odd multiples of π. As one integrates over f , these oscillations tend to cancel out, and thus the contribution to the SNR from these terms are numerically much smaller relative to the first two types of terms.\n\nC. Effect of higher harmonics for binaries with M < 4 × 10 7 M⊙\n\nIn the case of LISA, because of the polarisation factors, the amplitudes of none of the three types of terms is a simple power-law in f . The periodic variation of, for example, A (2,0) (period being one year) appears as an amplitude modulation A (2,0) (t(f /2)) in the Fourier transform, where the argument t(f /2) of A 2,0 is given by t(f /2) = -5 256π 8/3 M 5/3 1 f 8/3 + PN corrections. (4.1) Hence, in the frequency domain A (2,0) will undergo one complete oscillation as f varies from 2F in (see Eq. (3.3)) to 2F LSO . However, because of the inverse power-law dependence on f , the oscillation of A (2,0) is confined to a small frequency interval above F in and remains fairly constant over a major portion of the frequency span 2(F LSO -F in ) (see Fig. 3 ). For masses higher than the one shown in Fig. 3 , this region of significant variation moves to the left of the figure. On including in our analysis the effect of detector sensitivity (weighting down by S h (f )) this variation of 6 Note, however, that Ref. [23] , argues this in a somewhat different form.\n\n9 10 7 10 8 Total Mass (in Msun) 10 1 10 2 10 3 SNR Newtonian 0.5PN 1PN 1.5PN 2PN 2.5PN 10 7 10 8 Total Mass (in Msun) 10 1 10 2 10 3 SNR Newtonian 0.5PN 1PN 1.5PN 2PN 2.5PN FIG. 4: SNR versus total mass for successive PN amplitude-corrected waveforms and 3.5PN phasing. The left panel corresponds to a mass-ratio of 0.1 while the right panel plots the same quantities for mass-ratio of 1 (equal mass systems). The (2n + 2) th harmonic first appears at the nth PN order. For a binary of given total mass, the upper cut-off of the k th -harmonic of the orbital frequency in the frequency domain is proportional to k and inversely proportional to the total mass. As the mass increases the upper cut-off for the 2 nd harmonic falls below the lower cut-off of the LISA detector, leading to a zero value of SNR due to the RWF. The higher harmonics still enter the sensitive bandwidth of LISA and higher PN order waveforms produce significant SNR. The 2.5PN waveform has the highest mass-reach, being 3.5 times the mass-reach of the RWF. In the equal mass case displayed in the right panel, the differences in harmonic content of different PN order waveforms are more pronounced, as odd harmonics are absent. Sources are at a luminosity distance of 3 Gpc with fixed angles given by θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3.\n\nA (2,0) gets damped out when one evaluates the integral in Eq. (3.2). For masses satisfying 2F in ≪ 10 -4 Hz, the lower cut-off for LISA, this region of variation will fall below the LISA band. The polarisation phases determining the sign of the interference terms between the same harmonics also vary with f . However, as mentioned earlier, the phase relationships of the polarisation phases are independent of the parameter values. Thus the modulations which change the values of (θ, φ, ψ, ı) do not affect the trend of reduction of SNR with amplitude corrections. The Doppler modulations, which appear in only harmonic mixtures, are also not important as far as SNR is concerned.\n\nFinally, we would like to note an important point not explicitly mentioned in Ref. [23] . As the difference between the polarisation phases of successive PN corrections of the same harmonic tend to be nearly π, alternate PN corrections necessarily interfere constructively. Hence there are positive contributions also from the interference terms. Now, the numerical value of the contribution to SNR from each of these terms depends on the magnitude of the polarisation amplitude and the power of (M f ). It can be checked that for all allowed values of (ν, θ, φ, ψ, ı) , the polarisation amplitudes are roughly of the same order in magnitude. Consider the Newtonian term of the dominant harmonic and its interference with the first two corrections at 1PN and 1.5PN. The Newtonian term will be out of phase with the 1PN term, but in phase with the 1.5PN one. The two corresponding interference terms will contain powers of (M f ) 2/3 and (M f ), respectively, and since they have the same frequency span, the absolute numerical value of the contribution to SNR from the former will be more since (2πM f ) 1/3 will always be less than 1/ √ 6. Numerical values of contributions from interference between higher PN corrections of the second harmonic successively decrease. The same argument applies for all the other harmonics, and thus, inclusion of amplitude corrections will lead to an overall reduction in SNR.\n\nThe first column of Table I clearly demonstrates the effect of higher harmonics on SNRs. The increase in SNR for the 0.5PN waveform (with respect the RWF SNR) is also easily explained by noting that the 0.5PN correction only adds (apart from harmonic mixtures) two direct terms to | h(f )| 2 , corresponding to the first and third harmonics (n = 1, n = 3). Clearly, from the discussion in the previous subsection, the 0.5PN waveform will have a higher SNR than the RWF, independent of the binary parameters.\n\nFor 10 3 < ∼ M < ∼ 10 5 M ⊙ , the difference between the RWF and the FWF is not visible on the scale of Fig. 2 because for this mass range all the direct and interference terms corresponding to harmonics higher than the dominant ones, which are scaled by higher powers of (M f ), are negligible.\n\n10 D. Visibility of systems with M > 4 × 10 7 M⊙ In their analysis of the implications of the FWF for ground-based detectors Van Den Broeck and Sengupta [22, 23] pointed out an interesting effect due to higher harmonics. An analogous effect is found in the case of LISA in spite of the additional amplitude and Doppler modulations that exist in this case.\n\nNormally, the harmonic at twice the orbital frequency dominates the SNR. However, when the dominant harmonic fails to reach the LISA band the higher harmonics become important, which transpires for masses greater than 4 × 10 7 M ⊙ . Even though the second harmonic falls below the lower cut-off f s of the LISA bandwidth, the kth harmonic, k > 2, that has power up to a frequency k F LSO , might cross f s and produce a significant SNR. Of course, the kth harmonic would fall below the LISA sensitivity band for masses which satisfy the equality f s = k F LSO . Thus, higher PN order waveforms, which bring in higher harmonics, are capable of producing a significant SNR, even when the RWF fails to produce any.\n\nLet us examine this in a little more detail starting from the values of mass where the second harmonic dominates and the RWF is adequate. Eventually, for larger values of the total mass, the inequality f s ≥ 3F LSO becomes true. Then the 0.5PN waveform, which contains the first and the third harmonic, terminates before reaching f s and consequently the SNR due to the 0.5PN waveform goes to zero. SNRs for different PN waveforms for a binary whose dominant harmonic falls below f s and the third harmonic has a small span in the LISA bandwidth is given in the second column of Table I. Note that for the 5.5(10 6 -10 7 )M ⊙ system, the 1PN waveform has a higher SNR than the 0.5PN one. This is due to the absence of the first harmonic and the small span of the third harmonic in the LISA bandwidth. Further, the 2.5PN waveform has a slightly larger SNR compared to 2PN. This is due to the absence of the first and second harmonic and the small contribution from the third harmonic, all of which contribute interference terms due to their 2.5PN corrections. However, this increase is marginal, and is not generic. We have explicitly checked by choosing different angles that there can be a small decrease also. The detailed results for LISA are summarised in Fig 4 . We see that for masses for which the 1PN waveform fails to reach the LISA bandwidth, the higher PN order amplitudes are capable of producing SNRs as high as 100! Thus, the use of the FWF will enable LISA to make observations of SMBHs in the astrophysically interesting mass-regime, which would not be possible had one used only the standard RWF.\n\nUsing the expression for F LSO , it is simple to argue that the mass reach for the 2.5PN FWF, which has the seventh harmonic of the orbital frequency, is 7/2 times the RWF (around 1.5 × 10 8 M ⊙ ). The above ratio, of course, depends on the assumption that the Schwarzschild (test particle case) LSO frequency will not be very different from the LSO frequency in the comparable mass case.\n\nWe conclude with a discussion of a minor, but clear, feature seen in Fig. 2 for LISA, but not present for the ground-based detectors, concerning the relative values of the SNR obtained using the RWF and the FWF. For most of the mass range probed the RWF overestimates the SNR relative to the FWF; however, the figure clearly shows an anomaly for masses around ∼ 2 × 10 6 M ⊙ . To understand this, we first note that the dips in the two curves in Fig 2 , \nare due to the bump in the LISA noise-curve [6] just above 10 -3 Hz. This bump is due to the domination of whitedwarf confusion noise over instrumental noise and lies just below the most sensitive frequency region (∼ 3 × 10 -3 Hz -2 × 10 -2 Hz) of the LISA band. Below 3 × 10 -3 Hz, the noise increases sharply till one reaches the bump. For binaries of mass greater than 1.5 × 10 6 M ⊙ , the frequency span of the dominant harmonic ends just around the bump and the sensitive region of the LISA band is beyond the span of this harmonic. However, higher harmonics incorporated in the FWF are able to reach the sensitive part of the noise curve. This leads to higher SNR for the FWF relative to the RWF. This reversal of trend continues up to masses 4 × 10 4 M ⊙ . Above this mass, the frequency span of the seventh harmonic ends before the sensitive region of the LISA band and the general trend is restored.\n\nFor still higher mass values, the SNRs due to the RWF and the FWF both increase until the second harmonic fails to reach the LISA band. This is due to the overall scaling of the waveform with the total mass. At such high values, it is able to compensate both for the decreasing frequency span and the higher noise of the detector in this frequency range.\n\nIn contrast to asymmetric systems discussed so far, for systems of equal mass all odd harmonics are absent. Consequently, for symmetric systems the mass-reach of the 2.5PN FWF will be only 3 times the mass-reach of the RWF. Further, from the right panel of Fig. 4 , it is clear that the 0.5PN and the 0PN, or RWF, are identical, as are the 1PN and 1.5PN waveforms. Thus the decrease in SNR for the higher PN order waveforms with increasing total mass is more pronounced than in the unequal-mass case. We also note that for masses for which the second harmonic fails to reach the detector bandwidth, the 2PN waveform has a lower SNR than the 2.5PN waveform. This can be explained by noting that for these masses only the fourth and sixth harmonics enter the LISA bandwidth. 11 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Total Mass (in Msun) 10 2 10 3 10 4 10 5 10 6 Luminosity Distance (Mpc) FWF RWF FIG. 5: Luminosity Distance (in Mpc) versus total mass for a fixed SNR of 10. The systems have mass-ratio of 0.1. The distance reach can be as large as 500 Gpc for systems where the second harmonic enters the LISA bandwidth. Systems undetectable by the RWF (of mass around 10 8 M⊙) can be detected by the FWF at distances up to 10 Gpc. The location and orientation of the sources are the same as in the earlier figures.\n\nThe 2PN waveform contains the leading term of the fourth harmonic at 1PN and its 2PN correction, which interfere destructively. However, inclusion of the 2.5PN amplitude correction leads to a constructive interference term between the 2.5PN correction and the 1PN term which is responsible for increasing the SNR for the 2.5PN waveform.\n\nIt is interesting to note that the computation of the 3PN GW polarization which will introduce an harmonic at 8Ψ will be quantitatively more significant for the equal mass case as the mass reach will be better by 33% relative to the 2.5PN FWF as opposed to the unequal mass case where it is only 14%! This provides one motivation for work in progress towards the computation of the 3PN accurate GW polarizations [29].\n\nSince the mass reach depends only on the total mass, the trends remain the same for different values of mass ratios. Fig. 2 compares the variation of SNRs with mass for mass ratios of 0.1 and 0.01. If the SNR is dominated by the second harmonic, the SNR is smaller for more asymmetric systems by an overall factor of ν, where ν = m 1 m 2 /m 2 . However, once the second harmonic fails to reach the sensitive bandwidth of LISA, the more asymmetric systems have a dominant contribution from the odd harmonics which scale by a further factor of √ 1 -4ν, which is larger for more asymmetric systems. Thus the decrease in SNR for the FWF with an increase in the total mass is less steep for more asymmetric systems.\n\nG. Distance reach with the 2.5PN FWF\n\nNext, we compare the distance-reach of the RWF and the 2.5PN FWF. The results are shown graphically in Fig. 5 and are similar in appearance to the mass-reach plot. The mass-reach of the RWF is ≃ 4 × 10 7 M ⊙ . For a system of total mass 5×10 7 M ⊙ , the plot shows that LISA can detect such binaries with an SNR of 10 at a luminosity distance of 100 Gpc (z ≃ 15). SMBHs of total mass ∼ 10 8 M ⊙ , not even observable using RWF templates, have a distance-reach as high as 10 Gpc (z ≃ 1.5) with an SNR of 10.\n\nProposals to extend the frequency band-width of LISA up to 10 -5 Hz have been discussed. In that case, the FWF can increase the mass-reach of LISA to even around 10 9 M ⊙ . More specifically, LISA can then observe a 10 9 M ⊙ system with an SNR of about 30 at 3 Gpc, if it uses templates based on the 2.5PN FWF for data-analysis.\n\nAll the results for SNR using the amplitude-corrected waveforms quoted earlier in this paper have been for a fixed choice of location and orientation of the source [defined by the angles(θ S , φ S , θ L , φ L )] with respect to the barycentre coordinate system. To conclude our present analysis, in this section we look into the variation in the value of SNR 12 0 1000 2000 3000 4000 5000 6000 SNR 0 0.01 0.02 0.03 0.04 0.05 Probability RWF FWF 10 100 1000 SNR due to FWF 0.02 0.04 0.06 0.08 Probalility (10 5 -10 6 )M sun 5.5 (10 6 -10 7 ) M sun FIG. 6: Distribution of SNR with sources randomly located and oriented in the sky. The left figure plots SNRs due to both RWF and FWF for a binary of mass (10 5 -10 6 )M⊙. For this mass, the most probable SNR for the FWF is lower than the most probable SNR for the RWF, like the trend shown in Table I . The right figure compares the SNRs due to the FWF for binaries of mass (10 5 -10 6 )M⊙ and 5.5(10 6 -10 7 )M⊙\n\nfor sources at various locations in the sky and various orientations. To this end, we consider a collection of sources randomly oriented in the sky and study the probability distribution of their SNRs. The results of our simulations (consisting of 8000 random realisations of the angles involved) are shown in Fig. 6 . From the left panel of Fig. 6 we see that the most probable SNR due to the FWF for a (10 5 , 10 6 )M ⊙ binary is less than the most probable SNR due to the RWF, indicating that this trend is independent of the source location and orientation. In the right panel we see that a binary of mass 2 × (10 6 , 10 7 )M ⊙ , which is undetectable by the RWF, can be observed by the FWF with a most-probable SNR of around 220.\n\nThe implications of amplitude corrected 2.5PN full waveforms (FWF) for the construction of detection templates for LISA are investigated in detail. With the FWF, LISA can observe sources which are favoured by astronomical observations, but not observable with restricted waveforms (RWF). This includes binaries in the mass range 10 8 -10 9 M ⊙ , depending on whether the lower cut-off for LISA is chosen to be at 10 -4 Hz or 10 -5 Hz. With an SNR of 10, these systems can be observed up to a redshift of about 1.5. The computation of the 3PN polarization, which will introduce an harmonic at 8Ψ (i.e. four times the dominant harmonic), in addition to the existing harmonics, could enhance the mass reach for equal mass binaries by 33% and unequal mass binaries by 14.3%.\n\nThe implication of the FWF for parameter estimation will be far more important than the extension of LISA's mass-reach reported here. From the work of Van Den Broeck and Sengupta in the context of ground-based detectors [24] it is already clear that most parameters will be estimated with errors ∼ ten times smaller as compared to RWF. This raises the interesting possibility that binary SMBH coalescences might be located on the sky with accuracies good enough for optical observations to identify the galaxy cluster and measure its red-shift. Needless to say that this improved estimation of source properties will have important consequences in shedding light on the dark energy, better understanding of SMBH formation and evolution, structure formation, etc., and is currently under investigation.\n\nIn this work we have confined ourselves to only non-spinning black-holes ignoring the effect of spin-orbit coupling at 1.5PN [30] and 2.5PN [31] and spin-spin effect at 2PN order [32] . The effect of spin is expected to be astrophysically significant and it is important to revisit the present analysis including spin in the future. Though partial results for GW polarisations including spin do exist, a more exhaustive exercise would be necessary before the FWF required for this work is available. The problem will also be more complicated due to modulations arising from spin-orbit and spin-spin couplings which would need to be addressed.\n\nIn this work we also restricted to the inspiral phase and used a physical picture of the LSO that is based on the test-particle limit. For comparable masses, the notion of LSO is not as sharp, or unique, and hence our results are probably idealized limits of the real situation. Numerical relativity [33, 34, 35] is maturing over the past couple of years and could soon provide waveforms for late inspiral and merger. It should then be possible to compare the results of such numerical templates with those studied in this paper to provide a better understanding of how higher harmonics facilitate the mass reach of our detectors. 13 Acknowledgments KGA acknowledges the Cardiff university for hospitality during the initial stages of this work and thanks Chris Van Den Broeck and Anand Sengupta for useful discussions on data-analysis with the FWF for ground-based detectors. KGA also acknowledges VESF. BRI, BSS and SS thank the Institut Henri Poincaré and BRI the Institut des Hautes Etudes Scientifiques for hospitality during the initial stages of this work. All the calculations reported in this paper are performed with Mathematica.\n\n[1] L. Ferrarese and D. Merritt, Astrophys. J. 539, L9 (2000), astro-ph/0006053. [2] D. Richstone, E. A. Ajhar, R. Bender, G. Bower, A. Dressler, S. M. Faber, A. V. Filippenko, K. Gebhardt, R. Green, L. C. Ho, et al., Nature 395, A14 (1998), astro-ph/9810378. [3] S. Komossa et al., Astrophys. J. 582, L15 (2003). [4] http://lisa.jpl.nasa.gov . [5] S. A. Hughes, Mon. Not. R. Astron Soc. 331, 805 (2002), astro-ph/0108483. [6] E. Berti, A. Buonanno, and C. M. Will, Phys. Rev. D 71, 084025 (2005), gr-qc/0411129. [7] E. Berti, A. Buonanno, and C. M. Will, Class. Quantum Grav. 22, S943 (2005), gr-qc/0504017. [8] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Class. Quantum Grav. 23, L37 (2006), gr-qc/0604018. [9] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Phys. Rev. D 74, 024025 (2006), gr-qc/0604067. [10] S. A. Hughes and K. Menou, Astrophys. J. 623, 689 (2005), astro-ph/0410148. [11] C. Cutler et al., Phys. Rev. Lett. 70, 2984 (1993), astro-ph/9208005. [12] L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet, Phys. Rev. D 65, 061501(R) (2002), Erratum-ibid 71, 129902(E) (2005), gr-qc/0105099. [13] L. Blanchet, T. Damour, G. Esposito-Farèse, and B. R. Iyer, Phys. Rev. Lett. 93, 091101 (2004), gr-qc/0406012. [14] C. Cutler, T. Apostolatos, L. Bildsten, L. Finn, E. Flanagan, D. Kennefick, D. Markovic, A. Ori, E. Poisson, G. Sussman, et al., Phys. Rev. Lett. 70, 2984 (1993). [15] C. Cutler and E. Flanagan, Phys. Rev. D 49, 2658 (1994). [16] L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wiseman, Phys. Rev. Lett. 74, 3515 (1995), gr-qc/9501027. [17] L. Blanchet, Phys. Rev. D 54, 1417 (1996), Erratum-ibid.71, 129904(E) (2005), gr-qc/9603048. [18] L. Blanchet, B. R. Iyer, C. M. Will, and A. G. Wiseman, Class. Quantum Grav. 13, 575 (1996), gr-qc/9602024. [19] K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Class. Quantum Grav. 21, 3771 (2004), erratum-ibid. 22, 3115 (2005) , gr-qc/0404185. [20] A. M. Sintes and A. Vecchio, in Rencontres de Moriond:Gravitational waves and experimental gravity, edited by J. Dumarchez (Frontires, Paris, 2000), gr-qc/0005058. [21] A. M. Sintes and A. Vecchio, in Third Amaldi conference on Gravitational Waves, edited by S. Meshkov (American Institute of Physics Conference Series, New York, 2000), p. 403, gr-qc/0005059. [22] C. Van Den Broeck, Class. Quantum Grav. 23, L51 (2006), gr-qc/0604032. [23] C. Van Den Broeck and A. Sengupta, Class. Quantum Grav. 24, 155 (2007), gr-qc/0607092. [24] C. Van Den Broeck and A. S. Sengupta, Class. Quantum Grav. 24, 1089 (2007), gr-qc/0610126. [25] C. Cutler, Phys. Rev. D 57, 7089 (1998). [26] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 62, 084036 (2000), gr-qc/0001023. [27] J. G. Baker and J. Centrella, Class. Quantum Grav. 22, S355 (2005), astro-ph/0411616. [28] K. G. Arun, Phys. Rev. D 74, 024025 (2006) , gr-qc/0605021. [29] L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha (2007), work in progress. [30] L. Kidder, C. Will, and A. Wiseman, Phys. Rev. D 47, R4183 (1993). [31] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006), erratum-ibid. 75, 049903 (E) (2007), grqc/0605140. [32] E. Poisson and C. Will, Phys. Rev. D 52, 848 (1995). [33] B. Bruegmann, W. Tichy, and N. Jansen, Phys. Rev. Lett. 92, 211101 (2004), gr-qc/0312112. [34] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr-qc/0507014. [35] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. D 73, 104002 (2006) , gr-qc/0602026."},"sections":{"kind":"list like","value":[{"section_type":"OTHER","section_title":"A. Supermassive black hole binaries and LISA","text":"There is strong observational evidence for the existence of supermassive black holes (SMBHs) with masses in the range of 10 6 M ⊙ -10 9 M ⊙ (see e.g. Ref. [1] and references therein) in most galactic nuclei [2] . Therefore, mergers of galaxies, as evidenced by high-redshift surveys, should give rise to binaries containing SMBHs. Late stage evolution of a SMBH binary is dictated by the emission of gravitational radiation. The resulting loss of orbital energy and angular momentum would lead to the coalescence of the two holes. Indeed, X-ray observations have revealed the existence of at least one such system that would coalesce within the Hubble time [3] . Gravitational waves (GW) emitted in the process could be detected by the planned Laser Interferometer Space Antenna (LISA) [4] .\n\nObservation of SMBH binaries at high redshifts is one of the major science goals of LISA. These observations will allow us to probe the evolution of SMBHs and structure formation [5] and provide an unique opportunity to test General Relativity (and its alternatives) in the strong field regime of the theory [6, 7, 8, 9, 10] . Observing SMBH coalescences with high (100-1000) SNR [8, 9] is crucial for performing all the aforementioned tests."},{"section_type":"OTHER","section_title":"B. Restricted Vs Full Waveforms as Search templates in LISA","text":"Motivated by the fact that matched filtering is more sensitive to the phase of the signal than its amplitude [11] , search algorithms so far have deployed a waveform model involving only the dominant harmonic (at twice the orbital frequency), although the phase evolution itself is included to the maximum available post-Newtonian (PN) order 2 (currently 3.5PN, for non-spinning systems [12, 13] ). Waveforms in which all amplitude corrections are neglected, but the phase is treated to the maximum available order, are called restricted waveforms (RWF) and these are what are used so far in the analysis of data from ground-based detectors [14, 15, 16, 17] . This paper will consider the advantage of using the full wave forms (FWF) in the context of LISA.\n\nLISA is designed to detect gravitational waves in the frequency-band 0.1-100 mHz. This frequency range determines the range of masses accessible to LISA because the inspiral signal would end when the system's orbital frequency reaches the mass-dependent last stable orbit (LSO). In the test-mass approximation, the angular velocity ω LSO at LSO is given by ω LSO = 6 -3/2 M -1 , where M is the total mass of the binary. Search templates that contain only the dominant harmonic cannot extract power in the signal beyond f LSO = ω LSO /π ≃ 4.39(M/10 6 M ⊙ ) -1 mHz. This further implies that the frequency range [0.1, 100] mHz corresponds to the range ∼ 4.39 × [10 4 , 10 7 ]M ⊙ for the total mass of binary black holes that would be accessible to LISA 1 . However, as Table 1 of Ref. [1] would reveal, there is observational evidence for the existence of many SMBHs whose masses are of the order of 10 8 -10 9 M ⊙ . LISA will be unable to observe binaries containing SMBHs in this mass range if it used as search templates waveforms containing only the dominant harmonic.\n\nInclusion of higher-order amplitude terms in the waveform introduces the following two new features: (i) appearance of higher harmonics of the orbital phase and (ii) PN amplitude corrections to the leading as well as higher harmonics of the orbital frequency. For example, at 0.5PN order, which is the first-order correction, there are two new harmonics Ψ and 3Ψ, where Ψ is related to the orbital phase of the binary as in Refs. [18, 19] . More interestingly, in the expressions for the 'plus' and 'cross' polarizations, all odd harmonics of the orbital frequency are proportional to δm M , where δm is the difference in the masses of the binary components (see Eq. (5.7)-(5.10) of Refs. [19] ). Another important feature of the full waveform is that the (2n + 2) th harmonic first appears at the n th PN order in amplitude 2 . For example, the fourth harmonic first appears at 1PN, and has PN amplitude corrections to its dominant term at 2PN and 2.5PN (see Refs. [18, 19] for details).\n\nEarly investigations on the importance of amplitude-corrections to search templates were carried out by Sintes and Vecchio [20, 21] . Their study used only the first-order correction at 0.5 PN order. They concluded that the addition of the amplitude terms in the waveform did not improve the accuracy in the estimation of source's angular position and the distance, whereas the estimation of the chirp and reduced masses could be 10 times better when compared to the RWF. Recently, in the context of ground-based detectors, Van Den Broeck and Sengupta [22, 23, 24] examined the implications of going beyond the restricted PN approximation and employing instead the full waveform [18, 19] . The two main implications of the comprehensive analysis in Refs. [22, 23] for terrestrial GW detectors may be summarized as follows:\n\n1. For binary neutron stars and stellar mass black holes, restricted waveforms over-estimate the SNR as compared the full waveform.\n\n2. The use of the full waveforms significantly increases the mass-reach of second and third generation detectors, advanced LIGO and EGO being able to observe systems with total mass ∼ 400M ⊙ and a third generation detector as high as 10 3 M ⊙ .\n\nIn the present paper, we study in the context of LISA the implication of using templates based on the FWF (i.e. including all known harmonics of the orbital phase and all known amplitude corrections in the GW polarisations). Coalescences of SMBH binaries with masses ∼ 10 8-9 M ⊙ will not be observable by LISA if one uses only templates based on the RWF. Using templates based on amplitude corrected full waveforms, instead of the usual restricted waveforms, will enable LISA to observe coalescences of SMBH binaries with total mass ∼ 10 8 M ⊙ (10 9 M ⊙ ) if the lower frequency cut-off LISA can achieve is ∼ 10 -4 Hz (10 -5 Hz).\n\nThe rest of this paper is organized as follows: In Section II, we give the FWF in the frequency domain, by taking into account the orbital motion of LISA around the sun and its changing orientation. Section III discusses the results of our investigations where we compare the performances of the amplitude-corrected waveforms at different PN orders in terms of their mass-reach and distance-reach and correlate it to the 'observed' spectrum in LISA. Section V concludes with a brief summary of the main results and assumptions underlying their derivation.\n\n1 Although, binaries lighter than 10 4 M ⊙ would, in principle, evolve through the LISA band they would not be luminous enough to be visible in LISA unless they are close-by. 2 The 0.5PN term is an exception to this and also introduces a harmonic at the orbital frequency apart from the one at thrice the orbital frequency.\n\nII. TEMPLATE WAVEFORMS FOR LISA A. Amplitude corrected waveform\n\nFor non-spinning binaries in quasi-circular orbits inspiralling due to radiation-reaction, waveforms were computed in Refs. [18, 19] up to 2.5PN order in amplitude and 3.5PN in phase [12, 13] . This waveform h(t) is a linear combination of sine and cosine functions of multiples of the orbital phase Ψ(t). The expression for the 2.5PN polarization contains the first seven harmonics of the orbital phase, the dominant harmonic being the one at twice the orbital phase. The signal depends on the following parameters: D L , the luminosity distance to the binary, m the total (red-shifted) mass, ν the symmetric mass-ratio (reduced mass divided by total mass), the spherical polar angles (θ, φ) determining the direction of the \"line-of-sight\", the inclination angle ι of the angular momentum L of the binary with respect to the direction opposite to the line-of-sight, and the polarization angle ψ which determines the orientation of the projection of L in the plane normal to the line-of-sight.\n\nWe rewrite the waveform in terms of only cosines in a form similar to [23]: h(t) = 2M ν D L 7 k=1 5 n=0 A (k,n/2) cos[kΨ(t) + φ (k,n/2) ] x n 2 +1 (t), (2.1)\n\nwhere the coefficients A (k,n/2) and φ (k,n/2) are functions of (ν, θ, φ, ψ, ı), and x(t) = (2πM F (t)) 2/3 is the post-Newtonian parameter with F (t) the instantaneous orbital frequency. Terms 2Mν DL x n/2+1 (t) A (k,n/2) and φ (k,n/2) are the wave amplitude and polarization phase, respectively, corresponding to the k th harmonic and (n/2) th PN order. We call the coefficients A (k,n/2) the polarization amplitudes. The orbital phase Ψ(t) is a PN series in x, which, in the case of non-spinning binaries, is known to 3.5PN order [12, 13] . For a non-spinning source and a detector whose position and orientation are almost constant during the time of observation of the signal, all the above mentioned angles are constants. For ground-based GW detectors dealt with in Ref. [22, 23] , one is in this situation.\n\nB. Amplitude corrected waveform including modulations due to LISA's orbital motion -Time Domain\n\nLISA will be able to observe many sources from their early stages of inspiral and most would last for a pretty long time. We shall only consider binary sources that last for a year or less before merger. Since the LISA plane is tilted by 60 • with respect to the plane of the ecliptic, during the course of its heliocentric orbit its orientation and position varies periodically, with a period of one year and the signal in Eq. (2.1) will suffer additional amplitude and phase modulations. Thus in the case of LISA the angles θ, φ, and ψ (but not ı) appearing in Eq. 2.1 are functions of time.\n\nTo proceed further, in the frame of a non-rotating observer fixed to the solar-system barycentre, we denote by the location of the source on the sky by the spherical polar angles θ S and φ S and the orientation of the source by the spherical polar angles θ L and φ L determining the direction of the orbital angular momentum L of the binary. The transformation between the fixed set of angles 3 (θ S , φ S , θ L , φ L ) and the time-dependent angular coordinates of the source (θ(t), φ(t), ψ(t), ı) as measured by LISA are given in Ref. [25] .\n\nGeneralizing, Ref. [25] from the RWF to the FWF, the signal as seen in LISA is of the form,\n\nh(t) = √ 3 2 2M ν D L 7 k=1 5 n=0 A (k,n/2) (t) cos[kΨ(t) + φ (k,n/2) (t) + kφ D (t)] x n 2 +1 (t) . (2.2)\n\nThe PN parameter x(t) appearing in Eq. (2.2) is still equal to (2πM F (t)) 2/3 , where F (t), however, is the orbital frequency as measured by a non-rotating observer located at the solar-system barycentre. The term φ D (t) is the Doppler phase [25] , accounting for the phase difference of the gravitational wave-front between LISA and the solarsystem barycentre. The time-dependence of φ D (t) is due to the orbital motion of LISA about the barycentre. It is given by\n\nφ D (t) = 2 π F (t) R sin θ S cos[φ(t) -φ S ], (2.3)\n\nwhere R = 1 AU and φ(t) is the angular position of LISA with respect to the barycentre given by φ(t) = 2 π t T , T being equal to one year.\n\n3 This is a different notation from [25], where the source angles measured in the fixed barycentre frame are denoted by ( θS , φS , θL , φL ) 4 C. Amplitude corrected waveform including modulations due to LISA's orbit -Frequency Domain\n\nThe above waveform is valid in the adiabatic regime, where the radiation-reaction time-scale is much larger than the orbital time-scale. We also note that the additional amplitude and Doppler modulations in the waveform for LISA vary on time-scales of 1 yr (i.e. ∼ 3 × 10 7 s), while LISA can observe orbital periods at most up to 2 × 10 5 s, (i.e. gravitational wave frequencies of order 10 -5 Hz.). Consequently, the Doppler modulations change much more slowly (a hundredth) than the orbital phase. This permits the use of the stationary phase approximation (SPA) to obtain an analytical form for the Fourier transform (FT) h(f ) of the signal:\n\nh(f ) ≃ √ 3 2 2M ν D L 7 k=1 5 n=0\n\nA (k,n/2) (t(f /k)) x n 2 +1 (t(f /k)) e -iφ (k,n/2) (t(f /k))\n\n2 k Ḟ (t(f /k)) exp [i ψ f (t(f /k))] , (2.4)\n\nwhere an over dot denotes derivative with respect to time and ψ f (t(f /k)) is given by ψ f (t(f /k)) = 2πf t(f /k) -k Ψ(t(f /k)) -k φ D (t(f /k)) -π/4. (2.5)\n\nThe PN expansions for t(F ), Ψ(F ), Ḟ (F ) are given in Ref. [12] . The expression in Eq. (2.4) within the summation over k represents the FT due to the k th harmonic. It should be noted that the term Ḟ may be treated in different ways that could lead to numerically different results. In a numerical treatment, for instance, one could avoid performing a further re-expansion. Alternatively, one could re-expand the denominator in the amplitude and truncate the resulting expression at the n th PN order, to obtain the nPN amplitude-corrected waveform. Ref. [22, 23] choose the latter and we follow them in this work. Radiation reaction results in an increase in the orbital frequency F (t) which will ultimately drive the system beyond the adiabatic inspiral phase and the inspiral waveform given above will no longer be valid. In the first approximation this is expected to occur when the orbital frequency F (t) reaches F LSO -the orbital frequency of the LSO of a Schwarzschild solution with the same mass as the binary's total mass M ,\n\nF LSO = (2 π 6 3 2 M ) -1 .\n\n(2.6) Thus, we truncate the signal in the time domain at a time t LSO , given implicitly by F (t LSO ) = F LSO . In the SPA, the main contribution to the FT of the k th harmonic at a given Fourier frequency f , comes from the neighbourhood of the time when the instantaneous value of the k th harmonic sweeps past f . Thus the k th harmonic is not expected to contribute significant power to the FT for frequencies above k F LSO , if the signal is truncated in the time domain beyond t LSO . This motivates the truncation of the FT due to the k th harmonic at frequencies above k F LSO by a step function θ(k F LSO -f ) [θ(x) = 1, for x ≥ 0, and 0 for negative x]."},{"section_type":"OTHER","section_title":"III. SIGNAL TO NOISE RATIOS IN LISA WITH HIGHER HARMONICS","text":"In this Section we investigate the effect of the higher harmonics in LISA observations of supermassive black hole binaries. The LISA waveform discussed in the previous Section will be used for the analysis.\n\nGiven a waveform h, the best signal-to-noise ratio (SNR) achieved using an optimal filter is given by ρ[h] ≡ (h|h) 1/2 , where ( . | . ) is the usual inner product in terms of the one-sided noise power spectral density S h (f ) of the detector. With the convention for Fourier transforms, x(f ) = ∞ -∞ x(t) exp(-2πif t) dt, the inner product is given by:\n\n(x|y) ≡ 4 f end fs Re[x * (f )ỹ(f )] S h (f ) df. (3.1)\n\nFor an optimal filter, which maximises the overlap of the signal with template, one can write\n\nρ 2 = 4 f end fs | h(f )| 2 S h (f ) df. (3.2)\n\nWe use the non-sky-averaged noise-spectral-density as given in Eqs. (2.28)-(2.32) of Ref. [6]. 5 10 -4 10 -3 10 -2 frequency(in Hz) 0.0 2.0×10 6 4.0×10 6 6.0×10 6 8.0×10 6 Pf(f) RWF FWF 10 -4 frequency(in Hz) 0.0 5.0×10 5 1.0×10 6 1.5×10 6 2.0×10 6 Pf(f) RWF FWF FIG. 1: The observed spectrum, P(f f ) , in LISA using the full (solid, red) and restricted (dashed, blue) waveforms, for two archetypal binaries, (10 5 , 10 6 )M⊙ (left) and 2 × (10 6 , 10 7 )M⊙ (right). The sources are assumed to be at 3 Gpc and their orientation with respect to the solar-system barycentre is chosen to be θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3. The spectrum is much more complicated and highly oscillatory for the FWF than for the RWF, because of interference between various harmonics. The higher frequency reach of the FWF is due to presence of higher harmonics as apparent in the figure . The spectrum for the system in the left panel sharply rises at a frequency ∼ 2 × 10 -3 Hz. Beyond this frequency, the effective LISA noise decreases sharply with increasing frequency (as there are fewer galactic binaries per frequency bin) leading to the observed increase in the spectrum.\n\n) ≡ d(ρ 2 ) d(ln f ) = f | h(f )| 2 S h ("},{"section_type":"OTHER","section_title":"A. Choice of frequency cutoffs f end , fs","text":"The upper limit of integration f end is taken to be the minimum of 7 F LSO and 1 Hz, the latter being a conventional upper cut-off for the LISA noise curve. The lower limit f s is chosen assuming LISA observes the inspiral for a duration ∆t obs before it reaches the LSO. Using the quadrupole formula, we find that the orbital frequency at the epoch t LSO -∆t obs is given by\n\nF (t LSO -∆t obs ) = F LSO (1 + 256 ν 5 M ∆t obs v 8 LSO ) 3 8 , (3.3)\n\nwhere v LSO is the orbital velocity and t LSO the epoch at which the orbital frequency reaches the value F LSO . We take v LSO = 1/ √ 6, the orbital velocity at the LSO in the case of a test mass orbiting a Schwarzschild black hole. We designate F (t LSO -∆t obs ) as F in . Thus the k th harmonic will have a frequency k F in , ∆t obs before t LSO . The above formula reduces to the simpler form in Ref. [6] as v LSO → ∞. For the mass values explored in this work there is no significant dependence of the results on this choice. In all our calculations we take ∆t obs to be one year.\n\nThe lower cut-off for the k th harmonic should be the maximum of the lower cut-off of LISA (10 -4 Hz) and k F in and simply implemented by truncating the waveform due to the kth harmonic by another step-function θ(f -k F in ) and choosing f s to be 10 -4 Hz. It is worth noting that the k th harmonic probes a larger interval of the frequency domain i.e. k(F LSO -F in ) relative to the fundamental harmonic. For brevity, we refer to this as the span of the k th harmonic.\n\nThere is a caveat with regard to the use of higher harmonics that is worth mentioning: In the time-domain the waveform should begin when the highest harmonic reaches the lower cutoff. This has an implication on data analysis as the templates will be an order-of-magnitude longer than before. Thus, it might be sensible to use higher harmonics only in the case of higher masses."},{"section_type":"OTHER","section_title":"B. Observed signal spectrum with LISA","text":"To get some insight into the effect of higher harmonics via amplitude corrections let us first look at the SNR integrand, i.e., the \"noise-weighted signal power\" per unit logarithmic frequency interval [26] . Rewriting the expression for the SNR as\n\nρ 2 = 4 f end fs f | h(f )| 2 S h (f ) d ln(f ), (3.4) 6\n\nTABLE I: SNRs due to successive PN amplitude-corrected waveforms, with phase corrections to 3.5 PN order in all cases. The orientation of the source with respect to the solar-system barycentre is chosen to be θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3. For the (10 6 -10 7 )M⊙ binary system, all harmonics enter deep into the sensitive part of the LISA bandwidth. Apart from an increase at 0.5PN, we see a consistent reduction in the SNR on inclusion of higher PN order amplitude corrections. For the (5.5 × 10 6 , 5.5 × 10 7 )M⊙ binary system, the second harmonic fails to enter the LISA bandwidth, while the third harmonic spans a small insensitive region. Thus the SNR due to the RWF is zero, while the SNR due to the 0.5PN waveform is smaller than the SNRs due to higher order PN terms. Both sources are at a distance of 3 Gpc. PN SNR order (10 6 -10 7 )M⊙ 5.5 × (10 6 -10 7 )M⊙ 0 924.48 0 0.5 1025.8 211.98 1 928.48 343.17 1.5 869.78 319.34 2 824.65 266.65 2.5 809.51 277.34 the quantity of our interest is\n\nP(f ) ≡ d(ρ 2 ) d(ln f ) = f | h(f )| 2 S h (f ) , (3.5)\n\nwhich is designated as the \"observed spectrum\", following [24] . The observed spectrum is plotted versus frequency for given masses in Fig. 1 . As is the case for ground-based detectors [24], the spectrum due to the FWF has a lot more structure and is highly oscillatory because of interference between various harmonics. For the (10 5 , 10 6 )M ⊙ system, the mass being low, the second harmonic and hence the RWF extends up to frequencies ∼ 2 × 10 -3 Hz, where LISA is most sensitive. This leads to a rapid increase in the observed spectrum in this frequency region. The spectrum due to the FWF, containing higher harmonics continue beyond the RWF into the most sensitive part of the LISA band. For the 2(10 6 , 10 7 )M ⊙ system, the frequency span of the second harmonic is small and the sensitive region of the LISA band lies beyond its maximum reach."},{"section_type":"OTHER","section_title":"IV. THE EFFECT OF HIGHER HARMONICS","text":"Following the analysis of Ref. [22, 23] , we classify the sources into two types: In the first category are sources for which the dominant (second) harmonic has a large frequency span in the LISA band. The second category on the other hand comprise sources whose dominant harmonic fails to enter the LISA bandwidth but the higher harmonics do. Since the upper cut-off frequency for each harmonic is inversely proportional to the total mass (from the expression for F LSO ), we note that the sources of the first type will have total mass less than some value which we call the RWF mass-reach, the maximum mass detectable by the RWF, while the second type will have masses greater than this value. The condition that the upper cut-off of the dominant harmonic is less than or equal to the lower cut-off of LISA (i.e., by the inequality 2 F LSO ≤ f s ) determines the RWF mass-reach. The choice of f s for the LISA mission is still not clear and theoretical implications of this choice are explored in e.g. Ref. [27] . For f s in the range [10 -5 , 10 -4 ] Hz the RWF mass-reach varies over the range [4.39 , 43.9] × 10 7 M ⊙ , the lower end of the mass range corresponding to the higher end of the frequency range."},{"section_type":"OTHER","section_title":"A. How higher harmonics affect signal visibility","text":"In Fig. 2 we plot the SNRs computed using the restricted (RWF) and full (FWF) waveforms as a function of the binary's total mass for two values of the mass ratio 4 . We first consider systems whose total mass is less than 4 Our codes are calibrated by reproducing the results of [22, 23] , which considers ground-based detectors, and also of [28] , which computes SNRs in LISA using RWF.\n\n7 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Total Mass (in Msun) 10 0 10 1 10 2 10 3 SNR RWF; m 1 /m 2 =0.1 FWF; m 1 /m 2 =0.1 RWF; m 1 /m 2 =0.01 FWF; m 1 /m 2 =0.01 FIG. 2: SNR versus total mass for mass ratios of 0.1 and 0.01. The figure shows that apart from the dips due to white-dwarf confusion noise, for mass values where the RWF enters the LISA band, the corresponding SNR is consistently more than the SNR produced by the FWF. However, for mass values where the second harmonic terminates before it reaches the LISA bandwidth, the FWF which has higher harmonics that enter the LISA band produces significant SNRs. The frequency reach of a harmonic depends only on the total mass, and thus the mass reach of the FWF is independent of the mass-ratio. For more asymmetric systems, the magnitude of the SNR is low for all masses both for the RWF and the FWF. Sources are at a luminosity distance of 3 Gpc with fixed angles given by θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3.\n\n4 × 10 7 M ⊙ . For these systems, the SNRs computed using the two different approximations agree with each other to within 10%, with the RWF over-estimating the SNR, when compared to the FWF, in most of the range. This is explicitly shown for a (10 6 , 10 7 )M ⊙ binary in the first column of Table I . Indeed, but for the slight increase in SNR as we go from 0PN to 0.5PN, we find a steady decrease as one increases the PN order of the amplitude correction. The reduction in SNR at higher PN orders can be understood by studying the structure of | h(f )| 2 , the numerator in the integrand of the SNR in Eq. (3.2). There are basically three types of terms: 1. direct terms in which the phases in Eq. (2.4) cancel 2. interference terms between different PN corrections of the same harmonic,\n\nA 2 (k,n/2) (t(f /k)) f -7 3 (M f ) 2n 3 ,\n\nA (k,m/2) (t(f /k)) A (k,n/2) (t(f /k)) f -7 3 (M f ) m+n 3 cos[φ (k,m/2) (t(f /k)) -φ (k,n/2) (t(f /k))]\n\n3. harmonic mixtures 5 which are terms containing the interference between different PN corrections of different harmonics, e.g. the m/2 th PN correction of the k th harmonic and n/2 th PN correction of the l th harmonic.\n\nA (k,m/2) (t(f /k)) A (l,n/2) (t(f /l)) f -7 3 (M f ) m+n 3 cos[ψ f (t(f /k)) -φ (k,m/2) (t(f /k)) -ψ f (t(f /l)) + φ (l,n/2) (t(f /l))]\n\nwhere ψ f (t(f /k)) is given by Eq. (2.5), All these terms are scaled by M 5/3 , where M = M ν 3/5 is the chirp-mass. (Additional multiplicative factors have been omitted in the above expressions, among which are the step-functions mentioned earlier and PN expansion coefficients of the denominator of the Fourier amplitude in Eq. (2.4), the latter being time-independent.)"},{"section_type":"OTHER","section_title":"B. The effect of higher harmonics in ground-based detectors","text":"Before we explain the SNR trends in the context of LISA, we mention that for ground-based detectors a similar effect was found in Ref. [23] for a different but corresponding mass region. The lower cut-off for a typical ground-based 5 We use the term 'harmonic mixtures' at the risk of being mistaken to the well-known 'harmonic mixing' in music. Our use of the phrase 'harmonic mixtures' is simply to convey the physical effect of the interference between different harmonics 8 1×10 -4 1×10 -3 frequency(Hz) 0 0.5 1 polarisation amplitude A (2,0) 0.00 3.15×10 7 6.31×10 7 time (in sec) 0 0.5 1 polarisation amplitude A (2,0) polarisation amplitude A (2,0) time ( in years ) 0 1 2 0.5 1 FIG. 3: Variation of polarisation amplitude of the RWF with frequency and time (inset). The inset, plotted over a duration of two years clearly shows periodicity due to LISA's orbital motion around the Sun. The binary mass, (10 6 -10 4 )M⊙, has been chosen such that it can, in principle, be observed for two years. The plot in the frequency domain shows that the variation of the polarisation amplitude is confined to a very small part of the frequency span of the dominant harmonic, and essentially behaves as a constant in the frequency domain.\n\ndetector, say Advanced LIGO is 20Hz, and the effect of higher harmonics is seen for masses less than ∼ 220M ⊙ . In that case, as mentioned earlier, the polarisation amplitudes and phases are constants. The RWF contains only the Newtonian term of the second harmonic and thus | h(f )| 2 consists of a single direct term with n = 0 and k = 2. With the inclusion of higher-order amplitude terms in the waveform, PN corrections to the dominant harmonic, and higher harmonics and their PN corrections, also contribute to the SNR. In other words, the signal power spectrum | h(f )| 2 will contain all three types of terms discussed before. From the form of the direct terms, it is evident that their contribution to the SNR will be positive definite. We also note that, for ground-based detectors, the frequency dependence of the direct and interference terms will just be a power law. However, the sign of the interference terms (and consequently their contribution to the SNR) depends on the difference between the polarisation phases of different PN corrections for the same harmonic. Van Den Broeck and Sengupta showed that for a given harmonic, for all allowed values of the parameters (ν, θ, φ, ψ, ı), each PN correction is almost \"out of phase\" with both the PN correction preceding and succeeding it 6 . The resulting negative terms (representing destructive interferences) reduce the SNR as one includes higher PN amplitude corrections in the waveform.\n\nThe third type of terms, harmonic mixtures, however, are highly oscillatory functions of the frequency, as the phase difference ψ f (t(f /k)) -ψ f (t(f /l)) between the k th and the l th harmonic become even or odd multiples of π. As one integrates over f , these oscillations tend to cancel out, and thus the contribution to the SNR from these terms are numerically much smaller relative to the first two types of terms.\n\nC. Effect of higher harmonics for binaries with M < 4 × 10 7 M⊙\n\nIn the case of LISA, because of the polarisation factors, the amplitudes of none of the three types of terms is a simple power-law in f . The periodic variation of, for example, A (2,0) (period being one year) appears as an amplitude modulation A (2,0) (t(f /2)) in the Fourier transform, where the argument t(f /2) of A 2,0 is given by t(f /2) = -5 256π 8/3 M 5/3 1 f 8/3 + PN corrections. (4.1) Hence, in the frequency domain A (2,0) will undergo one complete oscillation as f varies from 2F in (see Eq. (3.3)) to 2F LSO . However, because of the inverse power-law dependence on f , the oscillation of A (2,0) is confined to a small frequency interval above F in and remains fairly constant over a major portion of the frequency span 2(F LSO -F in ) (see Fig. 3 ). For masses higher than the one shown in Fig. 3 , this region of significant variation moves to the left of the figure. On including in our analysis the effect of detector sensitivity (weighting down by S h (f )) this variation of 6 Note, however, that Ref. [23] , argues this in a somewhat different form.\n\n9 10 7 10 8 Total Mass (in Msun) 10 1 10 2 10 3 SNR Newtonian 0.5PN 1PN 1.5PN 2PN 2.5PN 10 7 10 8 Total Mass (in Msun) 10 1 10 2 10 3 SNR Newtonian 0.5PN 1PN 1.5PN 2PN 2.5PN FIG. 4: SNR versus total mass for successive PN amplitude-corrected waveforms and 3.5PN phasing. The left panel corresponds to a mass-ratio of 0.1 while the right panel plots the same quantities for mass-ratio of 1 (equal mass systems). The (2n + 2) th harmonic first appears at the nth PN order. For a binary of given total mass, the upper cut-off of the k th -harmonic of the orbital frequency in the frequency domain is proportional to k and inversely proportional to the total mass. As the mass increases the upper cut-off for the 2 nd harmonic falls below the lower cut-off of the LISA detector, leading to a zero value of SNR due to the RWF. The higher harmonics still enter the sensitive bandwidth of LISA and higher PN order waveforms produce significant SNR. The 2.5PN waveform has the highest mass-reach, being 3.5 times the mass-reach of the RWF. In the equal mass case displayed in the right panel, the differences in harmonic content of different PN order waveforms are more pronounced, as odd harmonics are absent. Sources are at a luminosity distance of 3 Gpc with fixed angles given by θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3.\n\nA (2,0) gets damped out when one evaluates the integral in Eq. (3.2). For masses satisfying 2F in ≪ 10 -4 Hz, the lower cut-off for LISA, this region of variation will fall below the LISA band. The polarisation phases determining the sign of the interference terms between the same harmonics also vary with f . However, as mentioned earlier, the phase relationships of the polarisation phases are independent of the parameter values. Thus the modulations which change the values of (θ, φ, ψ, ı) do not affect the trend of reduction of SNR with amplitude corrections. The Doppler modulations, which appear in only harmonic mixtures, are also not important as far as SNR is concerned.\n\nFinally, we would like to note an important point not explicitly mentioned in Ref. [23] . As the difference between the polarisation phases of successive PN corrections of the same harmonic tend to be nearly π, alternate PN corrections necessarily interfere constructively. Hence there are positive contributions also from the interference terms. Now, the numerical value of the contribution to SNR from each of these terms depends on the magnitude of the polarisation amplitude and the power of (M f ). It can be checked that for all allowed values of (ν, θ, φ, ψ, ı) , the polarisation amplitudes are roughly of the same order in magnitude. Consider the Newtonian term of the dominant harmonic and its interference with the first two corrections at 1PN and 1.5PN. The Newtonian term will be out of phase with the 1PN term, but in phase with the 1.5PN one. The two corresponding interference terms will contain powers of (M f ) 2/3 and (M f ), respectively, and since they have the same frequency span, the absolute numerical value of the contribution to SNR from the former will be more since (2πM f ) 1/3 will always be less than 1/ √ 6. Numerical values of contributions from interference between higher PN corrections of the second harmonic successively decrease. The same argument applies for all the other harmonics, and thus, inclusion of amplitude corrections will lead to an overall reduction in SNR.\n\nThe first column of Table I clearly demonstrates the effect of higher harmonics on SNRs. The increase in SNR for the 0.5PN waveform (with respect the RWF SNR) is also easily explained by noting that the 0.5PN correction only adds (apart from harmonic mixtures) two direct terms to | h(f )| 2 , corresponding to the first and third harmonics (n = 1, n = 3). Clearly, from the discussion in the previous subsection, the 0.5PN waveform will have a higher SNR than the RWF, independent of the binary parameters.\n\nFor 10 3 < ∼ M < ∼ 10 5 M ⊙ , the difference between the RWF and the FWF is not visible on the scale of Fig. 2 because for this mass range all the direct and interference terms corresponding to harmonics higher than the dominant ones, which are scaled by higher powers of (M f ), are negligible.\n\n10 D. Visibility of systems with M > 4 × 10 7 M⊙ In their analysis of the implications of the FWF for ground-based detectors Van Den Broeck and Sengupta [22, 23] pointed out an interesting effect due to higher harmonics. An analogous effect is found in the case of LISA in spite of the additional amplitude and Doppler modulations that exist in this case.\n\nNormally, the harmonic at twice the orbital frequency dominates the SNR. However, when the dominant harmonic fails to reach the LISA band the higher harmonics become important, which transpires for masses greater than 4 × 10 7 M ⊙ . Even though the second harmonic falls below the lower cut-off f s of the LISA bandwidth, the kth harmonic, k > 2, that has power up to a frequency k F LSO , might cross f s and produce a significant SNR. Of course, the kth harmonic would fall below the LISA sensitivity band for masses which satisfy the equality f s = k F LSO . Thus, higher PN order waveforms, which bring in higher harmonics, are capable of producing a significant SNR, even when the RWF fails to produce any.\n\nLet us examine this in a little more detail starting from the values of mass where the second harmonic dominates and the RWF is adequate. Eventually, for larger values of the total mass, the inequality f s ≥ 3F LSO becomes true. Then the 0.5PN waveform, which contains the first and the third harmonic, terminates before reaching f s and consequently the SNR due to the 0.5PN waveform goes to zero. SNRs for different PN waveforms for a binary whose dominant harmonic falls below f s and the third harmonic has a small span in the LISA bandwidth is given in the second column of Table I. Note that for the 5.5(10 6 -10 7 )M ⊙ system, the 1PN waveform has a higher SNR than the 0.5PN one. This is due to the absence of the first harmonic and the small span of the third harmonic in the LISA bandwidth. Further, the 2.5PN waveform has a slightly larger SNR compared to 2PN. This is due to the absence of the first and second harmonic and the small contribution from the third harmonic, all of which contribute interference terms due to their 2.5PN corrections. However, this increase is marginal, and is not generic. We have explicitly checked by choosing different angles that there can be a small decrease also. The detailed results for LISA are summarised in Fig 4 . We see that for masses for which the 1PN waveform fails to reach the LISA bandwidth, the higher PN order amplitudes are capable of producing SNRs as high as 100! Thus, the use of the FWF will enable LISA to make observations of SMBHs in the astrophysically interesting mass-regime, which would not be possible had one used only the standard RWF.\n\nUsing the expression for F LSO , it is simple to argue that the mass reach for the 2.5PN FWF, which has the seventh harmonic of the orbital frequency, is 7/2 times the RWF (around 1.5 × 10 8 M ⊙ ). The above ratio, of course, depends on the assumption that the Schwarzschild (test particle case) LSO frequency will not be very different from the LSO frequency in the comparable mass case.\n\nWe conclude with a discussion of a minor, but clear, feature seen in Fig. 2 for LISA, but not present for the ground-based detectors, concerning the relative values of the SNR obtained using the RWF and the FWF. For most of the mass range probed the RWF overestimates the SNR relative to the FWF; however, the figure clearly shows an anomaly for masses around ∼ 2 × 10 6 M ⊙ . To understand this, we first note that the dips in the two curves in Fig 2 , \nare due to the bump in the LISA noise-curve [6] just above 10 -3 Hz. This bump is due to the domination of whitedwarf confusion noise over instrumental noise and lies just below the most sensitive frequency region (∼ 3 × 10 -3 Hz -2 × 10 -2 Hz) of the LISA band. Below 3 × 10 -3 Hz, the noise increases sharply till one reaches the bump. For binaries of mass greater than 1.5 × 10 6 M ⊙ , the frequency span of the dominant harmonic ends just around the bump and the sensitive region of the LISA band is beyond the span of this harmonic. However, higher harmonics incorporated in the FWF are able to reach the sensitive part of the noise curve. This leads to higher SNR for the FWF relative to the RWF. This reversal of trend continues up to masses 4 × 10 4 M ⊙ . Above this mass, the frequency span of the seventh harmonic ends before the sensitive region of the LISA band and the general trend is restored.\n\nFor still higher mass values, the SNRs due to the RWF and the FWF both increase until the second harmonic fails to reach the LISA band. This is due to the overall scaling of the waveform with the total mass. At such high values, it is able to compensate both for the decreasing frequency span and the higher noise of the detector in this frequency range."},{"section_type":"OTHER","section_title":"E. Effect of higher harmonics in the equal mass case","text":"In contrast to asymmetric systems discussed so far, for systems of equal mass all odd harmonics are absent. Consequently, for symmetric systems the mass-reach of the 2.5PN FWF will be only 3 times the mass-reach of the RWF. Further, from the right panel of Fig. 4 , it is clear that the 0.5PN and the 0PN, or RWF, are identical, as are the 1PN and 1.5PN waveforms. Thus the decrease in SNR for the higher PN order waveforms with increasing total mass is more pronounced than in the unequal-mass case. We also note that for masses for which the second harmonic fails to reach the detector bandwidth, the 2PN waveform has a lower SNR than the 2.5PN waveform. This can be explained by noting that for these masses only the fourth and sixth harmonics enter the LISA bandwidth. 11 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Total Mass (in Msun) 10 2 10 3 10 4 10 5 10 6 Luminosity Distance (Mpc) FWF RWF FIG. 5: Luminosity Distance (in Mpc) versus total mass for a fixed SNR of 10. The systems have mass-ratio of 0.1. The distance reach can be as large as 500 Gpc for systems where the second harmonic enters the LISA bandwidth. Systems undetectable by the RWF (of mass around 10 8 M⊙) can be detected by the FWF at distances up to 10 Gpc. The location and orientation of the sources are the same as in the earlier figures.\n\nThe 2PN waveform contains the leading term of the fourth harmonic at 1PN and its 2PN correction, which interfere destructively. However, inclusion of the 2.5PN amplitude correction leads to a constructive interference term between the 2.5PN correction and the 1PN term which is responsible for increasing the SNR for the 2.5PN waveform.\n\nIt is interesting to note that the computation of the 3PN GW polarization which will introduce an harmonic at 8Ψ will be quantitatively more significant for the equal mass case as the mass reach will be better by 33% relative to the 2.5PN FWF as opposed to the unequal mass case where it is only 14%! This provides one motivation for work in progress towards the computation of the 3PN accurate GW polarizations [29]."},{"section_type":"OTHER","section_title":"F. Variation with mass ratio","text":"Since the mass reach depends only on the total mass, the trends remain the same for different values of mass ratios. Fig. 2 compares the variation of SNRs with mass for mass ratios of 0.1 and 0.01. If the SNR is dominated by the second harmonic, the SNR is smaller for more asymmetric systems by an overall factor of ν, where ν = m 1 m 2 /m 2 . However, once the second harmonic fails to reach the sensitive bandwidth of LISA, the more asymmetric systems have a dominant contribution from the odd harmonics which scale by a further factor of √ 1 -4ν, which is larger for more asymmetric systems. Thus the decrease in SNR for the FWF with an increase in the total mass is less steep for more asymmetric systems.\n\nG. Distance reach with the 2.5PN FWF\n\nNext, we compare the distance-reach of the RWF and the 2.5PN FWF. The results are shown graphically in Fig. 5 and are similar in appearance to the mass-reach plot. The mass-reach of the RWF is ≃ 4 × 10 7 M ⊙ . For a system of total mass 5×10 7 M ⊙ , the plot shows that LISA can detect such binaries with an SNR of 10 at a luminosity distance of 100 Gpc (z ≃ 15). SMBHs of total mass ∼ 10 8 M ⊙ , not even observable using RWF templates, have a distance-reach as high as 10 Gpc (z ≃ 1.5) with an SNR of 10.\n\nProposals to extend the frequency band-width of LISA up to 10 -5 Hz have been discussed. In that case, the FWF can increase the mass-reach of LISA to even around 10 9 M ⊙ . More specifically, LISA can then observe a 10 9 M ⊙ system with an SNR of about 30 at 3 Gpc, if it uses templates based on the 2.5PN FWF for data-analysis."},{"section_type":"OTHER","section_title":"H. Sensitivity of SNR to source location and orientation","text":"All the results for SNR using the amplitude-corrected waveforms quoted earlier in this paper have been for a fixed choice of location and orientation of the source [defined by the angles(θ S , φ S , θ L , φ L )] with respect to the barycentre coordinate system. To conclude our present analysis, in this section we look into the variation in the value of SNR 12 0 1000 2000 3000 4000 5000 6000 SNR 0 0.01 0.02 0.03 0.04 0.05 Probability RWF FWF 10 100 1000 SNR due to FWF 0.02 0.04 0.06 0.08 Probalility (10 5 -10 6 )M sun 5.5 (10 6 -10 7 ) M sun FIG. 6: Distribution of SNR with sources randomly located and oriented in the sky. The left figure plots SNRs due to both RWF and FWF for a binary of mass (10 5 -10 6 )M⊙. For this mass, the most probable SNR for the FWF is lower than the most probable SNR for the RWF, like the trend shown in Table I . The right figure compares the SNRs due to the FWF for binaries of mass (10 5 -10 6 )M⊙ and 5.5(10 6 -10 7 )M⊙\n\nfor sources at various locations in the sky and various orientations. To this end, we consider a collection of sources randomly oriented in the sky and study the probability distribution of their SNRs. The results of our simulations (consisting of 8000 random realisations of the angles involved) are shown in Fig. 6 . From the left panel of Fig. 6 we see that the most probable SNR due to the FWF for a (10 5 , 10 6 )M ⊙ binary is less than the most probable SNR due to the RWF, indicating that this trend is independent of the source location and orientation. In the right panel we see that a binary of mass 2 × (10 6 , 10 7 )M ⊙ , which is undetectable by the RWF, can be observed by the FWF with a most-probable SNR of around 220."},{"section_type":"OTHER","section_title":"V. SUMMARY","text":"The implications of amplitude corrected 2.5PN full waveforms (FWF) for the construction of detection templates for LISA are investigated in detail. With the FWF, LISA can observe sources which are favoured by astronomical observations, but not observable with restricted waveforms (RWF). This includes binaries in the mass range 10 8 -10 9 M ⊙ , depending on whether the lower cut-off for LISA is chosen to be at 10 -4 Hz or 10 -5 Hz. With an SNR of 10, these systems can be observed up to a redshift of about 1.5. The computation of the 3PN polarization, which will introduce an harmonic at 8Ψ (i.e. four times the dominant harmonic), in addition to the existing harmonics, could enhance the mass reach for equal mass binaries by 33% and unequal mass binaries by 14.3%.\n\nThe implication of the FWF for parameter estimation will be far more important than the extension of LISA's mass-reach reported here. From the work of Van Den Broeck and Sengupta in the context of ground-based detectors [24] it is already clear that most parameters will be estimated with errors ∼ ten times smaller as compared to RWF. This raises the interesting possibility that binary SMBH coalescences might be located on the sky with accuracies good enough for optical observations to identify the galaxy cluster and measure its red-shift. Needless to say that this improved estimation of source properties will have important consequences in shedding light on the dark energy, better understanding of SMBH formation and evolution, structure formation, etc., and is currently under investigation.\n\nIn this work we have confined ourselves to only non-spinning black-holes ignoring the effect of spin-orbit coupling at 1.5PN [30] and 2.5PN [31] and spin-spin effect at 2PN order [32] . The effect of spin is expected to be astrophysically significant and it is important to revisit the present analysis including spin in the future. Though partial results for GW polarisations including spin do exist, a more exhaustive exercise would be necessary before the FWF required for this work is available. The problem will also be more complicated due to modulations arising from spin-orbit and spin-spin couplings which would need to be addressed.\n\nIn this work we also restricted to the inspiral phase and used a physical picture of the LSO that is based on the test-particle limit. For comparable masses, the notion of LSO is not as sharp, or unique, and hence our results are probably idealized limits of the real situation. Numerical relativity [33, 34, 35] is maturing over the past couple of years and could soon provide waveforms for late inspiral and merger. It should then be possible to compare the results of such numerical templates with those studied in this paper to provide a better understanding of how higher harmonics facilitate the mass reach of our detectors. 13 Acknowledgments KGA acknowledges the Cardiff university for hospitality during the initial stages of this work and thanks Chris Van Den Broeck and Anand Sengupta for useful discussions on data-analysis with the FWF for ground-based detectors. KGA also acknowledges VESF. BRI, BSS and SS thank the Institut Henri Poincaré and BRI the Institut des Hautes Etudes Scientifiques for hospitality during the initial stages of this work. All the calculations reported in this paper are performed with Mathematica.\n\n[1] L. Ferrarese and D. Merritt, Astrophys. J. 539, L9 (2000), astro-ph/0006053. [2] D. Richstone, E. A. Ajhar, R. Bender, G. Bower, A. Dressler, S. M. Faber, A. V. Filippenko, K. Gebhardt, R. Green, L. C. Ho, et al., Nature 395, A14 (1998), astro-ph/9810378. [3] S. Komossa et al., Astrophys. J. 582, L15 (2003). [4] http://lisa.jpl.nasa.gov . [5] S. A. Hughes, Mon. Not. R. Astron Soc. 331, 805 (2002), astro-ph/0108483. [6] E. Berti, A. Buonanno, and C. M. Will, Phys. Rev. D 71, 084025 (2005), gr-qc/0411129. [7] E. Berti, A. Buonanno, and C. M. Will, Class. Quantum Grav. 22, S943 (2005), gr-qc/0504017. [8] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Class. Quantum Grav. 23, L37 (2006), gr-qc/0604018. [9] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Phys. Rev. D 74, 024025 (2006), gr-qc/0604067. [10] S. A. Hughes and K. Menou, Astrophys. J. 623, 689 (2005), astro-ph/0410148. [11] C. Cutler et al., Phys. Rev. Lett. 70, 2984 (1993), astro-ph/9208005. [12] L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet, Phys. Rev. D 65, 061501(R) (2002), Erratum-ibid 71, 129902(E) (2005), gr-qc/0105099. [13] L. Blanchet, T. Damour, G. Esposito-Farèse, and B. R. Iyer, Phys. Rev. Lett. 93, 091101 (2004), gr-qc/0406012. [14] C. Cutler, T. Apostolatos, L. Bildsten, L. Finn, E. Flanagan, D. Kennefick, D. Markovic, A. Ori, E. Poisson, G. Sussman, et al., Phys. Rev. Lett. 70, 2984 (1993). [15] C. Cutler and E. Flanagan, Phys. Rev. D 49, 2658 (1994). [16] L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wiseman, Phys. Rev. Lett. 74, 3515 (1995), gr-qc/9501027. [17] L. Blanchet, Phys. Rev. D 54, 1417 (1996), Erratum-ibid.71, 129904(E) (2005), gr-qc/9603048. [18] L. Blanchet, B. R. Iyer, C. M. Will, and A. G. Wiseman, Class. Quantum Grav. 13, 575 (1996), gr-qc/9602024. [19] K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Class. Quantum Grav. 21, 3771 (2004), erratum-ibid. 22, 3115 (2005) , gr-qc/0404185. [20] A. M. Sintes and A. Vecchio, in Rencontres de Moriond:Gravitational waves and experimental gravity, edited by J. Dumarchez (Frontires, Paris, 2000), gr-qc/0005058. [21] A. M. Sintes and A. Vecchio, in Third Amaldi conference on Gravitational Waves, edited by S. Meshkov (American Institute of Physics Conference Series, New York, 2000), p. 403, gr-qc/0005059. [22] C. Van Den Broeck, Class. Quantum Grav. 23, L51 (2006), gr-qc/0604032. [23] C. Van Den Broeck and A. Sengupta, Class. Quantum Grav. 24, 155 (2007), gr-qc/0607092. [24] C. Van Den Broeck and A. S. Sengupta, Class. Quantum Grav. 24, 1089 (2007), gr-qc/0610126. [25] C. Cutler, Phys. Rev. D 57, 7089 (1998). [26] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 62, 084036 (2000), gr-qc/0001023. [27] J. G. Baker and J. Centrella, Class. Quantum Grav. 22, S355 (2005), astro-ph/0411616. [28] K. G. Arun, Phys. Rev. D 74, 024025 (2006) , gr-qc/0605021. [29] L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha (2007), work in progress. [30] L. Kidder, C. Will, and A. Wiseman, Phys. Rev. D 47, R4183 (1993). [31] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006), erratum-ibid. 75, 049903 (E) (2007), grqc/0605140. [32] E. Poisson and C. Will, Phys. Rev. D 52, 848 (1995). [33] B. Bruegmann, W. Tichy, and N. Jansen, Phys. Rev. Lett. 92, 211101 (2004), gr-qc/0312112. [34] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr-qc/0507014. [35] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. D 73, 104002 (2006) , gr-qc/0602026."}],"string":"[\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Supermassive black hole binaries and LISA\",\n \"text\": \"There is strong observational evidence for the existence of supermassive black holes (SMBHs) with masses in the range of 10 6 M ⊙ -10 9 M ⊙ (see e.g. Ref. [1] and references therein) in most galactic nuclei [2] . Therefore, mergers of galaxies, as evidenced by high-redshift surveys, should give rise to binaries containing SMBHs. Late stage evolution of a SMBH binary is dictated by the emission of gravitational radiation. The resulting loss of orbital energy and angular momentum would lead to the coalescence of the two holes. Indeed, X-ray observations have revealed the existence of at least one such system that would coalesce within the Hubble time [3] . Gravitational waves (GW) emitted in the process could be detected by the planned Laser Interferometer Space Antenna (LISA) [4] .\\n\\nObservation of SMBH binaries at high redshifts is one of the major science goals of LISA. These observations will allow us to probe the evolution of SMBHs and structure formation [5] and provide an unique opportunity to test General Relativity (and its alternatives) in the strong field regime of the theory [6, 7, 8, 9, 10] . Observing SMBH coalescences with high (100-1000) SNR [8, 9] is crucial for performing all the aforementioned tests.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Restricted Vs Full Waveforms as Search templates in LISA\",\n \"text\": \"Motivated by the fact that matched filtering is more sensitive to the phase of the signal than its amplitude [11] , search algorithms so far have deployed a waveform model involving only the dominant harmonic (at twice the orbital frequency), although the phase evolution itself is included to the maximum available post-Newtonian (PN) order 2 (currently 3.5PN, for non-spinning systems [12, 13] ). Waveforms in which all amplitude corrections are neglected, but the phase is treated to the maximum available order, are called restricted waveforms (RWF) and these are what are used so far in the analysis of data from ground-based detectors [14, 15, 16, 17] . This paper will consider the advantage of using the full wave forms (FWF) in the context of LISA.\\n\\nLISA is designed to detect gravitational waves in the frequency-band 0.1-100 mHz. This frequency range determines the range of masses accessible to LISA because the inspiral signal would end when the system's orbital frequency reaches the mass-dependent last stable orbit (LSO). In the test-mass approximation, the angular velocity ω LSO at LSO is given by ω LSO = 6 -3/2 M -1 , where M is the total mass of the binary. Search templates that contain only the dominant harmonic cannot extract power in the signal beyond f LSO = ω LSO /π ≃ 4.39(M/10 6 M ⊙ ) -1 mHz. This further implies that the frequency range [0.1, 100] mHz corresponds to the range ∼ 4.39 × [10 4 , 10 7 ]M ⊙ for the total mass of binary black holes that would be accessible to LISA 1 . However, as Table 1 of Ref. [1] would reveal, there is observational evidence for the existence of many SMBHs whose masses are of the order of 10 8 -10 9 M ⊙ . LISA will be unable to observe binaries containing SMBHs in this mass range if it used as search templates waveforms containing only the dominant harmonic.\\n\\nInclusion of higher-order amplitude terms in the waveform introduces the following two new features: (i) appearance of higher harmonics of the orbital phase and (ii) PN amplitude corrections to the leading as well as higher harmonics of the orbital frequency. For example, at 0.5PN order, which is the first-order correction, there are two new harmonics Ψ and 3Ψ, where Ψ is related to the orbital phase of the binary as in Refs. [18, 19] . More interestingly, in the expressions for the 'plus' and 'cross' polarizations, all odd harmonics of the orbital frequency are proportional to δm M , where δm is the difference in the masses of the binary components (see Eq. (5.7)-(5.10) of Refs. [19] ). Another important feature of the full waveform is that the (2n + 2) th harmonic first appears at the n th PN order in amplitude 2 . For example, the fourth harmonic first appears at 1PN, and has PN amplitude corrections to its dominant term at 2PN and 2.5PN (see Refs. [18, 19] for details).\\n\\nEarly investigations on the importance of amplitude-corrections to search templates were carried out by Sintes and Vecchio [20, 21] . Their study used only the first-order correction at 0.5 PN order. They concluded that the addition of the amplitude terms in the waveform did not improve the accuracy in the estimation of source's angular position and the distance, whereas the estimation of the chirp and reduced masses could be 10 times better when compared to the RWF. Recently, in the context of ground-based detectors, Van Den Broeck and Sengupta [22, 23, 24] examined the implications of going beyond the restricted PN approximation and employing instead the full waveform [18, 19] . The two main implications of the comprehensive analysis in Refs. [22, 23] for terrestrial GW detectors may be summarized as follows:\\n\\n1. For binary neutron stars and stellar mass black holes, restricted waveforms over-estimate the SNR as compared the full waveform.\\n\\n2. The use of the full waveforms significantly increases the mass-reach of second and third generation detectors, advanced LIGO and EGO being able to observe systems with total mass ∼ 400M ⊙ and a third generation detector as high as 10 3 M ⊙ .\\n\\nIn the present paper, we study in the context of LISA the implication of using templates based on the FWF (i.e. including all known harmonics of the orbital phase and all known amplitude corrections in the GW polarisations). Coalescences of SMBH binaries with masses ∼ 10 8-9 M ⊙ will not be observable by LISA if one uses only templates based on the RWF. Using templates based on amplitude corrected full waveforms, instead of the usual restricted waveforms, will enable LISA to observe coalescences of SMBH binaries with total mass ∼ 10 8 M ⊙ (10 9 M ⊙ ) if the lower frequency cut-off LISA can achieve is ∼ 10 -4 Hz (10 -5 Hz).\\n\\nThe rest of this paper is organized as follows: In Section II, we give the FWF in the frequency domain, by taking into account the orbital motion of LISA around the sun and its changing orientation. Section III discusses the results of our investigations where we compare the performances of the amplitude-corrected waveforms at different PN orders in terms of their mass-reach and distance-reach and correlate it to the 'observed' spectrum in LISA. Section V concludes with a brief summary of the main results and assumptions underlying their derivation.\\n\\n1 Although, binaries lighter than 10 4 M ⊙ would, in principle, evolve through the LISA band they would not be luminous enough to be visible in LISA unless they are close-by. 2 The 0.5PN term is an exception to this and also introduces a harmonic at the orbital frequency apart from the one at thrice the orbital frequency.\\n\\nII. TEMPLATE WAVEFORMS FOR LISA A. Amplitude corrected waveform\\n\\nFor non-spinning binaries in quasi-circular orbits inspiralling due to radiation-reaction, waveforms were computed in Refs. [18, 19] up to 2.5PN order in amplitude and 3.5PN in phase [12, 13] . This waveform h(t) is a linear combination of sine and cosine functions of multiples of the orbital phase Ψ(t). The expression for the 2.5PN polarization contains the first seven harmonics of the orbital phase, the dominant harmonic being the one at twice the orbital phase. The signal depends on the following parameters: D L , the luminosity distance to the binary, m the total (red-shifted) mass, ν the symmetric mass-ratio (reduced mass divided by total mass), the spherical polar angles (θ, φ) determining the direction of the \\\"line-of-sight\\\", the inclination angle ι of the angular momentum L of the binary with respect to the direction opposite to the line-of-sight, and the polarization angle ψ which determines the orientation of the projection of L in the plane normal to the line-of-sight.\\n\\nWe rewrite the waveform in terms of only cosines in a form similar to [23]: h(t) = 2M ν D L 7 k=1 5 n=0 A (k,n/2) cos[kΨ(t) + φ (k,n/2) ] x n 2 +1 (t), (2.1)\\n\\nwhere the coefficients A (k,n/2) and φ (k,n/2) are functions of (ν, θ, φ, ψ, ı), and x(t) = (2πM F (t)) 2/3 is the post-Newtonian parameter with F (t) the instantaneous orbital frequency. Terms 2Mν DL x n/2+1 (t) A (k,n/2) and φ (k,n/2) are the wave amplitude and polarization phase, respectively, corresponding to the k th harmonic and (n/2) th PN order. We call the coefficients A (k,n/2) the polarization amplitudes. The orbital phase Ψ(t) is a PN series in x, which, in the case of non-spinning binaries, is known to 3.5PN order [12, 13] . For a non-spinning source and a detector whose position and orientation are almost constant during the time of observation of the signal, all the above mentioned angles are constants. For ground-based GW detectors dealt with in Ref. [22, 23] , one is in this situation.\\n\\nB. Amplitude corrected waveform including modulations due to LISA's orbital motion -Time Domain\\n\\nLISA will be able to observe many sources from their early stages of inspiral and most would last for a pretty long time. We shall only consider binary sources that last for a year or less before merger. Since the LISA plane is tilted by 60 • with respect to the plane of the ecliptic, during the course of its heliocentric orbit its orientation and position varies periodically, with a period of one year and the signal in Eq. (2.1) will suffer additional amplitude and phase modulations. Thus in the case of LISA the angles θ, φ, and ψ (but not ı) appearing in Eq. 2.1 are functions of time.\\n\\nTo proceed further, in the frame of a non-rotating observer fixed to the solar-system barycentre, we denote by the location of the source on the sky by the spherical polar angles θ S and φ S and the orientation of the source by the spherical polar angles θ L and φ L determining the direction of the orbital angular momentum L of the binary. The transformation between the fixed set of angles 3 (θ S , φ S , θ L , φ L ) and the time-dependent angular coordinates of the source (θ(t), φ(t), ψ(t), ı) as measured by LISA are given in Ref. [25] .\\n\\nGeneralizing, Ref. [25] from the RWF to the FWF, the signal as seen in LISA is of the form,\\n\\nh(t) = √ 3 2 2M ν D L 7 k=1 5 n=0 A (k,n/2) (t) cos[kΨ(t) + φ (k,n/2) (t) + kφ D (t)] x n 2 +1 (t) . (2.2)\\n\\nThe PN parameter x(t) appearing in Eq. (2.2) is still equal to (2πM F (t)) 2/3 , where F (t), however, is the orbital frequency as measured by a non-rotating observer located at the solar-system barycentre. The term φ D (t) is the Doppler phase [25] , accounting for the phase difference of the gravitational wave-front between LISA and the solarsystem barycentre. The time-dependence of φ D (t) is due to the orbital motion of LISA about the barycentre. It is given by\\n\\nφ D (t) = 2 π F (t) R sin θ S cos[φ(t) -φ S ], (2.3)\\n\\nwhere R = 1 AU and φ(t) is the angular position of LISA with respect to the barycentre given by φ(t) = 2 π t T , T being equal to one year.\\n\\n3 This is a different notation from [25], where the source angles measured in the fixed barycentre frame are denoted by ( θS , φS , θL , φL ) 4 C. Amplitude corrected waveform including modulations due to LISA's orbit -Frequency Domain\\n\\nThe above waveform is valid in the adiabatic regime, where the radiation-reaction time-scale is much larger than the orbital time-scale. We also note that the additional amplitude and Doppler modulations in the waveform for LISA vary on time-scales of 1 yr (i.e. ∼ 3 × 10 7 s), while LISA can observe orbital periods at most up to 2 × 10 5 s, (i.e. gravitational wave frequencies of order 10 -5 Hz.). Consequently, the Doppler modulations change much more slowly (a hundredth) than the orbital phase. This permits the use of the stationary phase approximation (SPA) to obtain an analytical form for the Fourier transform (FT) h(f ) of the signal:\\n\\nh(f ) ≃ √ 3 2 2M ν D L 7 k=1 5 n=0\\n\\nA (k,n/2) (t(f /k)) x n 2 +1 (t(f /k)) e -iφ (k,n/2) (t(f /k))\\n\\n2 k Ḟ (t(f /k)) exp [i ψ f (t(f /k))] , (2.4)\\n\\nwhere an over dot denotes derivative with respect to time and ψ f (t(f /k)) is given by ψ f (t(f /k)) = 2πf t(f /k) -k Ψ(t(f /k)) -k φ D (t(f /k)) -π/4. (2.5)\\n\\nThe PN expansions for t(F ), Ψ(F ), Ḟ (F ) are given in Ref. [12] . The expression in Eq. (2.4) within the summation over k represents the FT due to the k th harmonic. It should be noted that the term Ḟ may be treated in different ways that could lead to numerically different results. In a numerical treatment, for instance, one could avoid performing a further re-expansion. Alternatively, one could re-expand the denominator in the amplitude and truncate the resulting expression at the n th PN order, to obtain the nPN amplitude-corrected waveform. Ref. [22, 23] choose the latter and we follow them in this work. Radiation reaction results in an increase in the orbital frequency F (t) which will ultimately drive the system beyond the adiabatic inspiral phase and the inspiral waveform given above will no longer be valid. In the first approximation this is expected to occur when the orbital frequency F (t) reaches F LSO -the orbital frequency of the LSO of a Schwarzschild solution with the same mass as the binary's total mass M ,\\n\\nF LSO = (2 π 6 3 2 M ) -1 .\\n\\n(2.6) Thus, we truncate the signal in the time domain at a time t LSO , given implicitly by F (t LSO ) = F LSO . In the SPA, the main contribution to the FT of the k th harmonic at a given Fourier frequency f , comes from the neighbourhood of the time when the instantaneous value of the k th harmonic sweeps past f . Thus the k th harmonic is not expected to contribute significant power to the FT for frequencies above k F LSO , if the signal is truncated in the time domain beyond t LSO . This motivates the truncation of the FT due to the k th harmonic at frequencies above k F LSO by a step function θ(k F LSO -f ) [θ(x) = 1, for x ≥ 0, and 0 for negative x].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. SIGNAL TO NOISE RATIOS IN LISA WITH HIGHER HARMONICS\",\n \"text\": \"In this Section we investigate the effect of the higher harmonics in LISA observations of supermassive black hole binaries. The LISA waveform discussed in the previous Section will be used for the analysis.\\n\\nGiven a waveform h, the best signal-to-noise ratio (SNR) achieved using an optimal filter is given by ρ[h] ≡ (h|h) 1/2 , where ( . | . ) is the usual inner product in terms of the one-sided noise power spectral density S h (f ) of the detector. With the convention for Fourier transforms, x(f ) = ∞ -∞ x(t) exp(-2πif t) dt, the inner product is given by:\\n\\n(x|y) ≡ 4 f end fs Re[x * (f )ỹ(f )] S h (f ) df. (3.1)\\n\\nFor an optimal filter, which maximises the overlap of the signal with template, one can write\\n\\nρ 2 = 4 f end fs | h(f )| 2 S h (f ) df. (3.2)\\n\\nWe use the non-sky-averaged noise-spectral-density as given in Eqs. (2.28)-(2.32) of Ref. [6]. 5 10 -4 10 -3 10 -2 frequency(in Hz) 0.0 2.0×10 6 4.0×10 6 6.0×10 6 8.0×10 6 Pf(f) RWF FWF 10 -4 frequency(in Hz) 0.0 5.0×10 5 1.0×10 6 1.5×10 6 2.0×10 6 Pf(f) RWF FWF FIG. 1: The observed spectrum, P(f f ) , in LISA using the full (solid, red) and restricted (dashed, blue) waveforms, for two archetypal binaries, (10 5 , 10 6 )M⊙ (left) and 2 × (10 6 , 10 7 )M⊙ (right). The sources are assumed to be at 3 Gpc and their orientation with respect to the solar-system barycentre is chosen to be θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3. The spectrum is much more complicated and highly oscillatory for the FWF than for the RWF, because of interference between various harmonics. The higher frequency reach of the FWF is due to presence of higher harmonics as apparent in the figure . The spectrum for the system in the left panel sharply rises at a frequency ∼ 2 × 10 -3 Hz. Beyond this frequency, the effective LISA noise decreases sharply with increasing frequency (as there are fewer galactic binaries per frequency bin) leading to the observed increase in the spectrum.\\n\\n) ≡ d(ρ 2 ) d(ln f ) = f | h(f )| 2 S h (\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Choice of frequency cutoffs f end , fs\",\n \"text\": \"The upper limit of integration f end is taken to be the minimum of 7 F LSO and 1 Hz, the latter being a conventional upper cut-off for the LISA noise curve. The lower limit f s is chosen assuming LISA observes the inspiral for a duration ∆t obs before it reaches the LSO. Using the quadrupole formula, we find that the orbital frequency at the epoch t LSO -∆t obs is given by\\n\\nF (t LSO -∆t obs ) = F LSO (1 + 256 ν 5 M ∆t obs v 8 LSO ) 3 8 , (3.3)\\n\\nwhere v LSO is the orbital velocity and t LSO the epoch at which the orbital frequency reaches the value F LSO . We take v LSO = 1/ √ 6, the orbital velocity at the LSO in the case of a test mass orbiting a Schwarzschild black hole. We designate F (t LSO -∆t obs ) as F in . Thus the k th harmonic will have a frequency k F in , ∆t obs before t LSO . The above formula reduces to the simpler form in Ref. [6] as v LSO → ∞. For the mass values explored in this work there is no significant dependence of the results on this choice. In all our calculations we take ∆t obs to be one year.\\n\\nThe lower cut-off for the k th harmonic should be the maximum of the lower cut-off of LISA (10 -4 Hz) and k F in and simply implemented by truncating the waveform due to the kth harmonic by another step-function θ(f -k F in ) and choosing f s to be 10 -4 Hz. It is worth noting that the k th harmonic probes a larger interval of the frequency domain i.e. k(F LSO -F in ) relative to the fundamental harmonic. For brevity, we refer to this as the span of the k th harmonic.\\n\\nThere is a caveat with regard to the use of higher harmonics that is worth mentioning: In the time-domain the waveform should begin when the highest harmonic reaches the lower cutoff. This has an implication on data analysis as the templates will be an order-of-magnitude longer than before. Thus, it might be sensible to use higher harmonics only in the case of higher masses.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Observed signal spectrum with LISA\",\n \"text\": \"To get some insight into the effect of higher harmonics via amplitude corrections let us first look at the SNR integrand, i.e., the \\\"noise-weighted signal power\\\" per unit logarithmic frequency interval [26] . Rewriting the expression for the SNR as\\n\\nρ 2 = 4 f end fs f | h(f )| 2 S h (f ) d ln(f ), (3.4) 6\\n\\nTABLE I: SNRs due to successive PN amplitude-corrected waveforms, with phase corrections to 3.5 PN order in all cases. The orientation of the source with respect to the solar-system barycentre is chosen to be θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3. For the (10 6 -10 7 )M⊙ binary system, all harmonics enter deep into the sensitive part of the LISA bandwidth. Apart from an increase at 0.5PN, we see a consistent reduction in the SNR on inclusion of higher PN order amplitude corrections. For the (5.5 × 10 6 , 5.5 × 10 7 )M⊙ binary system, the second harmonic fails to enter the LISA bandwidth, while the third harmonic spans a small insensitive region. Thus the SNR due to the RWF is zero, while the SNR due to the 0.5PN waveform is smaller than the SNRs due to higher order PN terms. Both sources are at a distance of 3 Gpc. PN SNR order (10 6 -10 7 )M⊙ 5.5 × (10 6 -10 7 )M⊙ 0 924.48 0 0.5 1025.8 211.98 1 928.48 343.17 1.5 869.78 319.34 2 824.65 266.65 2.5 809.51 277.34 the quantity of our interest is\\n\\nP(f ) ≡ d(ρ 2 ) d(ln f ) = f | h(f )| 2 S h (f ) , (3.5)\\n\\nwhich is designated as the \\\"observed spectrum\\\", following [24] . The observed spectrum is plotted versus frequency for given masses in Fig. 1 . As is the case for ground-based detectors [24], the spectrum due to the FWF has a lot more structure and is highly oscillatory because of interference between various harmonics. For the (10 5 , 10 6 )M ⊙ system, the mass being low, the second harmonic and hence the RWF extends up to frequencies ∼ 2 × 10 -3 Hz, where LISA is most sensitive. This leads to a rapid increase in the observed spectrum in this frequency region. The spectrum due to the FWF, containing higher harmonics continue beyond the RWF into the most sensitive part of the LISA band. For the 2(10 6 , 10 7 )M ⊙ system, the frequency span of the second harmonic is small and the sensitive region of the LISA band lies beyond its maximum reach.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. THE EFFECT OF HIGHER HARMONICS\",\n \"text\": \"Following the analysis of Ref. [22, 23] , we classify the sources into two types: In the first category are sources for which the dominant (second) harmonic has a large frequency span in the LISA band. The second category on the other hand comprise sources whose dominant harmonic fails to enter the LISA bandwidth but the higher harmonics do. Since the upper cut-off frequency for each harmonic is inversely proportional to the total mass (from the expression for F LSO ), we note that the sources of the first type will have total mass less than some value which we call the RWF mass-reach, the maximum mass detectable by the RWF, while the second type will have masses greater than this value. The condition that the upper cut-off of the dominant harmonic is less than or equal to the lower cut-off of LISA (i.e., by the inequality 2 F LSO ≤ f s ) determines the RWF mass-reach. The choice of f s for the LISA mission is still not clear and theoretical implications of this choice are explored in e.g. Ref. [27] . For f s in the range [10 -5 , 10 -4 ] Hz the RWF mass-reach varies over the range [4.39 , 43.9] × 10 7 M ⊙ , the lower end of the mass range corresponding to the higher end of the frequency range.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. How higher harmonics affect signal visibility\",\n \"text\": \"In Fig. 2 we plot the SNRs computed using the restricted (RWF) and full (FWF) waveforms as a function of the binary's total mass for two values of the mass ratio 4 . We first consider systems whose total mass is less than 4 Our codes are calibrated by reproducing the results of [22, 23] , which considers ground-based detectors, and also of [28] , which computes SNRs in LISA using RWF.\\n\\n7 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Total Mass (in Msun) 10 0 10 1 10 2 10 3 SNR RWF; m 1 /m 2 =0.1 FWF; m 1 /m 2 =0.1 RWF; m 1 /m 2 =0.01 FWF; m 1 /m 2 =0.01 FIG. 2: SNR versus total mass for mass ratios of 0.1 and 0.01. The figure shows that apart from the dips due to white-dwarf confusion noise, for mass values where the RWF enters the LISA band, the corresponding SNR is consistently more than the SNR produced by the FWF. However, for mass values where the second harmonic terminates before it reaches the LISA bandwidth, the FWF which has higher harmonics that enter the LISA band produces significant SNRs. The frequency reach of a harmonic depends only on the total mass, and thus the mass reach of the FWF is independent of the mass-ratio. For more asymmetric systems, the magnitude of the SNR is low for all masses both for the RWF and the FWF. Sources are at a luminosity distance of 3 Gpc with fixed angles given by θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3.\\n\\n4 × 10 7 M ⊙ . For these systems, the SNRs computed using the two different approximations agree with each other to within 10%, with the RWF over-estimating the SNR, when compared to the FWF, in most of the range. This is explicitly shown for a (10 6 , 10 7 )M ⊙ binary in the first column of Table I . Indeed, but for the slight increase in SNR as we go from 0PN to 0.5PN, we find a steady decrease as one increases the PN order of the amplitude correction. The reduction in SNR at higher PN orders can be understood by studying the structure of | h(f )| 2 , the numerator in the integrand of the SNR in Eq. (3.2). There are basically three types of terms: 1. direct terms in which the phases in Eq. (2.4) cancel 2. interference terms between different PN corrections of the same harmonic,\\n\\nA 2 (k,n/2) (t(f /k)) f -7 3 (M f ) 2n 3 ,\\n\\nA (k,m/2) (t(f /k)) A (k,n/2) (t(f /k)) f -7 3 (M f ) m+n 3 cos[φ (k,m/2) (t(f /k)) -φ (k,n/2) (t(f /k))]\\n\\n3. harmonic mixtures 5 which are terms containing the interference between different PN corrections of different harmonics, e.g. the m/2 th PN correction of the k th harmonic and n/2 th PN correction of the l th harmonic.\\n\\nA (k,m/2) (t(f /k)) A (l,n/2) (t(f /l)) f -7 3 (M f ) m+n 3 cos[ψ f (t(f /k)) -φ (k,m/2) (t(f /k)) -ψ f (t(f /l)) + φ (l,n/2) (t(f /l))]\\n\\nwhere ψ f (t(f /k)) is given by Eq. (2.5), All these terms are scaled by M 5/3 , where M = M ν 3/5 is the chirp-mass. (Additional multiplicative factors have been omitted in the above expressions, among which are the step-functions mentioned earlier and PN expansion coefficients of the denominator of the Fourier amplitude in Eq. (2.4), the latter being time-independent.)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. The effect of higher harmonics in ground-based detectors\",\n \"text\": \"Before we explain the SNR trends in the context of LISA, we mention that for ground-based detectors a similar effect was found in Ref. [23] for a different but corresponding mass region. The lower cut-off for a typical ground-based 5 We use the term 'harmonic mixtures' at the risk of being mistaken to the well-known 'harmonic mixing' in music. Our use of the phrase 'harmonic mixtures' is simply to convey the physical effect of the interference between different harmonics 8 1×10 -4 1×10 -3 frequency(Hz) 0 0.5 1 polarisation amplitude A (2,0) 0.00 3.15×10 7 6.31×10 7 time (in sec) 0 0.5 1 polarisation amplitude A (2,0) polarisation amplitude A (2,0) time ( in years ) 0 1 2 0.5 1 FIG. 3: Variation of polarisation amplitude of the RWF with frequency and time (inset). The inset, plotted over a duration of two years clearly shows periodicity due to LISA's orbital motion around the Sun. The binary mass, (10 6 -10 4 )M⊙, has been chosen such that it can, in principle, be observed for two years. The plot in the frequency domain shows that the variation of the polarisation amplitude is confined to a very small part of the frequency span of the dominant harmonic, and essentially behaves as a constant in the frequency domain.\\n\\ndetector, say Advanced LIGO is 20Hz, and the effect of higher harmonics is seen for masses less than ∼ 220M ⊙ . In that case, as mentioned earlier, the polarisation amplitudes and phases are constants. The RWF contains only the Newtonian term of the second harmonic and thus | h(f )| 2 consists of a single direct term with n = 0 and k = 2. With the inclusion of higher-order amplitude terms in the waveform, PN corrections to the dominant harmonic, and higher harmonics and their PN corrections, also contribute to the SNR. In other words, the signal power spectrum | h(f )| 2 will contain all three types of terms discussed before. From the form of the direct terms, it is evident that their contribution to the SNR will be positive definite. We also note that, for ground-based detectors, the frequency dependence of the direct and interference terms will just be a power law. However, the sign of the interference terms (and consequently their contribution to the SNR) depends on the difference between the polarisation phases of different PN corrections for the same harmonic. Van Den Broeck and Sengupta showed that for a given harmonic, for all allowed values of the parameters (ν, θ, φ, ψ, ı), each PN correction is almost \\\"out of phase\\\" with both the PN correction preceding and succeeding it 6 . The resulting negative terms (representing destructive interferences) reduce the SNR as one includes higher PN amplitude corrections in the waveform.\\n\\nThe third type of terms, harmonic mixtures, however, are highly oscillatory functions of the frequency, as the phase difference ψ f (t(f /k)) -ψ f (t(f /l)) between the k th and the l th harmonic become even or odd multiples of π. As one integrates over f , these oscillations tend to cancel out, and thus the contribution to the SNR from these terms are numerically much smaller relative to the first two types of terms.\\n\\nC. Effect of higher harmonics for binaries with M < 4 × 10 7 M⊙\\n\\nIn the case of LISA, because of the polarisation factors, the amplitudes of none of the three types of terms is a simple power-law in f . The periodic variation of, for example, A (2,0) (period being one year) appears as an amplitude modulation A (2,0) (t(f /2)) in the Fourier transform, where the argument t(f /2) of A 2,0 is given by t(f /2) = -5 256π 8/3 M 5/3 1 f 8/3 + PN corrections. (4.1) Hence, in the frequency domain A (2,0) will undergo one complete oscillation as f varies from 2F in (see Eq. (3.3)) to 2F LSO . However, because of the inverse power-law dependence on f , the oscillation of A (2,0) is confined to a small frequency interval above F in and remains fairly constant over a major portion of the frequency span 2(F LSO -F in ) (see Fig. 3 ). For masses higher than the one shown in Fig. 3 , this region of significant variation moves to the left of the figure. On including in our analysis the effect of detector sensitivity (weighting down by S h (f )) this variation of 6 Note, however, that Ref. [23] , argues this in a somewhat different form.\\n\\n9 10 7 10 8 Total Mass (in Msun) 10 1 10 2 10 3 SNR Newtonian 0.5PN 1PN 1.5PN 2PN 2.5PN 10 7 10 8 Total Mass (in Msun) 10 1 10 2 10 3 SNR Newtonian 0.5PN 1PN 1.5PN 2PN 2.5PN FIG. 4: SNR versus total mass for successive PN amplitude-corrected waveforms and 3.5PN phasing. The left panel corresponds to a mass-ratio of 0.1 while the right panel plots the same quantities for mass-ratio of 1 (equal mass systems). The (2n + 2) th harmonic first appears at the nth PN order. For a binary of given total mass, the upper cut-off of the k th -harmonic of the orbital frequency in the frequency domain is proportional to k and inversely proportional to the total mass. As the mass increases the upper cut-off for the 2 nd harmonic falls below the lower cut-off of the LISA detector, leading to a zero value of SNR due to the RWF. The higher harmonics still enter the sensitive bandwidth of LISA and higher PN order waveforms produce significant SNR. The 2.5PN waveform has the highest mass-reach, being 3.5 times the mass-reach of the RWF. In the equal mass case displayed in the right panel, the differences in harmonic content of different PN order waveforms are more pronounced, as odd harmonics are absent. Sources are at a luminosity distance of 3 Gpc with fixed angles given by θS = cos -1 (-0.6), φS = 1, θL = cos -1 (0.2), φL = 3.\\n\\nA (2,0) gets damped out when one evaluates the integral in Eq. (3.2). For masses satisfying 2F in ≪ 10 -4 Hz, the lower cut-off for LISA, this region of variation will fall below the LISA band. The polarisation phases determining the sign of the interference terms between the same harmonics also vary with f . However, as mentioned earlier, the phase relationships of the polarisation phases are independent of the parameter values. Thus the modulations which change the values of (θ, φ, ψ, ı) do not affect the trend of reduction of SNR with amplitude corrections. The Doppler modulations, which appear in only harmonic mixtures, are also not important as far as SNR is concerned.\\n\\nFinally, we would like to note an important point not explicitly mentioned in Ref. [23] . As the difference between the polarisation phases of successive PN corrections of the same harmonic tend to be nearly π, alternate PN corrections necessarily interfere constructively. Hence there are positive contributions also from the interference terms. Now, the numerical value of the contribution to SNR from each of these terms depends on the magnitude of the polarisation amplitude and the power of (M f ). It can be checked that for all allowed values of (ν, θ, φ, ψ, ı) , the polarisation amplitudes are roughly of the same order in magnitude. Consider the Newtonian term of the dominant harmonic and its interference with the first two corrections at 1PN and 1.5PN. The Newtonian term will be out of phase with the 1PN term, but in phase with the 1.5PN one. The two corresponding interference terms will contain powers of (M f ) 2/3 and (M f ), respectively, and since they have the same frequency span, the absolute numerical value of the contribution to SNR from the former will be more since (2πM f ) 1/3 will always be less than 1/ √ 6. Numerical values of contributions from interference between higher PN corrections of the second harmonic successively decrease. The same argument applies for all the other harmonics, and thus, inclusion of amplitude corrections will lead to an overall reduction in SNR.\\n\\nThe first column of Table I clearly demonstrates the effect of higher harmonics on SNRs. The increase in SNR for the 0.5PN waveform (with respect the RWF SNR) is also easily explained by noting that the 0.5PN correction only adds (apart from harmonic mixtures) two direct terms to | h(f )| 2 , corresponding to the first and third harmonics (n = 1, n = 3). Clearly, from the discussion in the previous subsection, the 0.5PN waveform will have a higher SNR than the RWF, independent of the binary parameters.\\n\\nFor 10 3 < ∼ M < ∼ 10 5 M ⊙ , the difference between the RWF and the FWF is not visible on the scale of Fig. 2 because for this mass range all the direct and interference terms corresponding to harmonics higher than the dominant ones, which are scaled by higher powers of (M f ), are negligible.\\n\\n10 D. Visibility of systems with M > 4 × 10 7 M⊙ In their analysis of the implications of the FWF for ground-based detectors Van Den Broeck and Sengupta [22, 23] pointed out an interesting effect due to higher harmonics. An analogous effect is found in the case of LISA in spite of the additional amplitude and Doppler modulations that exist in this case.\\n\\nNormally, the harmonic at twice the orbital frequency dominates the SNR. However, when the dominant harmonic fails to reach the LISA band the higher harmonics become important, which transpires for masses greater than 4 × 10 7 M ⊙ . Even though the second harmonic falls below the lower cut-off f s of the LISA bandwidth, the kth harmonic, k > 2, that has power up to a frequency k F LSO , might cross f s and produce a significant SNR. Of course, the kth harmonic would fall below the LISA sensitivity band for masses which satisfy the equality f s = k F LSO . Thus, higher PN order waveforms, which bring in higher harmonics, are capable of producing a significant SNR, even when the RWF fails to produce any.\\n\\nLet us examine this in a little more detail starting from the values of mass where the second harmonic dominates and the RWF is adequate. Eventually, for larger values of the total mass, the inequality f s ≥ 3F LSO becomes true. Then the 0.5PN waveform, which contains the first and the third harmonic, terminates before reaching f s and consequently the SNR due to the 0.5PN waveform goes to zero. SNRs for different PN waveforms for a binary whose dominant harmonic falls below f s and the third harmonic has a small span in the LISA bandwidth is given in the second column of Table I. Note that for the 5.5(10 6 -10 7 )M ⊙ system, the 1PN waveform has a higher SNR than the 0.5PN one. This is due to the absence of the first harmonic and the small span of the third harmonic in the LISA bandwidth. Further, the 2.5PN waveform has a slightly larger SNR compared to 2PN. This is due to the absence of the first and second harmonic and the small contribution from the third harmonic, all of which contribute interference terms due to their 2.5PN corrections. However, this increase is marginal, and is not generic. We have explicitly checked by choosing different angles that there can be a small decrease also. The detailed results for LISA are summarised in Fig 4 . We see that for masses for which the 1PN waveform fails to reach the LISA bandwidth, the higher PN order amplitudes are capable of producing SNRs as high as 100! Thus, the use of the FWF will enable LISA to make observations of SMBHs in the astrophysically interesting mass-regime, which would not be possible had one used only the standard RWF.\\n\\nUsing the expression for F LSO , it is simple to argue that the mass reach for the 2.5PN FWF, which has the seventh harmonic of the orbital frequency, is 7/2 times the RWF (around 1.5 × 10 8 M ⊙ ). The above ratio, of course, depends on the assumption that the Schwarzschild (test particle case) LSO frequency will not be very different from the LSO frequency in the comparable mass case.\\n\\nWe conclude with a discussion of a minor, but clear, feature seen in Fig. 2 for LISA, but not present for the ground-based detectors, concerning the relative values of the SNR obtained using the RWF and the FWF. For most of the mass range probed the RWF overestimates the SNR relative to the FWF; however, the figure clearly shows an anomaly for masses around ∼ 2 × 10 6 M ⊙ . To understand this, we first note that the dips in the two curves in Fig 2 , \\nare due to the bump in the LISA noise-curve [6] just above 10 -3 Hz. This bump is due to the domination of whitedwarf confusion noise over instrumental noise and lies just below the most sensitive frequency region (∼ 3 × 10 -3 Hz -2 × 10 -2 Hz) of the LISA band. Below 3 × 10 -3 Hz, the noise increases sharply till one reaches the bump. For binaries of mass greater than 1.5 × 10 6 M ⊙ , the frequency span of the dominant harmonic ends just around the bump and the sensitive region of the LISA band is beyond the span of this harmonic. However, higher harmonics incorporated in the FWF are able to reach the sensitive part of the noise curve. This leads to higher SNR for the FWF relative to the RWF. This reversal of trend continues up to masses 4 × 10 4 M ⊙ . Above this mass, the frequency span of the seventh harmonic ends before the sensitive region of the LISA band and the general trend is restored.\\n\\nFor still higher mass values, the SNRs due to the RWF and the FWF both increase until the second harmonic fails to reach the LISA band. This is due to the overall scaling of the waveform with the total mass. At such high values, it is able to compensate both for the decreasing frequency span and the higher noise of the detector in this frequency range.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"E. Effect of higher harmonics in the equal mass case\",\n \"text\": \"In contrast to asymmetric systems discussed so far, for systems of equal mass all odd harmonics are absent. Consequently, for symmetric systems the mass-reach of the 2.5PN FWF will be only 3 times the mass-reach of the RWF. Further, from the right panel of Fig. 4 , it is clear that the 0.5PN and the 0PN, or RWF, are identical, as are the 1PN and 1.5PN waveforms. Thus the decrease in SNR for the higher PN order waveforms with increasing total mass is more pronounced than in the unequal-mass case. We also note that for masses for which the second harmonic fails to reach the detector bandwidth, the 2PN waveform has a lower SNR than the 2.5PN waveform. This can be explained by noting that for these masses only the fourth and sixth harmonics enter the LISA bandwidth. 11 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Total Mass (in Msun) 10 2 10 3 10 4 10 5 10 6 Luminosity Distance (Mpc) FWF RWF FIG. 5: Luminosity Distance (in Mpc) versus total mass for a fixed SNR of 10. The systems have mass-ratio of 0.1. The distance reach can be as large as 500 Gpc for systems where the second harmonic enters the LISA bandwidth. Systems undetectable by the RWF (of mass around 10 8 M⊙) can be detected by the FWF at distances up to 10 Gpc. The location and orientation of the sources are the same as in the earlier figures.\\n\\nThe 2PN waveform contains the leading term of the fourth harmonic at 1PN and its 2PN correction, which interfere destructively. However, inclusion of the 2.5PN amplitude correction leads to a constructive interference term between the 2.5PN correction and the 1PN term which is responsible for increasing the SNR for the 2.5PN waveform.\\n\\nIt is interesting to note that the computation of the 3PN GW polarization which will introduce an harmonic at 8Ψ will be quantitatively more significant for the equal mass case as the mass reach will be better by 33% relative to the 2.5PN FWF as opposed to the unequal mass case where it is only 14%! This provides one motivation for work in progress towards the computation of the 3PN accurate GW polarizations [29].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"F. Variation with mass ratio\",\n \"text\": \"Since the mass reach depends only on the total mass, the trends remain the same for different values of mass ratios. Fig. 2 compares the variation of SNRs with mass for mass ratios of 0.1 and 0.01. If the SNR is dominated by the second harmonic, the SNR is smaller for more asymmetric systems by an overall factor of ν, where ν = m 1 m 2 /m 2 . However, once the second harmonic fails to reach the sensitive bandwidth of LISA, the more asymmetric systems have a dominant contribution from the odd harmonics which scale by a further factor of √ 1 -4ν, which is larger for more asymmetric systems. Thus the decrease in SNR for the FWF with an increase in the total mass is less steep for more asymmetric systems.\\n\\nG. Distance reach with the 2.5PN FWF\\n\\nNext, we compare the distance-reach of the RWF and the 2.5PN FWF. The results are shown graphically in Fig. 5 and are similar in appearance to the mass-reach plot. The mass-reach of the RWF is ≃ 4 × 10 7 M ⊙ . For a system of total mass 5×10 7 M ⊙ , the plot shows that LISA can detect such binaries with an SNR of 10 at a luminosity distance of 100 Gpc (z ≃ 15). SMBHs of total mass ∼ 10 8 M ⊙ , not even observable using RWF templates, have a distance-reach as high as 10 Gpc (z ≃ 1.5) with an SNR of 10.\\n\\nProposals to extend the frequency band-width of LISA up to 10 -5 Hz have been discussed. In that case, the FWF can increase the mass-reach of LISA to even around 10 9 M ⊙ . More specifically, LISA can then observe a 10 9 M ⊙ system with an SNR of about 30 at 3 Gpc, if it uses templates based on the 2.5PN FWF for data-analysis.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"H. Sensitivity of SNR to source location and orientation\",\n \"text\": \"All the results for SNR using the amplitude-corrected waveforms quoted earlier in this paper have been for a fixed choice of location and orientation of the source [defined by the angles(θ S , φ S , θ L , φ L )] with respect to the barycentre coordinate system. To conclude our present analysis, in this section we look into the variation in the value of SNR 12 0 1000 2000 3000 4000 5000 6000 SNR 0 0.01 0.02 0.03 0.04 0.05 Probability RWF FWF 10 100 1000 SNR due to FWF 0.02 0.04 0.06 0.08 Probalility (10 5 -10 6 )M sun 5.5 (10 6 -10 7 ) M sun FIG. 6: Distribution of SNR with sources randomly located and oriented in the sky. The left figure plots SNRs due to both RWF and FWF for a binary of mass (10 5 -10 6 )M⊙. For this mass, the most probable SNR for the FWF is lower than the most probable SNR for the RWF, like the trend shown in Table I . The right figure compares the SNRs due to the FWF for binaries of mass (10 5 -10 6 )M⊙ and 5.5(10 6 -10 7 )M⊙\\n\\nfor sources at various locations in the sky and various orientations. To this end, we consider a collection of sources randomly oriented in the sky and study the probability distribution of their SNRs. The results of our simulations (consisting of 8000 random realisations of the angles involved) are shown in Fig. 6 . From the left panel of Fig. 6 we see that the most probable SNR due to the FWF for a (10 5 , 10 6 )M ⊙ binary is less than the most probable SNR due to the RWF, indicating that this trend is independent of the source location and orientation. In the right panel we see that a binary of mass 2 × (10 6 , 10 7 )M ⊙ , which is undetectable by the RWF, can be observed by the FWF with a most-probable SNR of around 220.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"V. SUMMARY\",\n \"text\": \"The implications of amplitude corrected 2.5PN full waveforms (FWF) for the construction of detection templates for LISA are investigated in detail. With the FWF, LISA can observe sources which are favoured by astronomical observations, but not observable with restricted waveforms (RWF). This includes binaries in the mass range 10 8 -10 9 M ⊙ , depending on whether the lower cut-off for LISA is chosen to be at 10 -4 Hz or 10 -5 Hz. With an SNR of 10, these systems can be observed up to a redshift of about 1.5. The computation of the 3PN polarization, which will introduce an harmonic at 8Ψ (i.e. four times the dominant harmonic), in addition to the existing harmonics, could enhance the mass reach for equal mass binaries by 33% and unequal mass binaries by 14.3%.\\n\\nThe implication of the FWF for parameter estimation will be far more important than the extension of LISA's mass-reach reported here. From the work of Van Den Broeck and Sengupta in the context of ground-based detectors [24] it is already clear that most parameters will be estimated with errors ∼ ten times smaller as compared to RWF. This raises the interesting possibility that binary SMBH coalescences might be located on the sky with accuracies good enough for optical observations to identify the galaxy cluster and measure its red-shift. Needless to say that this improved estimation of source properties will have important consequences in shedding light on the dark energy, better understanding of SMBH formation and evolution, structure formation, etc., and is currently under investigation.\\n\\nIn this work we have confined ourselves to only non-spinning black-holes ignoring the effect of spin-orbit coupling at 1.5PN [30] and 2.5PN [31] and spin-spin effect at 2PN order [32] . The effect of spin is expected to be astrophysically significant and it is important to revisit the present analysis including spin in the future. Though partial results for GW polarisations including spin do exist, a more exhaustive exercise would be necessary before the FWF required for this work is available. The problem will also be more complicated due to modulations arising from spin-orbit and spin-spin couplings which would need to be addressed.\\n\\nIn this work we also restricted to the inspiral phase and used a physical picture of the LSO that is based on the test-particle limit. For comparable masses, the notion of LSO is not as sharp, or unique, and hence our results are probably idealized limits of the real situation. Numerical relativity [33, 34, 35] is maturing over the past couple of years and could soon provide waveforms for late inspiral and merger. It should then be possible to compare the results of such numerical templates with those studied in this paper to provide a better understanding of how higher harmonics facilitate the mass reach of our detectors. 13 Acknowledgments KGA acknowledges the Cardiff university for hospitality during the initial stages of this work and thanks Chris Van Den Broeck and Anand Sengupta for useful discussions on data-analysis with the FWF for ground-based detectors. KGA also acknowledges VESF. BRI, BSS and SS thank the Institut Henri Poincaré and BRI the Institut des Hautes Etudes Scientifiques for hospitality during the initial stages of this work. All the calculations reported in this paper are performed with Mathematica.\\n\\n[1] L. Ferrarese and D. Merritt, Astrophys. J. 539, L9 (2000), astro-ph/0006053. [2] D. Richstone, E. A. Ajhar, R. Bender, G. Bower, A. Dressler, S. M. Faber, A. V. Filippenko, K. Gebhardt, R. Green, L. C. Ho, et al., Nature 395, A14 (1998), astro-ph/9810378. [3] S. Komossa et al., Astrophys. J. 582, L15 (2003). [4] http://lisa.jpl.nasa.gov . [5] S. A. Hughes, Mon. Not. R. Astron Soc. 331, 805 (2002), astro-ph/0108483. [6] E. Berti, A. Buonanno, and C. M. Will, Phys. Rev. D 71, 084025 (2005), gr-qc/0411129. [7] E. Berti, A. Buonanno, and C. M. Will, Class. Quantum Grav. 22, S943 (2005), gr-qc/0504017. [8] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Class. Quantum Grav. 23, L37 (2006), gr-qc/0604018. [9] K. G. Arun, B. R. Iyer, M. S. S. Qusailah, and B. S. Sathyaprakash, Phys. Rev. D 74, 024025 (2006), gr-qc/0604067. [10] S. A. Hughes and K. Menou, Astrophys. J. 623, 689 (2005), astro-ph/0410148. [11] C. Cutler et al., Phys. Rev. Lett. 70, 2984 (1993), astro-ph/9208005. [12] L. Blanchet, G. Faye, B. R. Iyer, and B. Joguet, Phys. Rev. D 65, 061501(R) (2002), Erratum-ibid 71, 129902(E) (2005), gr-qc/0105099. [13] L. Blanchet, T. Damour, G. Esposito-Farèse, and B. R. Iyer, Phys. Rev. Lett. 93, 091101 (2004), gr-qc/0406012. [14] C. Cutler, T. Apostolatos, L. Bildsten, L. Finn, E. Flanagan, D. Kennefick, D. Markovic, A. Ori, E. Poisson, G. Sussman, et al., Phys. Rev. Lett. 70, 2984 (1993). [15] C. Cutler and E. Flanagan, Phys. Rev. D 49, 2658 (1994). [16] L. Blanchet, T. Damour, B. R. Iyer, C. M. Will, and A. G. Wiseman, Phys. Rev. Lett. 74, 3515 (1995), gr-qc/9501027. [17] L. Blanchet, Phys. Rev. D 54, 1417 (1996), Erratum-ibid.71, 129904(E) (2005), gr-qc/9603048. [18] L. Blanchet, B. R. Iyer, C. M. Will, and A. G. Wiseman, Class. Quantum Grav. 13, 575 (1996), gr-qc/9602024. [19] K. G. Arun, L. Blanchet, B. R. Iyer, and M. S. S. Qusailah, Class. Quantum Grav. 21, 3771 (2004), erratum-ibid. 22, 3115 (2005) , gr-qc/0404185. [20] A. M. Sintes and A. Vecchio, in Rencontres de Moriond:Gravitational waves and experimental gravity, edited by J. Dumarchez (Frontires, Paris, 2000), gr-qc/0005058. [21] A. M. Sintes and A. Vecchio, in Third Amaldi conference on Gravitational Waves, edited by S. Meshkov (American Institute of Physics Conference Series, New York, 2000), p. 403, gr-qc/0005059. [22] C. Van Den Broeck, Class. Quantum Grav. 23, L51 (2006), gr-qc/0604032. [23] C. Van Den Broeck and A. Sengupta, Class. Quantum Grav. 24, 155 (2007), gr-qc/0607092. [24] C. Van Den Broeck and A. S. Sengupta, Class. Quantum Grav. 24, 1089 (2007), gr-qc/0610126. [25] C. Cutler, Phys. Rev. D 57, 7089 (1998). [26] T. Damour, B. R. Iyer, and B. S. Sathyaprakash, Phys. Rev. D 62, 084036 (2000), gr-qc/0001023. [27] J. G. Baker and J. Centrella, Class. Quantum Grav. 22, S355 (2005), astro-ph/0411616. [28] K. G. Arun, Phys. Rev. D 74, 024025 (2006) , gr-qc/0605021. [29] L. Blanchet, G. Faye, B. R. Iyer, and S. Sinha (2007), work in progress. [30] L. Kidder, C. Will, and A. Wiseman, Phys. Rev. D 47, R4183 (1993). [31] L. Blanchet, A. Buonanno, and G. Faye, Phys. Rev. D 74, 104034 (2006), erratum-ibid. 75, 049903 (E) (2007), grqc/0605140. [32] E. Poisson and C. Will, Phys. Rev. D 52, 848 (1995). [33] B. Bruegmann, W. Tichy, and N. Jansen, Phys. Rev. Lett. 92, 211101 (2004), gr-qc/0312112. [34] F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005), gr-qc/0507014. [35] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. D 73, 104002 (2006) , gr-qc/0602026.\"\n }\n]","isConversational":false}}},{"rowIdx":193,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1087"},"arxiv_id":{"kind":"string","value":"0704.1087"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":""},"content_hash":{"kind":"string","value":"57ae57ad802e32ae02271f91071064ea51c6198a0a0e354c32d1360cad7aef45"},"title":{"kind":"string","value":"EPR, Bell, Schrodinger's cat, and the Monty Hall Paradox"},"abstract":{"kind":"string","value":"The purpose of this manuscript is to provide a short pedagogical explanation\nwhy \"quantum collapse\" is not a metaphysical event, by pointing out the analogy\nwith a \"classical collapse\" which is associated with the Monty Hall Paradox."},"authors":{"kind":"list like","value":["Doron Cohen"],"string":"[\n \"Doron Cohen\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"This manuscript constitutes a short self-contained version of some selected sections taken from \"lecture notes in Quantum Mechanics\" [quant-ph/0605180](∼250p). In particular section 47.4 regarding the notion of collapse has attracted some attention. In this section I suggest to use the \"Monty Hall Paradox\" as a pedagogical introduction to the discussion of \"quantum collapse\". From my experience it is the most effective way to convince students and other non-experts that \"quantum collapse\" is not a metaphysical event.\n\n[8] Quantum States ====== [8.1] Is the world classical? (EPR, Bell)\n\nWe would like to examine whether the world we live in is \"classical\" or not. The notion of classical world includes mainly two ingredients: (i) realism (ii) determinism. By realism we means that any quantity that can be measured is well defined even if we do not measure it in practice. By determinism we mean that the result of a measurement is determined in a definite way by the state of the system and by the measurement setup. We shall see later that quantum mechanics is not classical in both respects: In the case of spin 1/2 we cannot associate a definite value of σy for a spin which has been polarized in the σx direction. Moreover, if we measure the σy of a σx polarized spin, we get with equal probability ±1 as the result.\n\nIn this section we would like to assume that our world is \"classical\". Also we would like to assume that interactions cannot travel faster than light. In some textbooks the latter is called \"locality of the interactions\" or \"causality\". It has been found by Bell that the two assumptions lead to an inequality that can be tested experimentally. It turns out from actual experiments that Bell's inequality are violated. This means that our world is either non-classical or else we have to assume that interactions can travel faster than light.\n\nIf the world is classical it follows that for any set of initial conditions a given measurement would yield a definite result. Whether or not we know how to predict or calculate the outcome of a possible measurement is not assumed. To be specific let us consider a particle of zero spin, which disintegrates into two particles going in opposite directions, each with spin 1/2. Let us assume that each spin is described by a set of state variables. state of particle A = x A 1 , x A 2 , ... (1) state of particle B = x B 1 , x B 2 , ...\n\nThe number of state variables might be very big, but it is assumed to be a finite set. Possibly we are not aware or not able to measure some of these \"hidden\" variables. Since we possibly do not have total control over the disintegration, the emerging state of the two particles is described by a joint probability function ρ x A 1 , ..., x B 1 , ... . We assume that the particles do not affect each other after the disintegration (\"causality\" assumption). We measure the spin of each of the particles using a Stern-Gerlach apparatus. The measurement can yield either 1 or -1. For the first particle the measurement outcome will be denoted as a, and for the second particle it will be denoted as b. It is assumed that the outcomes a and b are determined in a deterministic fashion. Namely, given the state variables of the particle and the orientation θ of the apparatus we have a = a(θ A ) = f (θ A , x A 1 , x A 2 , ...) = ±1 (2) b = b(θ B ) = f (θ B , x B 1 , x B 2 , ...) = ±1 2 where the function f () is possibly very complicated. If we put the Stern-Gerlach machine in a different orientation then we will get different results:\n\na ′ = a(θ ′ A ) = f θ ′ A , x A 1 , x A 2 , ... = ±1 ( 3\n\n) b ′ = b(θ ′ B ) = f θ ′ B , x B 1 , x B 2 , ... = ±1\n\nWe have following innocent identity:\n\nab + ab ′ + a ′ b -a ′ b ′ = ±2 ( 4\n\n)\n\nThe proof is as follows: if b = b ′ the sum is ±2a, while if b = -b ′ the sum is ±2a ′ . Though this identity looks innocent, it is completely non trivial. It assumes both \"reality\" and \"causality\" This becomes more manifest if we write this identity as\n\na(θ A )b(θ B ) + a(θ A )b(θ ′ B ) + a(θ ′ A )b(θ B ) -a(θ ′ A )b(θ ′ B ) = ±2 ( 5\n\n)\n\nThe realism is reflected by the assumption that both a(θ A ) and a(θ ′ A ) have definite values, though it is clear that in practice we can measure either a(θ A ) or a(θ ′ A ), but not both. The causality is reflected by assuming that a depends on θ A but not on the distant setup parameter θ B .\n\nLet us assume that we have conducted this experiment many times. Since we have a joint probability distribution ρ, we can calculate average values, for instance:\n\nab = ρ x A 1 , ..., x B 1 , ... f θ A , x A 1 , ... f θ B , x B 1 , ... ( 6\n\n)\n\nThus we get that the following inequality should hold:\n\n| ab + ab ′ + a ′ b -a ′ b ′ | ≤ 2 ( 7\n\n)\n\nThis is called Bell's inequality. Let us see whether it is consistent with quantum mechanics. We assume that all the pairs are generated in a singlet (zero angular momentum) state. It is not difficult to calculate the expectation values. The result is\n\nab = -cos(θ A -θ B ) ≡ C(θ A -θ B ) ( 8\n\n)\n\nwe have for example\n\nC(0 o ) = -1, C(45 o ) = - 1 √ 2 , C(90 o ) = 0, C(180 o ) = +1. ( 9\n\n)\n\nIf the world were classical the Bell's inequality would imply\n\n|C(θ A -θ B ) + C(θ A -θ ′ B ) + C(θ ′ A -θ B ) + C(θ ′ A -θ ′ B )| ≤ 2 ( 10\n\n)\n\nLet us take θ A = 0 o and θ B = 45 o and θ ′ A = 90 o and θ ′ B = -45 o . Assuming that quantum mechanics holds we get\n\n- 1 √ 2 + - 1 √ 2 + - 1 √ 2 -+ 1 √ 2 = 2 √ 2 > 2 ( 11\n\n)\n\nIt turns out, on the basis of celebrated experiments that Nature has chosen to violate Bell's inequality. Furthermore it seems that the results of the experiments are consistent with the predictions of quantum mechanics. Assuming that we do not want to admit that interactions can travel faster than light it follows that our world is not classical. 3 ====== [8.2] The four Postulates of Quantum Mechanics\n\nThe 18th century version classical mechanics can be derived from three postulates: The three laws of Newton. The better formulated 19th century version of classical mechanics can be derived from three postulates: (1) The state of classical particles is determined by the specification of their positions and its velocities; (2) The trajectories are determined by a minimum action principle. (3) The form of the Lagrangian of the theory is determined by symmetry considerations, namely Galilei invariance in the non-relativistic case. See the Classical Mechanics book of Landau and Lifshitz for details.\n\nQuantum mechanically requires four postulates: Two postulates define the notion of quantum state, while the other two postulates, in analogy with classical mechanics, are about the laws that govern the evolution of quantum mechanical systems. [The rest of this section can be found in the lecture notes].\n\n\"Pure states\" are states that have been filtered. The filtering is called \"preparation\". For example: we take a beam of electrons. Without \"filtering\" the beam is not polarized. If we measure the spin we will find (in any orientation of the measurement apparatus) that the polarization is zero. On the other hand, if we \"filter\" the beam (e.g. in the left direction) then there is a direction for which we will get a definite result (in the above example, in the right/left direction). In that case we say that there is full polarization -a pure state. The \"uncertainty principle\" tells us that if in a specific measurement we get a definite result (in the above example, in the right/left direction), then there are different measurements (in the above example, in the up/down direction) for which the result is uncertain. The uncertainty principle is implied by the first postulate.\n\nIn contrast with classical mechanics, in quantum mechanics measurement only has meaning in a statistical sense. We measure \"states\" in the following way: we prepare a collection of systems that were all prepared in the same way. We make the measurement on all the \"copies\". The outcome of the measurement is an event x = x that can be characterized by a distribution function. The single event has no statistical meaning. For example, if we measured the spin of a single electron and get σz = 1, it does not mean that the state is polarized \"up\". In order to know if the electron is polarized we must measure a large number of electrons that were prepared in an identical way. If only 50% of the events give σz = 1 we should conclude that there is no definite polarization in the direction we measured!\n\n====== [8.5] Random Variables\n\nA random variable is an object that can have any numerical value. In other words x = x is an event. Let's assume, for example, that we have a particle that can be in one of five sites: = 1, 2, 3, 4, 5. An experimentalist could measure Prob(x = 3) or Prob(p = 3(2π/5)). Another example is a measurement of the probability Prob( σz = 1) that the particle will have spin up.\n\nx\n\nThe collection of values of x is called the spectrum of values of the random variable. We make the distinction between random variables with a discrete spectrum, and random variables with a continuous spectrum. [The rest of this section can be found in the lecture notes].\n\nQuantum mechanics stands opposite classical statistical mechanics. A particle is described in classical statistical mechanics by a probability function (for presentation purpose we treat x and p as having discrete spectrum):\n\nρ(x, p) = Prob{x = x, p = p} ( 12\n\n)\n\n4 The expectation value of a random variable  = A(x, p) is calculated using the definition:  = x,p ρ(x, p)A(x, p) ≡ trace(ρA) (13)\n\nIn particular we can write:\n\nρ(x, p) = δ(p -p) δ(x -x) ( 14\n\n)\n\nFrom the definition of the expectation value follows the linear relation α  + β B = α  + β B for any pair of observables. This linear relation is a trivial result of classical probability theory. It assumes that the joint probability function Eq.(12) can be defined. But in quantum mechanics we cannot define a \"quantum state\" using a joint probability function, as implied by the observation that our world is not \"classical\". For this reason, we have to use a more sophisticated approach. Loosely speaking one may say that Quantum Mechanics takes Eq.( 14 ) as the definition of ρ, and use the linear relation of the expectation values as a second postulate in order to deduce Eq.(13).\n\n[The rest of this section an be found in the lecture notes].\n\n[47] Theory of Quantum Measurements ====== [47.4] Measurements, the notion of collapse\n\nIn elementary textbooks the quantum measurement process is described as inducing \"collapse\" of the wavefunction. Assume that the system is prepared in state ρ initial = |ψ ψ| and that one measures P = |ϕ ϕ|. If the result of the measurement is P = 1 then it is said that the system has collapsed into the state ρ final = |ϕ ϕ|. The probability for this \"collapse\" is given by the projection formula Prob(ϕ|ψ) = | ϕ|ψ | 2 .\n\nIf one regard ρ(x, x ′ ) or ψ(x) as representing physical reality, rather than a probability matrix or a probability amplitude, then one immediately gets into puzzles. Recalling the EPR experiment this world imply that once the state of one spin is measured at Earth, then immediately the state of the other spin (at the Moon) would change from unpolarized to polarized. This would suggest that some spooky type of \"interaction\" over distance has occurred.\n\nIn fact we shall see that the quantum theory of measurement does not involve any assumption of spooky \"collapse\" mechanism. Once we recall that the notion of quantum state has a statistical interpretation the mystery fades away.\n\nIn fact we explain (see below) that there is \"collapse\" also in classical physics! To avoid potential miss-understanding it should be clear that I do not claim that the classical \"collapse\" which is described below is an explanation of the the quantum collapse. The explanation of quantum collapse using a quantum measurement (probabilistic) point of view will be presented in a later section. The only claim of this section is that in probability theory a correlation is frequently mistaken to be a causal relation: \"smokers are less likely to have Alzheimer\" not because cigarettes help to their health, but simply because their life span is smaller. Similarly quantum collapse is frequently mistaken to be a spooky interaction between well separated systems.\n\nConsider the thought experiment which is known as the \"Monty Hall Paradox\". There is a car behind one of three doors. The car is like a classical \"particle\", and each door is like a \"site\". The initial classical state is such that the car has equal probability to be behind any of the three doors. You are asked to make a guess. Let us say that you peak door #1. Now the organizer opens door #2 and you see that there is no car behind it. This is like a measurement. Now the organizer allows you to change your mind. The naive reasoning is that now the car has equal probability to be behind either of the two remaining doors. So you may claim that it does not matter. But it turns out that this simple answer is very very wrong! The car is no longer in a state of equal probabilities: Now the probability to find it behind door #3 has increased. A standard calculation reveals that the probability to find it behind door #3 is twice large compared with the probability to find it behind door #2. So we have here an example for a classical collapse.\n\nIf the reader is not familiar with this well known \"paradox\", the following may help to understand why we have this collapse (I thank my colleague Eitan Bachmat for providing this explanation). Imagine that there are billion doors. You peak door #1. The organizer opens all the other doors except door #234123. So now you know that the car is either behind door #1 or behind door #234123. You want the car. What are you going to do? It is quite obvious that the car is almost definitely behind door #234123. It is also clear the that the collapse of the car into site #234123 does not imply any physical change in the position of the car. 5 ====== [47.5] Quantum measurements, Schroedinger's cat What do we mean by quantum measurement? In order to clarify this notion let us consider a system and a detector which are prepared independently as\n\nΨ = a ψ a |a ⊗ |q = 0 ( 15\n\n)\n\nAs a result of an interaction we assume that the detector correlates with the system as follows:\n\nÛmeasurement Ψ = ψ a |a ⊗ |q = a ( 16\n\n)\n\nWe call such type of unitary evolution \"ideal measurement\". If the system is in a definite a state, then it is not affected by the detector. Rather, we gain information on the state of the system. One can think of q as representing a memory device in which the information is stored. This memory device can be of course the brain of a human observer. Form the point of view of the observer, the result at the end of the measurement process is to have a definite a. This is interpreted as a \"collapse\" of the state of the system. Some people wrongly think that \"collapse\" is something that goes beyond unitary evolution. But in fact this term just makes over dramatization of the above unitary process.\n\nThe concept of measurement in quantum mechanics involves psychological difficulties which are best illustrated by considering the \"Schroedinger's cat\" experiment. This thought experiment involves a radioactive nucleus, a cat, and a human being. The half life time of the nucleus is an hour. If the radioactive nucleus decays it triggers a poison which kills the cat. The radioactive nucleus and the cat are inside an isolated box. At some stage the human observer may open the box to see what happens with the cat... Let us translate the story into a mathematical language. A time t = 0 the state of the universe (nucleus⊗cat⊗observer) is\n\nΨ = | ↑= radioactive ⊗ |q = 1 = alive ⊗ |Q = 0 = ignorant ( 17\n\n)\n\nwhere q is the state of the cat, and Q is the state of the memory bit inside the human observer. If we wait a very long time the nucleus would definitely decay, and as a result we will have a definitely dead cat:\n\nU waiting Ψ = | ↓= decayed ⊗ |q = -1 = dead ⊗ |Q = 0 = ignorant ( 18\n\n)\n\nIf the observer opens the box he/she would see a dead cat:\n\nU seeing U waiting Ψ = | ↑= decayed ⊗ |q = -1 = dead ⊗ |Q = -1 = shocked ( 19\n\n)\n\nBut if we wait only one hour then\n\nU waiting Ψ = 1 √ 2 | ↑ ⊗ |q = +1 + | ↓ ⊗ |q = -1 ⊗ |Q = 0 = ignorant ( 20\n\n)\n\nwhich means that from the point of view of the observer the system (nucleus+cat) is in a superposition. The cat at this stage is neither definitely alive nor definitely dead. But now the observer open the box and we have:\n\nU seeing U waiting Ψ = 1 √ 2 | ↑ ⊗ |q = +1 ⊗ |Q = +1 = happy + | ↓ ⊗ |q = -1 ⊗ |Q = -1 = shocked ( 21\n\n)\n\nWe see that now, form the point of view of the observer, the cat is in a definite(!) state. This is regarded by the observer as \"collapse\" of the superposition. We have of course two possibilities: one possibility is that the observer sees a definitely dead cat, while the other possibility is that the observer sees a definitely alive cat. The two possibilities 6 \"exist\" in parallel, which leads to the \"many worlds\" interpretation. Equivalently one may say that only one of the two possible scenarios is realized from the point of view of the observer, which leads to the \"relative state\" concept of Everett. Whatever terminology we use, \"collapse\" or \"many worlds\" or \"relative state\", the bottom line is that we have here merely a unitary evolution. ====== [47.6] Measurements, formal treatment\n\nIn this section we describe mathematically how an ideal measurement affects the state of the system. First of all let us write how the U of a measurement process looks like. The formal expression is Ûmeasurement = a P (a) ⊗ D(a) (\n\n) 22\n\nwhere P (a) = |a a| is the projection operator on the state |a , and D(a) is a translation operator. Assuming that the measurement device is prepared in a state of ignorance |q = 0 , the effect of D(a) is to get |q = a . Hence Û Ψ = a P (a) ⊗ D(a)\n\na ′ ψ a ′ |a ′ ⊗ |q = 0 = a ψ a |a ⊗ D(a) |q = 0 = a ψ a |a ⊗ |q = a ( 23\n\n)\n\nA more appropriate way to describe the state of the system is using the probability matrix. Let us describe the above measurement process using this language. After \"reset\" the state of the measurement apparatus is σ (0) = |q=0 q=0|. The system is initially in an arbitrary state ρ. The measurement process correlates that state of the measurement apparatus with the state of the system as follows:\n\nÛ ρ ⊗ σ (0) Û † =\n\na,b P (a) ρ P (b) ⊗ [ D(a) ]σ (0) [ D(b) ] † = a,b\n\nP (a) ρ P (b) ⊗ |q=a q=b| ( 24\n\n)\n\nTracing out the measurement apparatus we get\n\nρ system = a P (a) ρ preparation P (a) = a p a ρ (a) ( 25\n\n)\n\nWhere p a is the trace of the projected probability matrix P (a) ρ P (a) , while ρ (a) is its normalized version. We see that the effect of the measurement is to turn the superposition into a mixture of a states, unlike unitary evolution for which ρ system = U system ρ preparation U † system . So indeed a measurement process looks like a non-unitary process: it turns a pure superposition into a mixture. A simple example is in order. Let us assume that the system is a spin 1/2 particle. The spin is prepared in a pure polarization state ρ =| ψ ψ | which is represented by the matrix\n\nρ ab = ψ a ψ * b = | ψ 1 | 2 ψ 1 ψ * 2 ψ 2 ψ * 1 | ψ 2 | 2 ( 26\n\n)\n\nwhere 1 and 2 are (say) the \"up\" and \"down\" states. Using a Stern-Gerlech apparatus we can measure the polarization of the spin in the up/down direction. This means that the measurement apparatus projects the state of the spin using P (1) = 1 0 0 0 and P (2) = 0 0 0 1 (27) leading after the measurement to the state\n\nρ system = P (1) ρ preparation P (1) + P (2) ρ preparation P (2) = | ψ 1 | 2 0 0 | ψ 2 | 2 ( 28\n\n)\n\nThus the measurement process has eliminated the off-diagonal terms in ρ and hence turned a pure state into a mixture. It is important to remember that this non-unitary non-coherent evolution arise because we look only on the state of the system. On a universal scale the evolution is in fact unitary."},"sections":{"kind":"list like","value":[{"section_type":"OTHER","section_title":"Untitled Section","text":"This manuscript constitutes a short self-contained version of some selected sections taken from \"lecture notes in Quantum Mechanics\" [quant-ph/0605180](∼250p). In particular section 47.4 regarding the notion of collapse has attracted some attention. In this section I suggest to use the \"Monty Hall Paradox\" as a pedagogical introduction to the discussion of \"quantum collapse\". From my experience it is the most effective way to convince students and other non-experts that \"quantum collapse\" is not a metaphysical event.\n\n[8] Quantum States ====== [8.1] Is the world classical? (EPR, Bell)\n\nWe would like to examine whether the world we live in is \"classical\" or not. The notion of classical world includes mainly two ingredients: (i) realism (ii) determinism. By realism we means that any quantity that can be measured is well defined even if we do not measure it in practice. By determinism we mean that the result of a measurement is determined in a definite way by the state of the system and by the measurement setup. We shall see later that quantum mechanics is not classical in both respects: In the case of spin 1/2 we cannot associate a definite value of σy for a spin which has been polarized in the σx direction. Moreover, if we measure the σy of a σx polarized spin, we get with equal probability ±1 as the result.\n\nIn this section we would like to assume that our world is \"classical\". Also we would like to assume that interactions cannot travel faster than light. In some textbooks the latter is called \"locality of the interactions\" or \"causality\". It has been found by Bell that the two assumptions lead to an inequality that can be tested experimentally. It turns out from actual experiments that Bell's inequality are violated. This means that our world is either non-classical or else we have to assume that interactions can travel faster than light.\n\nIf the world is classical it follows that for any set of initial conditions a given measurement would yield a definite result. Whether or not we know how to predict or calculate the outcome of a possible measurement is not assumed. To be specific let us consider a particle of zero spin, which disintegrates into two particles going in opposite directions, each with spin 1/2. Let us assume that each spin is described by a set of state variables. state of particle A = x A 1 , x A 2 , ... (1) state of particle B = x B 1 , x B 2 , ...\n\nThe number of state variables might be very big, but it is assumed to be a finite set. Possibly we are not aware or not able to measure some of these \"hidden\" variables. Since we possibly do not have total control over the disintegration, the emerging state of the two particles is described by a joint probability function ρ x A 1 , ..., x B 1 , ... . We assume that the particles do not affect each other after the disintegration (\"causality\" assumption). We measure the spin of each of the particles using a Stern-Gerlach apparatus. The measurement can yield either 1 or -1. For the first particle the measurement outcome will be denoted as a, and for the second particle it will be denoted as b. It is assumed that the outcomes a and b are determined in a deterministic fashion. Namely, given the state variables of the particle and the orientation θ of the apparatus we have a = a(θ A ) = f (θ A , x A 1 , x A 2 , ...) = ±1 (2) b = b(θ B ) = f (θ B , x B 1 , x B 2 , ...) = ±1 2 where the function f () is possibly very complicated. If we put the Stern-Gerlach machine in a different orientation then we will get different results:\n\na ′ = a(θ ′ A ) = f θ ′ A , x A 1 , x A 2 , ... = ±1 ( 3\n\n) b ′ = b(θ ′ B ) = f θ ′ B , x B 1 , x B 2 , ... = ±1\n\nWe have following innocent identity:\n\nab + ab ′ + a ′ b -a ′ b ′ = ±2 ( 4\n\n)\n\nThe proof is as follows: if b = b ′ the sum is ±2a, while if b = -b ′ the sum is ±2a ′ . Though this identity looks innocent, it is completely non trivial. It assumes both \"reality\" and \"causality\" This becomes more manifest if we write this identity as\n\na(θ A )b(θ B ) + a(θ A )b(θ ′ B ) + a(θ ′ A )b(θ B ) -a(θ ′ A )b(θ ′ B ) = ±2 ( 5\n\n)\n\nThe realism is reflected by the assumption that both a(θ A ) and a(θ ′ A ) have definite values, though it is clear that in practice we can measure either a(θ A ) or a(θ ′ A ), but not both. The causality is reflected by assuming that a depends on θ A but not on the distant setup parameter θ B .\n\nLet us assume that we have conducted this experiment many times. Since we have a joint probability distribution ρ, we can calculate average values, for instance:\n\nab = ρ x A 1 , ..., x B 1 , ... f θ A , x A 1 , ... f θ B , x B 1 , ... ( 6\n\n)\n\nThus we get that the following inequality should hold:\n\n| ab + ab ′ + a ′ b -a ′ b ′ | ≤ 2 ( 7\n\n)\n\nThis is called Bell's inequality. Let us see whether it is consistent with quantum mechanics. We assume that all the pairs are generated in a singlet (zero angular momentum) state. It is not difficult to calculate the expectation values. The result is\n\nab = -cos(θ A -θ B ) ≡ C(θ A -θ B ) ( 8\n\n)\n\nwe have for example\n\nC(0 o ) = -1, C(45 o ) = - 1 √ 2 , C(90 o ) = 0, C(180 o ) = +1. ( 9\n\n)\n\nIf the world were classical the Bell's inequality would imply\n\n|C(θ A -θ B ) + C(θ A -θ ′ B ) + C(θ ′ A -θ B ) + C(θ ′ A -θ ′ B )| ≤ 2 ( 10\n\n)\n\nLet us take θ A = 0 o and θ B = 45 o and θ ′ A = 90 o and θ ′ B = -45 o . Assuming that quantum mechanics holds we get\n\n- 1 √ 2 + - 1 √ 2 + - 1 √ 2 -+ 1 √ 2 = 2 √ 2 > 2 ( 11\n\n)\n\nIt turns out, on the basis of celebrated experiments that Nature has chosen to violate Bell's inequality. Furthermore it seems that the results of the experiments are consistent with the predictions of quantum mechanics. Assuming that we do not want to admit that interactions can travel faster than light it follows that our world is not classical. 3 ====== [8.2] The four Postulates of Quantum Mechanics\n\nThe 18th century version classical mechanics can be derived from three postulates: The three laws of Newton. The better formulated 19th century version of classical mechanics can be derived from three postulates: (1) The state of classical particles is determined by the specification of their positions and its velocities; (2) The trajectories are determined by a minimum action principle. (3) The form of the Lagrangian of the theory is determined by symmetry considerations, namely Galilei invariance in the non-relativistic case. See the Classical Mechanics book of Landau and Lifshitz for details.\n\nQuantum mechanically requires four postulates: Two postulates define the notion of quantum state, while the other two postulates, in analogy with classical mechanics, are about the laws that govern the evolution of quantum mechanical systems. [The rest of this section can be found in the lecture notes]."},{"section_type":"OTHER","section_title":"====== [8.3] What is a Pure State","text":"\"Pure states\" are states that have been filtered. The filtering is called \"preparation\". For example: we take a beam of electrons. Without \"filtering\" the beam is not polarized. If we measure the spin we will find (in any orientation of the measurement apparatus) that the polarization is zero. On the other hand, if we \"filter\" the beam (e.g. in the left direction) then there is a direction for which we will get a definite result (in the above example, in the right/left direction). In that case we say that there is full polarization -a pure state. The \"uncertainty principle\" tells us that if in a specific measurement we get a definite result (in the above example, in the right/left direction), then there are different measurements (in the above example, in the up/down direction) for which the result is uncertain. The uncertainty principle is implied by the first postulate."},{"section_type":"OTHER","section_title":"====== [8.4] What is a Measurement","text":"In contrast with classical mechanics, in quantum mechanics measurement only has meaning in a statistical sense. We measure \"states\" in the following way: we prepare a collection of systems that were all prepared in the same way. We make the measurement on all the \"copies\". The outcome of the measurement is an event x = x that can be characterized by a distribution function. The single event has no statistical meaning. For example, if we measured the spin of a single electron and get σz = 1, it does not mean that the state is polarized \"up\". In order to know if the electron is polarized we must measure a large number of electrons that were prepared in an identical way. If only 50% of the events give σz = 1 we should conclude that there is no definite polarization in the direction we measured!\n\n====== [8.5] Random Variables\n\nA random variable is an object that can have any numerical value. In other words x = x is an event. Let's assume, for example, that we have a particle that can be in one of five sites: = 1, 2, 3, 4, 5. An experimentalist could measure Prob(x = 3) or Prob(p = 3(2π/5)). Another example is a measurement of the probability Prob( σz = 1) that the particle will have spin up.\n\nx\n\nThe collection of values of x is called the spectrum of values of the random variable. We make the distinction between random variables with a discrete spectrum, and random variables with a continuous spectrum. [The rest of this section can be found in the lecture notes]."},{"section_type":"OTHER","section_title":"====== [8.6] Quantum Versus Statistical Mechanics","text":"Quantum mechanics stands opposite classical statistical mechanics. A particle is described in classical statistical mechanics by a probability function (for presentation purpose we treat x and p as having discrete spectrum):\n\nρ(x, p) = Prob{x = x, p = p} ( 12\n\n)\n\n4 The expectation value of a random variable  = A(x, p) is calculated using the definition:  = x,p ρ(x, p)A(x, p) ≡ trace(ρA) (13)\n\nIn particular we can write:\n\nρ(x, p) = δ(p -p) δ(x -x) ( 14\n\n)\n\nFrom the definition of the expectation value follows the linear relation α  + β B = α  + β B for any pair of observables. This linear relation is a trivial result of classical probability theory. It assumes that the joint probability function Eq.(12) can be defined. But in quantum mechanics we cannot define a \"quantum state\" using a joint probability function, as implied by the observation that our world is not \"classical\". For this reason, we have to use a more sophisticated approach. Loosely speaking one may say that Quantum Mechanics takes Eq.( 14 ) as the definition of ρ, and use the linear relation of the expectation values as a second postulate in order to deduce Eq.(13).\n\n[The rest of this section an be found in the lecture notes].\n\n[47] Theory of Quantum Measurements ====== [47.4] Measurements, the notion of collapse\n\nIn elementary textbooks the quantum measurement process is described as inducing \"collapse\" of the wavefunction. Assume that the system is prepared in state ρ initial = |ψ ψ| and that one measures P = |ϕ ϕ|. If the result of the measurement is P = 1 then it is said that the system has collapsed into the state ρ final = |ϕ ϕ|. The probability for this \"collapse\" is given by the projection formula Prob(ϕ|ψ) = | ϕ|ψ | 2 .\n\nIf one regard ρ(x, x ′ ) or ψ(x) as representing physical reality, rather than a probability matrix or a probability amplitude, then one immediately gets into puzzles. Recalling the EPR experiment this world imply that once the state of one spin is measured at Earth, then immediately the state of the other spin (at the Moon) would change from unpolarized to polarized. This would suggest that some spooky type of \"interaction\" over distance has occurred.\n\nIn fact we shall see that the quantum theory of measurement does not involve any assumption of spooky \"collapse\" mechanism. Once we recall that the notion of quantum state has a statistical interpretation the mystery fades away.\n\nIn fact we explain (see below) that there is \"collapse\" also in classical physics! To avoid potential miss-understanding it should be clear that I do not claim that the classical \"collapse\" which is described below is an explanation of the the quantum collapse. The explanation of quantum collapse using a quantum measurement (probabilistic) point of view will be presented in a later section. The only claim of this section is that in probability theory a correlation is frequently mistaken to be a causal relation: \"smokers are less likely to have Alzheimer\" not because cigarettes help to their health, but simply because their life span is smaller. Similarly quantum collapse is frequently mistaken to be a spooky interaction between well separated systems.\n\nConsider the thought experiment which is known as the \"Monty Hall Paradox\". There is a car behind one of three doors. The car is like a classical \"particle\", and each door is like a \"site\". The initial classical state is such that the car has equal probability to be behind any of the three doors. You are asked to make a guess. Let us say that you peak door #1. Now the organizer opens door #2 and you see that there is no car behind it. This is like a measurement. Now the organizer allows you to change your mind. The naive reasoning is that now the car has equal probability to be behind either of the two remaining doors. So you may claim that it does not matter. But it turns out that this simple answer is very very wrong! The car is no longer in a state of equal probabilities: Now the probability to find it behind door #3 has increased. A standard calculation reveals that the probability to find it behind door #3 is twice large compared with the probability to find it behind door #2. So we have here an example for a classical collapse.\n\nIf the reader is not familiar with this well known \"paradox\", the following may help to understand why we have this collapse (I thank my colleague Eitan Bachmat for providing this explanation). Imagine that there are billion doors. You peak door #1. The organizer opens all the other doors except door #234123. So now you know that the car is either behind door #1 or behind door #234123. You want the car. What are you going to do? It is quite obvious that the car is almost definitely behind door #234123. It is also clear the that the collapse of the car into site #234123 does not imply any physical change in the position of the car. 5 ====== [47.5] Quantum measurements, Schroedinger's cat What do we mean by quantum measurement? In order to clarify this notion let us consider a system and a detector which are prepared independently as\n\nΨ = a ψ a |a ⊗ |q = 0 ( 15\n\n)\n\nAs a result of an interaction we assume that the detector correlates with the system as follows:\n\nÛmeasurement Ψ = ψ a |a ⊗ |q = a ( 16\n\n)\n\nWe call such type of unitary evolution \"ideal measurement\". If the system is in a definite a state, then it is not affected by the detector. Rather, we gain information on the state of the system. One can think of q as representing a memory device in which the information is stored. This memory device can be of course the brain of a human observer. Form the point of view of the observer, the result at the end of the measurement process is to have a definite a. This is interpreted as a \"collapse\" of the state of the system. Some people wrongly think that \"collapse\" is something that goes beyond unitary evolution. But in fact this term just makes over dramatization of the above unitary process.\n\nThe concept of measurement in quantum mechanics involves psychological difficulties which are best illustrated by considering the \"Schroedinger's cat\" experiment. This thought experiment involves a radioactive nucleus, a cat, and a human being. The half life time of the nucleus is an hour. If the radioactive nucleus decays it triggers a poison which kills the cat. The radioactive nucleus and the cat are inside an isolated box. At some stage the human observer may open the box to see what happens with the cat... Let us translate the story into a mathematical language. A time t = 0 the state of the universe (nucleus⊗cat⊗observer) is\n\nΨ = | ↑= radioactive ⊗ |q = 1 = alive ⊗ |Q = 0 = ignorant ( 17\n\n)\n\nwhere q is the state of the cat, and Q is the state of the memory bit inside the human observer. If we wait a very long time the nucleus would definitely decay, and as a result we will have a definitely dead cat:\n\nU waiting Ψ = | ↓= decayed ⊗ |q = -1 = dead ⊗ |Q = 0 = ignorant ( 18\n\n)\n\nIf the observer opens the box he/she would see a dead cat:\n\nU seeing U waiting Ψ = | ↑= decayed ⊗ |q = -1 = dead ⊗ |Q = -1 = shocked ( 19\n\n)\n\nBut if we wait only one hour then\n\nU waiting Ψ = 1 √ 2 | ↑ ⊗ |q = +1 + | ↓ ⊗ |q = -1 ⊗ |Q = 0 = ignorant ( 20\n\n)\n\nwhich means that from the point of view of the observer the system (nucleus+cat) is in a superposition. The cat at this stage is neither definitely alive nor definitely dead. But now the observer open the box and we have:\n\nU seeing U waiting Ψ = 1 √ 2 | ↑ ⊗ |q = +1 ⊗ |Q = +1 = happy + | ↓ ⊗ |q = -1 ⊗ |Q = -1 = shocked ( 21\n\n)\n\nWe see that now, form the point of view of the observer, the cat is in a definite(!) state. This is regarded by the observer as \"collapse\" of the superposition. We have of course two possibilities: one possibility is that the observer sees a definitely dead cat, while the other possibility is that the observer sees a definitely alive cat. The two possibilities 6 \"exist\" in parallel, which leads to the \"many worlds\" interpretation. Equivalently one may say that only one of the two possible scenarios is realized from the point of view of the observer, which leads to the \"relative state\" concept of Everett. Whatever terminology we use, \"collapse\" or \"many worlds\" or \"relative state\", the bottom line is that we have here merely a unitary evolution. ====== [47.6] Measurements, formal treatment\n\nIn this section we describe mathematically how an ideal measurement affects the state of the system. First of all let us write how the U of a measurement process looks like. The formal expression is Ûmeasurement = a P (a) ⊗ D(a) (\n\n) 22\n\nwhere P (a) = |a a| is the projection operator on the state |a , and D(a) is a translation operator. Assuming that the measurement device is prepared in a state of ignorance |q = 0 , the effect of D(a) is to get |q = a . Hence Û Ψ = a P (a) ⊗ D(a)\n\na ′ ψ a ′ |a ′ ⊗ |q = 0 = a ψ a |a ⊗ D(a) |q = 0 = a ψ a |a ⊗ |q = a ( 23\n\n)\n\nA more appropriate way to describe the state of the system is using the probability matrix. Let us describe the above measurement process using this language. After \"reset\" the state of the measurement apparatus is σ (0) = |q=0 q=0|. The system is initially in an arbitrary state ρ. The measurement process correlates that state of the measurement apparatus with the state of the system as follows:\n\nÛ ρ ⊗ σ (0) Û † =\n\na,b P (a) ρ P (b) ⊗ [ D(a) ]σ (0) [ D(b) ] † = a,b\n\nP (a) ρ P (b) ⊗ |q=a q=b| ( 24\n\n)\n\nTracing out the measurement apparatus we get\n\nρ system = a P (a) ρ preparation P (a) = a p a ρ (a) ( 25\n\n)\n\nWhere p a is the trace of the projected probability matrix P (a) ρ P (a) , while ρ (a) is its normalized version. We see that the effect of the measurement is to turn the superposition into a mixture of a states, unlike unitary evolution for which ρ system = U system ρ preparation U † system . So indeed a measurement process looks like a non-unitary process: it turns a pure superposition into a mixture. A simple example is in order. Let us assume that the system is a spin 1/2 particle. The spin is prepared in a pure polarization state ρ =| ψ ψ | which is represented by the matrix\n\nρ ab = ψ a ψ * b = | ψ 1 | 2 ψ 1 ψ * 2 ψ 2 ψ * 1 | ψ 2 | 2 ( 26\n\n)\n\nwhere 1 and 2 are (say) the \"up\" and \"down\" states. Using a Stern-Gerlech apparatus we can measure the polarization of the spin in the up/down direction. This means that the measurement apparatus projects the state of the spin using P (1) = 1 0 0 0 and P (2) = 0 0 0 1 (27) leading after the measurement to the state\n\nρ system = P (1) ρ preparation P (1) + P (2) ρ preparation P (2) = | ψ 1 | 2 0 0 | ψ 2 | 2 ( 28\n\n)\n\nThus the measurement process has eliminated the off-diagonal terms in ρ and hence turned a pure state into a mixture. It is important to remember that this non-unitary non-coherent evolution arise because we look only on the state of the system. On a universal scale the evolution is in fact unitary."}],"string":"[\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"This manuscript constitutes a short self-contained version of some selected sections taken from \\\"lecture notes in Quantum Mechanics\\\" [quant-ph/0605180](∼250p). In particular section 47.4 regarding the notion of collapse has attracted some attention. In this section I suggest to use the \\\"Monty Hall Paradox\\\" as a pedagogical introduction to the discussion of \\\"quantum collapse\\\". From my experience it is the most effective way to convince students and other non-experts that \\\"quantum collapse\\\" is not a metaphysical event.\\n\\n[8] Quantum States ====== [8.1] Is the world classical? (EPR, Bell)\\n\\nWe would like to examine whether the world we live in is \\\"classical\\\" or not. The notion of classical world includes mainly two ingredients: (i) realism (ii) determinism. By realism we means that any quantity that can be measured is well defined even if we do not measure it in practice. By determinism we mean that the result of a measurement is determined in a definite way by the state of the system and by the measurement setup. We shall see later that quantum mechanics is not classical in both respects: In the case of spin 1/2 we cannot associate a definite value of σy for a spin which has been polarized in the σx direction. Moreover, if we measure the σy of a σx polarized spin, we get with equal probability ±1 as the result.\\n\\nIn this section we would like to assume that our world is \\\"classical\\\". Also we would like to assume that interactions cannot travel faster than light. In some textbooks the latter is called \\\"locality of the interactions\\\" or \\\"causality\\\". It has been found by Bell that the two assumptions lead to an inequality that can be tested experimentally. It turns out from actual experiments that Bell's inequality are violated. This means that our world is either non-classical or else we have to assume that interactions can travel faster than light.\\n\\nIf the world is classical it follows that for any set of initial conditions a given measurement would yield a definite result. Whether or not we know how to predict or calculate the outcome of a possible measurement is not assumed. To be specific let us consider a particle of zero spin, which disintegrates into two particles going in opposite directions, each with spin 1/2. Let us assume that each spin is described by a set of state variables. state of particle A = x A 1 , x A 2 , ... (1) state of particle B = x B 1 , x B 2 , ...\\n\\nThe number of state variables might be very big, but it is assumed to be a finite set. Possibly we are not aware or not able to measure some of these \\\"hidden\\\" variables. Since we possibly do not have total control over the disintegration, the emerging state of the two particles is described by a joint probability function ρ x A 1 , ..., x B 1 , ... . We assume that the particles do not affect each other after the disintegration (\\\"causality\\\" assumption). We measure the spin of each of the particles using a Stern-Gerlach apparatus. The measurement can yield either 1 or -1. For the first particle the measurement outcome will be denoted as a, and for the second particle it will be denoted as b. It is assumed that the outcomes a and b are determined in a deterministic fashion. Namely, given the state variables of the particle and the orientation θ of the apparatus we have a = a(θ A ) = f (θ A , x A 1 , x A 2 , ...) = ±1 (2) b = b(θ B ) = f (θ B , x B 1 , x B 2 , ...) = ±1 2 where the function f () is possibly very complicated. If we put the Stern-Gerlach machine in a different orientation then we will get different results:\\n\\na ′ = a(θ ′ A ) = f θ ′ A , x A 1 , x A 2 , ... = ±1 ( 3\\n\\n) b ′ = b(θ ′ B ) = f θ ′ B , x B 1 , x B 2 , ... = ±1\\n\\nWe have following innocent identity:\\n\\nab + ab ′ + a ′ b -a ′ b ′ = ±2 ( 4\\n\\n)\\n\\nThe proof is as follows: if b = b ′ the sum is ±2a, while if b = -b ′ the sum is ±2a ′ . Though this identity looks innocent, it is completely non trivial. It assumes both \\\"reality\\\" and \\\"causality\\\" This becomes more manifest if we write this identity as\\n\\na(θ A )b(θ B ) + a(θ A )b(θ ′ B ) + a(θ ′ A )b(θ B ) -a(θ ′ A )b(θ ′ B ) = ±2 ( 5\\n\\n)\\n\\nThe realism is reflected by the assumption that both a(θ A ) and a(θ ′ A ) have definite values, though it is clear that in practice we can measure either a(θ A ) or a(θ ′ A ), but not both. The causality is reflected by assuming that a depends on θ A but not on the distant setup parameter θ B .\\n\\nLet us assume that we have conducted this experiment many times. Since we have a joint probability distribution ρ, we can calculate average values, for instance:\\n\\nab = ρ x A 1 , ..., x B 1 , ... f θ A , x A 1 , ... f θ B , x B 1 , ... ( 6\\n\\n)\\n\\nThus we get that the following inequality should hold:\\n\\n| ab + ab ′ + a ′ b -a ′ b ′ | ≤ 2 ( 7\\n\\n)\\n\\nThis is called Bell's inequality. Let us see whether it is consistent with quantum mechanics. We assume that all the pairs are generated in a singlet (zero angular momentum) state. It is not difficult to calculate the expectation values. The result is\\n\\nab = -cos(θ A -θ B ) ≡ C(θ A -θ B ) ( 8\\n\\n)\\n\\nwe have for example\\n\\nC(0 o ) = -1, C(45 o ) = - 1 √ 2 , C(90 o ) = 0, C(180 o ) = +1. ( 9\\n\\n)\\n\\nIf the world were classical the Bell's inequality would imply\\n\\n|C(θ A -θ B ) + C(θ A -θ ′ B ) + C(θ ′ A -θ B ) + C(θ ′ A -θ ′ B )| ≤ 2 ( 10\\n\\n)\\n\\nLet us take θ A = 0 o and θ B = 45 o and θ ′ A = 90 o and θ ′ B = -45 o . Assuming that quantum mechanics holds we get\\n\\n- 1 √ 2 + - 1 √ 2 + - 1 √ 2 -+ 1 √ 2 = 2 √ 2 > 2 ( 11\\n\\n)\\n\\nIt turns out, on the basis of celebrated experiments that Nature has chosen to violate Bell's inequality. Furthermore it seems that the results of the experiments are consistent with the predictions of quantum mechanics. Assuming that we do not want to admit that interactions can travel faster than light it follows that our world is not classical. 3 ====== [8.2] The four Postulates of Quantum Mechanics\\n\\nThe 18th century version classical mechanics can be derived from three postulates: The three laws of Newton. The better formulated 19th century version of classical mechanics can be derived from three postulates: (1) The state of classical particles is determined by the specification of their positions and its velocities; (2) The trajectories are determined by a minimum action principle. (3) The form of the Lagrangian of the theory is determined by symmetry considerations, namely Galilei invariance in the non-relativistic case. See the Classical Mechanics book of Landau and Lifshitz for details.\\n\\nQuantum mechanically requires four postulates: Two postulates define the notion of quantum state, while the other two postulates, in analogy with classical mechanics, are about the laws that govern the evolution of quantum mechanical systems. [The rest of this section can be found in the lecture notes].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"====== [8.3] What is a Pure State\",\n \"text\": \"\\\"Pure states\\\" are states that have been filtered. The filtering is called \\\"preparation\\\". For example: we take a beam of electrons. Without \\\"filtering\\\" the beam is not polarized. If we measure the spin we will find (in any orientation of the measurement apparatus) that the polarization is zero. On the other hand, if we \\\"filter\\\" the beam (e.g. in the left direction) then there is a direction for which we will get a definite result (in the above example, in the right/left direction). In that case we say that there is full polarization -a pure state. The \\\"uncertainty principle\\\" tells us that if in a specific measurement we get a definite result (in the above example, in the right/left direction), then there are different measurements (in the above example, in the up/down direction) for which the result is uncertain. The uncertainty principle is implied by the first postulate.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"====== [8.4] What is a Measurement\",\n \"text\": \"In contrast with classical mechanics, in quantum mechanics measurement only has meaning in a statistical sense. We measure \\\"states\\\" in the following way: we prepare a collection of systems that were all prepared in the same way. We make the measurement on all the \\\"copies\\\". The outcome of the measurement is an event x = x that can be characterized by a distribution function. The single event has no statistical meaning. For example, if we measured the spin of a single electron and get σz = 1, it does not mean that the state is polarized \\\"up\\\". In order to know if the electron is polarized we must measure a large number of electrons that were prepared in an identical way. If only 50% of the events give σz = 1 we should conclude that there is no definite polarization in the direction we measured!\\n\\n====== [8.5] Random Variables\\n\\nA random variable is an object that can have any numerical value. In other words x = x is an event. Let's assume, for example, that we have a particle that can be in one of five sites: = 1, 2, 3, 4, 5. An experimentalist could measure Prob(x = 3) or Prob(p = 3(2π/5)). Another example is a measurement of the probability Prob( σz = 1) that the particle will have spin up.\\n\\nx\\n\\nThe collection of values of x is called the spectrum of values of the random variable. We make the distinction between random variables with a discrete spectrum, and random variables with a continuous spectrum. [The rest of this section can be found in the lecture notes].\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"====== [8.6] Quantum Versus Statistical Mechanics\",\n \"text\": \"Quantum mechanics stands opposite classical statistical mechanics. A particle is described in classical statistical mechanics by a probability function (for presentation purpose we treat x and p as having discrete spectrum):\\n\\nρ(x, p) = Prob{x = x, p = p} ( 12\\n\\n)\\n\\n4 The expectation value of a random variable  = A(x, p) is calculated using the definition:  = x,p ρ(x, p)A(x, p) ≡ trace(ρA) (13)\\n\\nIn particular we can write:\\n\\nρ(x, p) = δ(p -p) δ(x -x) ( 14\\n\\n)\\n\\nFrom the definition of the expectation value follows the linear relation α  + β B = α  + β B for any pair of observables. This linear relation is a trivial result of classical probability theory. It assumes that the joint probability function Eq.(12) can be defined. But in quantum mechanics we cannot define a \\\"quantum state\\\" using a joint probability function, as implied by the observation that our world is not \\\"classical\\\". For this reason, we have to use a more sophisticated approach. Loosely speaking one may say that Quantum Mechanics takes Eq.( 14 ) as the definition of ρ, and use the linear relation of the expectation values as a second postulate in order to deduce Eq.(13).\\n\\n[The rest of this section an be found in the lecture notes].\\n\\n[47] Theory of Quantum Measurements ====== [47.4] Measurements, the notion of collapse\\n\\nIn elementary textbooks the quantum measurement process is described as inducing \\\"collapse\\\" of the wavefunction. Assume that the system is prepared in state ρ initial = |ψ ψ| and that one measures P = |ϕ ϕ|. If the result of the measurement is P = 1 then it is said that the system has collapsed into the state ρ final = |ϕ ϕ|. The probability for this \\\"collapse\\\" is given by the projection formula Prob(ϕ|ψ) = | ϕ|ψ | 2 .\\n\\nIf one regard ρ(x, x ′ ) or ψ(x) as representing physical reality, rather than a probability matrix or a probability amplitude, then one immediately gets into puzzles. Recalling the EPR experiment this world imply that once the state of one spin is measured at Earth, then immediately the state of the other spin (at the Moon) would change from unpolarized to polarized. This would suggest that some spooky type of \\\"interaction\\\" over distance has occurred.\\n\\nIn fact we shall see that the quantum theory of measurement does not involve any assumption of spooky \\\"collapse\\\" mechanism. Once we recall that the notion of quantum state has a statistical interpretation the mystery fades away.\\n\\nIn fact we explain (see below) that there is \\\"collapse\\\" also in classical physics! To avoid potential miss-understanding it should be clear that I do not claim that the classical \\\"collapse\\\" which is described below is an explanation of the the quantum collapse. The explanation of quantum collapse using a quantum measurement (probabilistic) point of view will be presented in a later section. The only claim of this section is that in probability theory a correlation is frequently mistaken to be a causal relation: \\\"smokers are less likely to have Alzheimer\\\" not because cigarettes help to their health, but simply because their life span is smaller. Similarly quantum collapse is frequently mistaken to be a spooky interaction between well separated systems.\\n\\nConsider the thought experiment which is known as the \\\"Monty Hall Paradox\\\". There is a car behind one of three doors. The car is like a classical \\\"particle\\\", and each door is like a \\\"site\\\". The initial classical state is such that the car has equal probability to be behind any of the three doors. You are asked to make a guess. Let us say that you peak door #1. Now the organizer opens door #2 and you see that there is no car behind it. This is like a measurement. Now the organizer allows you to change your mind. The naive reasoning is that now the car has equal probability to be behind either of the two remaining doors. So you may claim that it does not matter. But it turns out that this simple answer is very very wrong! The car is no longer in a state of equal probabilities: Now the probability to find it behind door #3 has increased. A standard calculation reveals that the probability to find it behind door #3 is twice large compared with the probability to find it behind door #2. So we have here an example for a classical collapse.\\n\\nIf the reader is not familiar with this well known \\\"paradox\\\", the following may help to understand why we have this collapse (I thank my colleague Eitan Bachmat for providing this explanation). Imagine that there are billion doors. You peak door #1. The organizer opens all the other doors except door #234123. So now you know that the car is either behind door #1 or behind door #234123. You want the car. What are you going to do? It is quite obvious that the car is almost definitely behind door #234123. It is also clear the that the collapse of the car into site #234123 does not imply any physical change in the position of the car. 5 ====== [47.5] Quantum measurements, Schroedinger's cat What do we mean by quantum measurement? In order to clarify this notion let us consider a system and a detector which are prepared independently as\\n\\nΨ = a ψ a |a ⊗ |q = 0 ( 15\\n\\n)\\n\\nAs a result of an interaction we assume that the detector correlates with the system as follows:\\n\\nÛmeasurement Ψ = ψ a |a ⊗ |q = a ( 16\\n\\n)\\n\\nWe call such type of unitary evolution \\\"ideal measurement\\\". If the system is in a definite a state, then it is not affected by the detector. Rather, we gain information on the state of the system. One can think of q as representing a memory device in which the information is stored. This memory device can be of course the brain of a human observer. Form the point of view of the observer, the result at the end of the measurement process is to have a definite a. This is interpreted as a \\\"collapse\\\" of the state of the system. Some people wrongly think that \\\"collapse\\\" is something that goes beyond unitary evolution. But in fact this term just makes over dramatization of the above unitary process.\\n\\nThe concept of measurement in quantum mechanics involves psychological difficulties which are best illustrated by considering the \\\"Schroedinger's cat\\\" experiment. This thought experiment involves a radioactive nucleus, a cat, and a human being. The half life time of the nucleus is an hour. If the radioactive nucleus decays it triggers a poison which kills the cat. The radioactive nucleus and the cat are inside an isolated box. At some stage the human observer may open the box to see what happens with the cat... Let us translate the story into a mathematical language. A time t = 0 the state of the universe (nucleus⊗cat⊗observer) is\\n\\nΨ = | ↑= radioactive ⊗ |q = 1 = alive ⊗ |Q = 0 = ignorant ( 17\\n\\n)\\n\\nwhere q is the state of the cat, and Q is the state of the memory bit inside the human observer. If we wait a very long time the nucleus would definitely decay, and as a result we will have a definitely dead cat:\\n\\nU waiting Ψ = | ↓= decayed ⊗ |q = -1 = dead ⊗ |Q = 0 = ignorant ( 18\\n\\n)\\n\\nIf the observer opens the box he/she would see a dead cat:\\n\\nU seeing U waiting Ψ = | ↑= decayed ⊗ |q = -1 = dead ⊗ |Q = -1 = shocked ( 19\\n\\n)\\n\\nBut if we wait only one hour then\\n\\nU waiting Ψ = 1 √ 2 | ↑ ⊗ |q = +1 + | ↓ ⊗ |q = -1 ⊗ |Q = 0 = ignorant ( 20\\n\\n)\\n\\nwhich means that from the point of view of the observer the system (nucleus+cat) is in a superposition. The cat at this stage is neither definitely alive nor definitely dead. But now the observer open the box and we have:\\n\\nU seeing U waiting Ψ = 1 √ 2 | ↑ ⊗ |q = +1 ⊗ |Q = +1 = happy + | ↓ ⊗ |q = -1 ⊗ |Q = -1 = shocked ( 21\\n\\n)\\n\\nWe see that now, form the point of view of the observer, the cat is in a definite(!) state. This is regarded by the observer as \\\"collapse\\\" of the superposition. We have of course two possibilities: one possibility is that the observer sees a definitely dead cat, while the other possibility is that the observer sees a definitely alive cat. The two possibilities 6 \\\"exist\\\" in parallel, which leads to the \\\"many worlds\\\" interpretation. Equivalently one may say that only one of the two possible scenarios is realized from the point of view of the observer, which leads to the \\\"relative state\\\" concept of Everett. Whatever terminology we use, \\\"collapse\\\" or \\\"many worlds\\\" or \\\"relative state\\\", the bottom line is that we have here merely a unitary evolution. ====== [47.6] Measurements, formal treatment\\n\\nIn this section we describe mathematically how an ideal measurement affects the state of the system. First of all let us write how the U of a measurement process looks like. The formal expression is Ûmeasurement = a P (a) ⊗ D(a) (\\n\\n) 22\\n\\nwhere P (a) = |a a| is the projection operator on the state |a , and D(a) is a translation operator. Assuming that the measurement device is prepared in a state of ignorance |q = 0 , the effect of D(a) is to get |q = a . Hence Û Ψ = a P (a) ⊗ D(a)\\n\\na ′ ψ a ′ |a ′ ⊗ |q = 0 = a ψ a |a ⊗ D(a) |q = 0 = a ψ a |a ⊗ |q = a ( 23\\n\\n)\\n\\nA more appropriate way to describe the state of the system is using the probability matrix. Let us describe the above measurement process using this language. After \\\"reset\\\" the state of the measurement apparatus is σ (0) = |q=0 q=0|. The system is initially in an arbitrary state ρ. The measurement process correlates that state of the measurement apparatus with the state of the system as follows:\\n\\nÛ ρ ⊗ σ (0) Û † =\\n\\na,b P (a) ρ P (b) ⊗ [ D(a) ]σ (0) [ D(b) ] † = a,b\\n\\nP (a) ρ P (b) ⊗ |q=a q=b| ( 24\\n\\n)\\n\\nTracing out the measurement apparatus we get\\n\\nρ system = a P (a) ρ preparation P (a) = a p a ρ (a) ( 25\\n\\n)\\n\\nWhere p a is the trace of the projected probability matrix P (a) ρ P (a) , while ρ (a) is its normalized version. We see that the effect of the measurement is to turn the superposition into a mixture of a states, unlike unitary evolution for which ρ system = U system ρ preparation U † system . So indeed a measurement process looks like a non-unitary process: it turns a pure superposition into a mixture. A simple example is in order. Let us assume that the system is a spin 1/2 particle. The spin is prepared in a pure polarization state ρ =| ψ ψ | which is represented by the matrix\\n\\nρ ab = ψ a ψ * b = | ψ 1 | 2 ψ 1 ψ * 2 ψ 2 ψ * 1 | ψ 2 | 2 ( 26\\n\\n)\\n\\nwhere 1 and 2 are (say) the \\\"up\\\" and \\\"down\\\" states. Using a Stern-Gerlech apparatus we can measure the polarization of the spin in the up/down direction. This means that the measurement apparatus projects the state of the spin using P (1) = 1 0 0 0 and P (2) = 0 0 0 1 (27) leading after the measurement to the state\\n\\nρ system = P (1) ρ preparation P (1) + P (2) ρ preparation P (2) = | ψ 1 | 2 0 0 | ψ 2 | 2 ( 28\\n\\n)\\n\\nThus the measurement process has eliminated the off-diagonal terms in ρ and hence turned a pure state into a mixture. It is important to remember that this non-unitary non-coherent evolution arise because we look only on the state of the system. On a universal scale the evolution is in fact unitary.\"\n }\n]","isConversational":false}}},{"rowIdx":194,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1090"},"arxiv_id":{"kind":"string","value":"0704.1090"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1088/1126-6708/2007/10/071"},"content_hash":{"kind":"string","value":"2623c387ee17c729321fff1ce4314279a5e01c6ab9c899752e543243c7e44ed3"},"title":{"kind":"string","value":"Bouncing Universe with Quintom Matter"},"abstract":{"kind":"string","value":"The bouncing universe provides a possible solution to the Big Bang\nsingularity problem. In this paper we study the bouncing solution in the\nuniverse dominated by the Quintom matter with an equation of state (EoS)\ncrossing the cosmological constant boundary. We will show explicitly the\nanalytical and numerical bouncing solutions in three types of models for the\nQuintom matter with an phenomenological EoS, the two scalar fields and a scalar\nfield with a modified Born-Infeld action."},"authors":{"kind":"list like","value":["Yi-Fu Cai","Taotao Qiu","Yun-Song Piao","Mingzhe Li","Xinmin Zhang"],"string":"[\n \"Yi-Fu Cai\",\n \"Taotao Qiu\",\n \"Yun-Song Piao\",\n \"Mingzhe Li\",\n \"Xinmin Zhang\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["gr-qc","astro-ph","hep-ph","hep-th"],"string":"[\n \"gr-qc\",\n \"astro-ph\",\n \"hep-ph\",\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"gr-qc"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The bouncing universe provides a possible solution to the Big Bang singularity problem. In this paper we study the bouncing solution in the universe dominated by the Quintom matter with an equation of state (EoS) crossing the cosmological constant boundary. We will show explicitly the analytical and numerical bouncing solutions in three types of models for the Quintom matter with an phenomenological EoS, the two scalar fields and a scalar field with a modified Born-Infeld action.\n\nA bouncing universe with an initial contraction to a non-vanishing minimal radius, then subsequent an expanding phase provides a possible solution to the singularity problem of the standard Big Bang cosmology. For a successful bounce, it can be shown that within the framework of the standard 4-dimensional Friedmann-Robertson-Walker (FRW) cosmology with Einstein gravity the null energy condition (NEC) is violated for a period of time around the bouncing point. Moreover, for the universe entering into the hot Big Bang era after the bouncing, the EoS of the matter content w in the universe must transit from w < -1 to w > -1.\n\nThe Quintom model [1], proposed to understand the behavior of dark energy with an EoS of w > -1 in the past and w < -1 at present, has been supported by the observational data [2] . Quintom is a dynamical model of dark energy. It differs from the cosmological constant, Quintessence, Phantom, K-essence and so on in the determination of the cosmological evolution. A salient feature of the Quintom model is that its EoS can smoothly cross over w = -1.\n\nIn the recent years there has been a lot of proposals for the Quintom-like models in the literature. In this paper we study the bouncing solution in the universe dominated by the Quintom matter and working with three specific models we will show explicitly the analytical and numerical solutions of the bounce.\n\nWe will start with a detailed examination on the necessary conditions required for a successful bounce. During the contracting phase, the scale factor a(t) is decreasing, i.e., ȧ(t) < 0, and in the expanding phase we have ȧ(t) > 0. At the bouncing point, ȧ(t) = 0, and around this point ä(t) > 0 for a period of time. Equivalently in the bouncing cosmology the hubble parameter H runs across zero from H < 0 to H > 0 and H = 0 at the bouncing point. A successful bounce requires around this point, Ḣ = -4πGρ(1 + w) > 0 . (1)\n\nFrom (1) one can see that w < -1 in a neighborhood of the bouncing point. After the bounce the universe needs to enter into the hot Big Bang era, otherwise the universe filled with the matter with an EoS w < -1 will reach the big rip singularity as what happens to the Phantom dark energy[3]. This requires the EoS of the matter to transit from w < -1 to w > -1.\n\nIn this paper, we study the bouncing solutions in the Quintom models. The paper is organized as follows. In section II, we present the analytical and numerical solutions for different types of models of the Quintom matter. Specifically we consider three models: i)a phenomenological Quintom fluid with a parameterized EoS crossing the cosmological constant boundary; ii)the two-field models of Quintom matter with one being the quintessence-like scalar and another the phantom-like scalar; iii) a single scalar with a Born-Infeld type action. III is the summary of the paper. 2 II. BOUNCING SOLUTION IN THE PRESENCE OF QUINTOM MATTER A. A phenomenological Quintom model\n\nWe start with a study on the possibility of obtaining the bouncing solution in a phenomenological Quintom matter described by the following EoS:\n\nw(t) = -r - s t 2 . ( 2\n\n)\n\nIn (2) r and s are parameters and we require that r < 1 and s > 0. One can see from (2) that w runs from negative infinity at t = 0 to the cosmological constant boundary at t = s 1-r and then crosses this boundary. Assuming that the universe is dominated by the matter with the EoS given by (2), we solve the Friedmann equation and obtain the corresponding evolution of hubble parameter H(t) and scale factor a(t) as follows,\n\nH(t) = 2 3 t (1 -r)t 2 + s , ( 3\n\n)\n\na(t) = (t 2 + s 1 -r ) 1 3(1-r) . ( 4\n\n)\n\nHere we choose t = 0 as the bouncing point and normalize a = 1 at this point. One can see that our solution provides a picture of the universe evolution with contracting for t < 0, and then bouncing at t = 0 to the expanding phase for t > 0. In Fig. 1 we plot the evolution of the EoS, the hubble parameter and the scale factor.\n\n-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -10 -8 -6 -4 -2 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 11 12 t w H t a t FIG. 1: Plot of the evolution of the EoS w, the hubble parameter H and the scale factor a as a function of the cosmic time t.\n\nHere in the numerical calculation we have taken r = 0.6 and s = 1.\n\nFig. 1 that a non-singular bouncing happens at t = 0 with the hubble parameter H running across zero and a minimal non-vanishing scale factor a. At the bouncing point w approaches negative infinity. 3 B. Two-field Quintom model Having presented the bouncing solution with the phenomenological Quintom matter, we now study the bounce in the scalar field models of Quintom matter. However it is not easy to build a Quintom model theoretically. The No-Go theorem proven in Ref. [4] (also see Ref. [1, 5, 6, 7, 8, 9] ) forbids the traditional scalar field model with a lagrangian of general form L = L(φ, ∇ µ φ∇ µ φ) to have its EoS cross over the cosmological constant boundary. Therefore, to realize a viable Quintom field model in the framework of Einstein's gravity theory, it needs to introduce extra degree of freedom to the conventional theory with a single scalar field. The simplest Quintom model involves two scalars with one being the Quintessence-like and another the Phantom-like [1, 10] . This model has been studied in detail later on in the literature. In the recent years there have been a lot of activities in the theoretical study on Quintom-like models such as a single scalar with high-derivative [11, 12] , vector field [13] , extended theory of gravity [14] and so on, see e.g. [15] .\n\nIn this section we consider a two-field Quintom model with the action given by\n\nS = d 4 x √ -g 1 2 ∂ µ φ 1 ∂ µ φ 1 - 1 2 ∂ µ φ 2 ∂ µ φ 2 -V (φ 1 , φ 2 ) , ( 5\n\n)\n\nwhere the metric is in form of (+, -, -, -). Here the field φ 1 has a canonical kinetic term, but φ 2 is a ghost field. In the framework of FRW cosmology, we can easily obtain the energy density and the pressure of this model,\n\nρ = 1 2 φ2 1 - 1 2 φ2 2 + V , p = 1 2 φ2 1 - 1 2 φ2 2 -V , ( 6\n\n)\n\nand the Einstein equations are given by\n\nH 2 = 8πG 3 ( 1 2 φ2 1 - 1 2 φ2 2 + V ) , ( 7\n\n) φ1 + 3H φ1 + dV dφ 1 = 0 , ( 8\n\n) φ2 + 3H φ2 - dV dφ 2 = 0 . ( 9\n\n)\n\nFrom Eq. ( 1 ), we can see that a bouncing solution requires φ2 2 = φ2 1 + 2V when H crosses zero; and the Quintom behavior requires φ2 2 = φ2 1 when w crosses -1. These constraints can be easily satisfied in the parameter space of this model.\n\nIn Fig. 2 and Fig. 3 , we show the bouncing solution for two different type of potentials. In Fig. 2 we take\n\nV (φ 1 , φ 2 ) = V 1 e -λ1 φ 2 1 M 2 + V 2 e -λ2 φ 2 2 M 2 .\n\nIn the numerical calculation we normalize the dimensional parameters such as V 1 , V 2 , φ 1 and φ 2 by a mass scale M which we take specifically to be 10 -2 M pl . And the hubble parameter is normalized with M 2 M pl . One can see from this figure the non-singular behavior of the Hubble parameter and the scale factor for a bounce. The EoS w crosses over the w = -1 and approaches to negative infinity at the bouncing point. And due to the oscillatory behavior of the field φ 1 in the evolution, the EoS w is also oscillating around bouncing point.\n\nIn Fig. 3 , we take\n\nV (φ) = 1 2 m 2 φ 1 2 + V 0 φ 2 -2 .\n\nThis model also provides a bouncing solution, however the detailed evolution of the universe differs from the one shown in Fig. 2 . Fig. 3 shows that the EoS of the Quintom matter will approach w = 1 asymptotically.\n\nIn this section we consider a class of Quintom models described by an effective lagrangian with higher derivative operators. Starting with a canonical scalar field with the lagrangian φ) . This type of models has been considered as a candidate for dark energy, however as shown by the No-Go theorem it does not give w crossing -1. As an effective theory we know that the lagrangian should include more operators, especially if these operators involve the term ✷φ, as pointed in Ref. [11] it will give rise to an EoS across w = -1. A connection of this type of Quintom theory to the string theory has been considered in Ref. [16] and [12] . In this paper we take the string-inspired model in [16] for the detailed study on the bouncing solution, where the action is given by\n\nL = 1 2 ∂ µ φ∂ µ φ -V (\n\nS = d 4 x √ -g -V (φ) 1 -α ′ ∇ µ φ∇ µ φ + β ′ φ✷φ . ( 10\n\n) 4 -30 -20 -10 0 10 20 30 -5 -4 -3 -2 -1 0 -30 -20 -10 0 10 20 30 -15 -12 -9 -6 -3 0 3 6 9 12 15 -1.0 -0.5 0.0 0.5 1.0 0 1 2 3 4 5 6 7 8 9 10 t w t H t a FIG. 2: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose V (φ1, φ2) = V1e -λ 1 φ 2 1 M 2 + V2e -λ 2 φ 2 2 M 2 with parameters: V1 = 15, V2 = 1, λ1 = -1.0, λ2 = 1.0, and for the initial conditions φ1 = 0.5, φ1 = 0.1, φ2 = 0.3, φ2 = 4.\n\n0.6 0.7 0.8 0.9 1.0 1.1 -5 -4 -3 -2 -1 0 1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -4 -2 0 2 4 6 8 10 12 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0 1 2 3 4 5 6 7 8 w t w=1 H t a t FIG. 3: The same plots as Fig. 2 with different potential and model parameters and for the initial conditions φ1 = 2, φ1 = 3, φ2 = 1, φ2 = 2. This is a generalized version of \"Born-Infeld\" action [17, 18] with the introduction of the β ′ term. To the lowest order, the Box-operator term φ✷φ is equivalent to the term ∇ µ φ∇ µ φ when the tachyon is on the top of its potential. However when the tachyon rolls down from the top of the potential, these two terms exhibit different dynamical behavior. The two parameters α ′ and β ′ in (10) could be arbitrary in the case of the background flux being turned on [19] . One interesting feature of this model is that it provides the possibility of its EoS w running across the cosmological constant boundary. In the analytical and numerical studies below to make two parameters (α ′ , β ′ ) dimensionless, it is convenient to redefine α = α ′ M 4 and β = β ′ M 4 where M is an energy scale of the effective theory of tachyon.\n\nV (φ) = 1 2 m 2 φ1 2 + V0φ2 -2 , m = 2, V0 = 0.4,\n\n5 From (10) we obtain the equation of motion for the scalar field φ:\n\nβ 2 ✷( V φ f ) + α∇ µ ( V ∇ µ φ f ) + M 4 V φ f + βV 2f ✷φ = 0 , ( 11\n\n)\n\nwhere f = 1 -α ′ ∇ µ φ∇ µ φ + β ′ φ✷φ and V φ = dV /dφ. Correspondingly, the stress energy tensor of the model is given by\n\nT µν = g µν [V f - β 2M 4 ∇ ρ ( φV f ∇ ρ φ)] + α M 4 V f ∇ µ φ∇ ν φ + β 2M 4 ∇ µ ( φV f )∇ ν φ + β 2M 4 ∇ ν ( φV f )∇ µ φ . ( 12\n\n)\n\nTechnically, it is very useful to define a parameter ψ ≡ ∂L ∂✷φ = -βφV 2M 4 f to solve (11) and (12) . In the framework of a flat FRW universe filled with a homogenous scalar field φ, we have the equations of motion in form of\n\nφ + 3H φ = βφ 4M 4 ψ 2 V 2 - M 4 βφ + α βφ φ2 , ( 13\n\n) ψ + 3H ψ = (2α + β)( M 4 ψ β 2 φ 2 - V 2 4M 4 ψ ) - βφ 2M 4 ψ V V φ -(2α -β) αψ β 2 φ 2 φ2 - 2α βφ ψ φ . ( 14\n\n)\n\nMoreover, the energy density and the pressure of this field can be written as\n\nρ = - αψ βφ φ2 -ψ φ - βφ 4M 4 ψ V 2 - M 4 ψ βφ , ( 15\n\n) p = - αψ βφ φ2 -ψ φ + βφ 4M 4 ψ V 2 + M 4 ψ βφ . ( 16\n\n)\n\nFrom Eq.( 1 ), one can see that a successful bounce requires:\n\nβφ 4M 4 ψ V 2 + M 4 ψ βφ = - αψ βφ φ2 -ψ φ < 0 . ( 17\n\n)\n\nWe will show below that (17) can be satisfied easily for our model. In the numerical study on the bouncing solution, we constrain the parameters α and β so that the model in (10) when expanding the derivative terms in the square root to the lowest order gives rise to a canonical kinetic term for the scalar field φ [16] , i.e., α + β > 0. In Fig. 4 and Fig. 5 we show the bounce solution for different potentials. In Fig. 4 , we take V (φ) = V 0 e -λφ 2 /M 2 with λ being a dimensionless parameter. One can see from this figure the scale factor initially decreases, then passes through its minimum and increases. Moreover, away from the bouncing point in the expanding phase, the EoS of the scalar field crosses w = -1 and approaches w = -0.6, which gives rise to a possible inflationary phase after the bouncing. In the numerical calculation we take the energy scale M to be 10 -2 M pl , and the hubble parameter is normalized with M 2 M pl . In Fig. 5 , we consider the model with potential V (φ) = V 0 /φ and then show another example of the bouncing solution 1 . Here the energy scale M is chosen to be 10 -2 M pl as well. One can see from this figure the clear picture of the bouncing, however the detailed evolution of the universe differs from the one shown in Fig. 4 . After entering the expanding phase, the EoS w crosses the cosmological constant boundary and approaches w = 1 3 , which is equivalent to the EoS of the radiation.\n\nIn this paper we have studied the possibility of obtaining a non-singular bounce in the presence of the Quintom matter. In the literature there have been a lot of efforts in constructing the bouncing universe, for instance, the Pre Big Bang scenario [21] , and the Ekpyrotic scenario [22] . In Refs. [23, 24, 25] and [26, 27] the authors have considered models with the modifications of gravity with the high order terms. In general these models modify the 4-dimensional 1 A Born-Infeld lagrangian with this potential provides a scaling solution, see Ref. [20] .\n\n-1 0 1 2 3 4 5 6 7 -5 -4 -3 -2 -1 0 1 0 1 2 3 -6 -4 -2 0 2 4 6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 6 w t H t a t FIG. 4: The plots of the evolution of the EoS w, the hubble parameter H and the scale factor a. Here in the numerical calculation we take the potential V (φ) = V0e -λφ 2 , α = -0.2, β = 2, λ = 2, V0 = 5, and the initial values are φ = 1, φ = 3, H = -1, and ψ = -80.\n\n1.0 1.5 2.0 2.5 3.0 -5 -4 -3 -2 -1 0 1 -3 -2 -1 0 1 2 3 -5 0 5 10 15 20 25 -2 -1 0 1 0 1 2 3 4 5 6 7 w t w=1/3 H t a t FIG. 5: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose the potential as V (φ) = V 0 φ , α = -0.2, β = 2, V0 = 0.7, and for the initial conditions φ = 10, φ = -3, H = -1, ψ = -40.\n\nEinstein gravity. However, the models we consider for bounce universe in this paper are restricted to be within the standard 4-dimensional FRW framework. Recently two papers [28, 29] have studied the possibilities of having a bounce universe with the ghost condensate. In the original formulation the ghost condensate [30] will not be able to give EoS crossing w = -1. The authors of these papers [28, 29] have considered a generalized model of ghost condensate [31] and shown the bouncing solutions. In this paper we have studied the general issue of obtaining a bouncing universe with the Quintom matter. Our results show that a universe in the presence of the Quintom matter will avoid the problem of the Big Bang singularity. Explicitly for the analytical and numerical studies we have considered three models: the phenomenological model, 7 the two-field model and the string-inspired Quintom model. The latter one is a generalization of the idea in Ref. [11] by introducing higher derivative terms to realize the EoS crossing w = -1. In this regard, this model for the bounce solution has the similarity with a recent paper [32] where the authors presented a bouncing solution with non-local SFT [12] .\n\nWe thank Jie Liu, Jian-Xin Lu, Anupam Mazumdar, Shinji Tsujikawa, Jun-Qing Xia, and Gong-Bo Zhao, for discussions. This work is supported in part by National Natural Science Foundation of China under Grant Nos. 90303004, 10533010 and 10675136 and by the Chinese Academy of Science under Grant No. KJCX3-SYW-N2.\n\n[1] B. Feng, X. Wang and X. Zhang, Phys. Lett. B607, 35 (2005). [2] For example, see G.-B. Zhao, J.-Q. Xia, H. Li, C. Tao, J. M. Virey, Z.-H. Zhu and X. Zhang, arXiv:astro-ph/0612728. [3] R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003) [4] J.-Q. Xia, Y.-F. Cai, T.-T. Qiu, G.-B. Zhao and X.-M. Zhang, arXiv:astro-ph/0703202. [5] G.-B. Zhao, J.-Q. Xia, M. Li, B. Feng, and X. Zhang, Phys. Rev. D72, 123515 (2005). [6] R. R. Caldwell, M. Doran, Phys. Rev. D72, 043527 (2005). [7] A. Vikman, Phys. Rev. D71, 023515 (2005). [8] W. Hu, Phys. Rev. D71, 047301 (2005). [9] M. Kunz, and D. Sapone, Phys. Rev. D74, 123503 (2006), arXiv:astro-ph/0609040. [10] Z. Guo, Y. Piao, X. Zhang, and Y.-Z. Zhang, Phys. Lett. B608, 177 (2005). [11] M. Li, B. Feng, and X. Zhang, JCAP 0512, 002 (2005). [12] I. Y. Aref'eva, A. S. Koshelev and S. Y. Vernov, Phys. Rev. D 72, 064017 (2005); I. Y. Aref'eva and A. S. Koshelev, JHEP 0702, 041 (2007) [13] C. Armendariz-Picon, JCAP 0407, 007 (2004); H. Wei, and R.-G. Cai, Phys. Rev. D73, 083002 (2006). [14] R.-G. Cai, H.-S. Zhang, and A. Wang, Commun. Theor. Phys. 44, 948 (2005); P. S. Apostolopoulos, and N. Tetradis, Phys. Rev. D74, 064021 (2006); H.-S. Zhang, and Z.-H. Zhu, Phys. Rev. D75, 023510 (2007); A. Sheykhi, B. Wang, and N. Riazi, e-Print: arXiv:0704.0666 [hep-th]. [15] B. Feng, M. Li, Y. Piao, and X. Zhang, Phys. Lett. B634, 101 (2006); X.-F. Zhang, H. Li, Y.-S. Piao, and X. M. Zhang, Mod. Phys. Lett. A21, 231 (2006); H. Li, B. Feng, J.-Q. Xia, and X. Zhang, Phys. Rev. D73, 103503 (2006); X.-F. Zhang, and T.-T. Qiu, Phys. Lett. B642, 187 (2006); H. M. Sadjadi, and M. Alimohammadi, Phys. Rev. D74, 043506 (2006); Z.-K. Guo, Y.-S. Piao, X. Zhang, and Y.-Z. Zhang, astro-ph/0608165; Y.-F. Cai, H. Li, Y.-S. Piao, and X. Zhang, Phys. Lett. B646, 141 (2007), arXiv:gr-qc/0609039; W. Zhao, and Y. Zhang, Phys. Rev. D73, 123509 (2006); M. R. Setare, Phys. Lett. B641, 130 (2006); E. O. Kahya, and V. K. Onemli, gr-qc/0612026; X. Zhang, and F.-Q. Wu, Phys. Rev. D72, 043524 (2005); B. Wang, J. Zang, C.-Y. Lin, E. Abdalla, and S. Micheletti, e-Print: astro-ph/0607126; B. Wang, Y.-G. Gong, and E. Abdalla, Phys. Lett. B624, 141 (2005). [16] Y.-F. Cai, M.-Z. Li, J.-X. Lu, Y.-S. Piao, T.-T. Qiu and X.-M. Zhang, arXiv:hep-th/0701016. [17] A. A. Gerasimov and S. L. Shatashvili, JHEP 0010, 034 (2000). [18] D. Kutasov, M. Marino and G. W. Moore, JHEP 0010, 045 (2000); D. Kutasov, M. Marino and G. W. Moore, arXiv:hep-th/0010108. [19] P. Mukhopadhyay and A. Sen, JHEP 0211, 047 (2002). [20] M. R. Garousi, M. Sami and S. Tsujikawa, Phys. Rev. D 71, 083005 (2005). [21] G. Veneziano, Phys. Lett. B265, 287 (1991); M. Gasperini, and G. Veneziano, Astropart. Phys. 1, 317 (1993). [22] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Phys. Rev. D64, 123522 (2001); J. Khoury, B. A. Ovrut, N.\n\nSeiberg, P. J. Steinhardt, and N. Turok, Phys. Rev. D65, 086007 (2002). [23] R. Brustein, and R. Madden, Phys. Rev. D57, 712 (1998). [24] C. Cartier, E. J. Copeland, and R. Madden, JHEP 0001, 035 (2000). [25] S. Tsujikawa, R. Brandenberger, and F. Finelli, Phys. Rev. D66, 083513 (2002). [26] T. Biswas, A. Mazumdar, W. Siegel, JCAP 0603, 009 (2006). [27] T. Biswas, R. Brandenberger, A. Mazumdar, W. Siegel, arXiv:hep-th/0610274. [28] P. Creminelli and L. Senatore, arXiv:hep-th/0702165. [29] E. I. Buchbinder, J. Khoury and B. A. Ovrut, arXiv:hep-th/0702154. [30] N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, JHEP 0405, 074 (2004). [31] S. Tsujikawa, Phys. Rev. D72, 083512 (2005). [32] I. Y. Aref'eva, L. V. Joukovskaya, and S. Y. Vernov, arXiv:hep-th/0701184."},"sections":{"kind":"list like","value":[{"section_type":"OTHER","section_title":"Untitled Section","text":"The bouncing universe provides a possible solution to the Big Bang singularity problem. In this paper we study the bouncing solution in the universe dominated by the Quintom matter with an equation of state (EoS) crossing the cosmological constant boundary. We will show explicitly the analytical and numerical bouncing solutions in three types of models for the Quintom matter with an phenomenological EoS, the two scalar fields and a scalar field with a modified Born-Infeld action."},{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"A bouncing universe with an initial contraction to a non-vanishing minimal radius, then subsequent an expanding phase provides a possible solution to the singularity problem of the standard Big Bang cosmology. For a successful bounce, it can be shown that within the framework of the standard 4-dimensional Friedmann-Robertson-Walker (FRW) cosmology with Einstein gravity the null energy condition (NEC) is violated for a period of time around the bouncing point. Moreover, for the universe entering into the hot Big Bang era after the bouncing, the EoS of the matter content w in the universe must transit from w < -1 to w > -1.\n\nThe Quintom model [1], proposed to understand the behavior of dark energy with an EoS of w > -1 in the past and w < -1 at present, has been supported by the observational data [2] . Quintom is a dynamical model of dark energy. It differs from the cosmological constant, Quintessence, Phantom, K-essence and so on in the determination of the cosmological evolution. A salient feature of the Quintom model is that its EoS can smoothly cross over w = -1.\n\nIn the recent years there has been a lot of proposals for the Quintom-like models in the literature. In this paper we study the bouncing solution in the universe dominated by the Quintom matter and working with three specific models we will show explicitly the analytical and numerical solutions of the bounce.\n\nWe will start with a detailed examination on the necessary conditions required for a successful bounce. During the contracting phase, the scale factor a(t) is decreasing, i.e., ȧ(t) < 0, and in the expanding phase we have ȧ(t) > 0. At the bouncing point, ȧ(t) = 0, and around this point ä(t) > 0 for a period of time. Equivalently in the bouncing cosmology the hubble parameter H runs across zero from H < 0 to H > 0 and H = 0 at the bouncing point. A successful bounce requires around this point, Ḣ = -4πGρ(1 + w) > 0 . (1)\n\nFrom (1) one can see that w < -1 in a neighborhood of the bouncing point. After the bounce the universe needs to enter into the hot Big Bang era, otherwise the universe filled with the matter with an EoS w < -1 will reach the big rip singularity as what happens to the Phantom dark energy[3]. This requires the EoS of the matter to transit from w < -1 to w > -1.\n\nIn this paper, we study the bouncing solutions in the Quintom models. The paper is organized as follows. In section II, we present the analytical and numerical solutions for different types of models of the Quintom matter. Specifically we consider three models: i)a phenomenological Quintom fluid with a parameterized EoS crossing the cosmological constant boundary; ii)the two-field models of Quintom matter with one being the quintessence-like scalar and another the phantom-like scalar; iii) a single scalar with a Born-Infeld type action. III is the summary of the paper. 2 II. BOUNCING SOLUTION IN THE PRESENCE OF QUINTOM MATTER A. A phenomenological Quintom model\n\nWe start with a study on the possibility of obtaining the bouncing solution in a phenomenological Quintom matter described by the following EoS:\n\nw(t) = -r - s t 2 . ( 2\n\n)\n\nIn (2) r and s are parameters and we require that r < 1 and s > 0. One can see from (2) that w runs from negative infinity at t = 0 to the cosmological constant boundary at t = s 1-r and then crosses this boundary. Assuming that the universe is dominated by the matter with the EoS given by (2), we solve the Friedmann equation and obtain the corresponding evolution of hubble parameter H(t) and scale factor a(t) as follows,\n\nH(t) = 2 3 t (1 -r)t 2 + s , ( 3\n\n)\n\na(t) = (t 2 + s 1 -r ) 1 3(1-r) . ( 4\n\n)\n\nHere we choose t = 0 as the bouncing point and normalize a = 1 at this point. One can see that our solution provides a picture of the universe evolution with contracting for t < 0, and then bouncing at t = 0 to the expanding phase for t > 0. In Fig. 1 we plot the evolution of the EoS, the hubble parameter and the scale factor.\n\n-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -10 -8 -6 -4 -2 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 11 12 t w H t a t FIG. 1: Plot of the evolution of the EoS w, the hubble parameter H and the scale factor a as a function of the cosmic time t.\n\nHere in the numerical calculation we have taken r = 0.6 and s = 1."},{"section_type":"OTHER","section_title":"One can see from","text":"Fig. 1 that a non-singular bouncing happens at t = 0 with the hubble parameter H running across zero and a minimal non-vanishing scale factor a. At the bouncing point w approaches negative infinity. 3 B. Two-field Quintom model Having presented the bouncing solution with the phenomenological Quintom matter, we now study the bounce in the scalar field models of Quintom matter. However it is not easy to build a Quintom model theoretically. The No-Go theorem proven in Ref. [4] (also see Ref. [1, 5, 6, 7, 8, 9] ) forbids the traditional scalar field model with a lagrangian of general form L = L(φ, ∇ µ φ∇ µ φ) to have its EoS cross over the cosmological constant boundary. Therefore, to realize a viable Quintom field model in the framework of Einstein's gravity theory, it needs to introduce extra degree of freedom to the conventional theory with a single scalar field. The simplest Quintom model involves two scalars with one being the Quintessence-like and another the Phantom-like [1, 10] . This model has been studied in detail later on in the literature. In the recent years there have been a lot of activities in the theoretical study on Quintom-like models such as a single scalar with high-derivative [11, 12] , vector field [13] , extended theory of gravity [14] and so on, see e.g. [15] .\n\nIn this section we consider a two-field Quintom model with the action given by\n\nS = d 4 x √ -g 1 2 ∂ µ φ 1 ∂ µ φ 1 - 1 2 ∂ µ φ 2 ∂ µ φ 2 -V (φ 1 , φ 2 ) , ( 5\n\n)\n\nwhere the metric is in form of (+, -, -, -). Here the field φ 1 has a canonical kinetic term, but φ 2 is a ghost field. In the framework of FRW cosmology, we can easily obtain the energy density and the pressure of this model,\n\nρ = 1 2 φ2 1 - 1 2 φ2 2 + V , p = 1 2 φ2 1 - 1 2 φ2 2 -V , ( 6\n\n)\n\nand the Einstein equations are given by\n\nH 2 = 8πG 3 ( 1 2 φ2 1 - 1 2 φ2 2 + V ) , ( 7\n\n) φ1 + 3H φ1 + dV dφ 1 = 0 , ( 8\n\n) φ2 + 3H φ2 - dV dφ 2 = 0 . ( 9\n\n)\n\nFrom Eq. ( 1 ), we can see that a bouncing solution requires φ2 2 = φ2 1 + 2V when H crosses zero; and the Quintom behavior requires φ2 2 = φ2 1 when w crosses -1. These constraints can be easily satisfied in the parameter space of this model.\n\nIn Fig. 2 and Fig. 3 , we show the bouncing solution for two different type of potentials. In Fig. 2 we take\n\nV (φ 1 , φ 2 ) = V 1 e -λ1 φ 2 1 M 2 + V 2 e -λ2 φ 2 2 M 2 .\n\nIn the numerical calculation we normalize the dimensional parameters such as V 1 , V 2 , φ 1 and φ 2 by a mass scale M which we take specifically to be 10 -2 M pl . And the hubble parameter is normalized with M 2 M pl . One can see from this figure the non-singular behavior of the Hubble parameter and the scale factor for a bounce. The EoS w crosses over the w = -1 and approaches to negative infinity at the bouncing point. And due to the oscillatory behavior of the field φ 1 in the evolution, the EoS w is also oscillating around bouncing point.\n\nIn Fig. 3 , we take\n\nV (φ) = 1 2 m 2 φ 1 2 + V 0 φ 2 -2 .\n\nThis model also provides a bouncing solution, however the detailed evolution of the universe differs from the one shown in Fig. 2 . Fig. 3 shows that the EoS of the Quintom matter will approach w = 1 asymptotically."},{"section_type":"OTHER","section_title":"C. A single scalar with high-derivative terms","text":"In this section we consider a class of Quintom models described by an effective lagrangian with higher derivative operators. Starting with a canonical scalar field with the lagrangian φ) . This type of models has been considered as a candidate for dark energy, however as shown by the No-Go theorem it does not give w crossing -1. As an effective theory we know that the lagrangian should include more operators, especially if these operators involve the term ✷φ, as pointed in Ref. [11] it will give rise to an EoS across w = -1. A connection of this type of Quintom theory to the string theory has been considered in Ref. [16] and [12] . In this paper we take the string-inspired model in [16] for the detailed study on the bouncing solution, where the action is given by\n\nL = 1 2 ∂ µ φ∂ µ φ -V (\n\nS = d 4 x √ -g -V (φ) 1 -α ′ ∇ µ φ∇ µ φ + β ′ φ✷φ . ( 10\n\n) 4 -30 -20 -10 0 10 20 30 -5 -4 -3 -2 -1 0 -30 -20 -10 0 10 20 30 -15 -12 -9 -6 -3 0 3 6 9 12 15 -1.0 -0.5 0.0 0.5 1.0 0 1 2 3 4 5 6 7 8 9 10 t w t H t a FIG. 2: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose V (φ1, φ2) = V1e -λ 1 φ 2 1 M 2 + V2e -λ 2 φ 2 2 M 2 with parameters: V1 = 15, V2 = 1, λ1 = -1.0, λ2 = 1.0, and for the initial conditions φ1 = 0.5, φ1 = 0.1, φ2 = 0.3, φ2 = 4.\n\n0.6 0.7 0.8 0.9 1.0 1.1 -5 -4 -3 -2 -1 0 1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -4 -2 0 2 4 6 8 10 12 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0 1 2 3 4 5 6 7 8 w t w=1 H t a t FIG. 3: The same plots as Fig. 2 with different potential and model parameters and for the initial conditions φ1 = 2, φ1 = 3, φ2 = 1, φ2 = 2. This is a generalized version of \"Born-Infeld\" action [17, 18] with the introduction of the β ′ term. To the lowest order, the Box-operator term φ✷φ is equivalent to the term ∇ µ φ∇ µ φ when the tachyon is on the top of its potential. However when the tachyon rolls down from the top of the potential, these two terms exhibit different dynamical behavior. The two parameters α ′ and β ′ in (10) could be arbitrary in the case of the background flux being turned on [19] . One interesting feature of this model is that it provides the possibility of its EoS w running across the cosmological constant boundary. In the analytical and numerical studies below to make two parameters (α ′ , β ′ ) dimensionless, it is convenient to redefine α = α ′ M 4 and β = β ′ M 4 where M is an energy scale of the effective theory of tachyon.\n\nV (φ) = 1 2 m 2 φ1 2 + V0φ2 -2 , m = 2, V0 = 0.4,\n\n5 From (10) we obtain the equation of motion for the scalar field φ:\n\nβ 2 ✷( V φ f ) + α∇ µ ( V ∇ µ φ f ) + M 4 V φ f + βV 2f ✷φ = 0 , ( 11\n\n)\n\nwhere f = 1 -α ′ ∇ µ φ∇ µ φ + β ′ φ✷φ and V φ = dV /dφ. Correspondingly, the stress energy tensor of the model is given by\n\nT µν = g µν [V f - β 2M 4 ∇ ρ ( φV f ∇ ρ φ)] + α M 4 V f ∇ µ φ∇ ν φ + β 2M 4 ∇ µ ( φV f )∇ ν φ + β 2M 4 ∇ ν ( φV f )∇ µ φ . ( 12\n\n)\n\nTechnically, it is very useful to define a parameter ψ ≡ ∂L ∂✷φ = -βφV 2M 4 f to solve (11) and (12) . In the framework of a flat FRW universe filled with a homogenous scalar field φ, we have the equations of motion in form of\n\nφ + 3H φ = βφ 4M 4 ψ 2 V 2 - M 4 βφ + α βφ φ2 , ( 13\n\n) ψ + 3H ψ = (2α + β)( M 4 ψ β 2 φ 2 - V 2 4M 4 ψ ) - βφ 2M 4 ψ V V φ -(2α -β) αψ β 2 φ 2 φ2 - 2α βφ ψ φ . ( 14\n\n)\n\nMoreover, the energy density and the pressure of this field can be written as\n\nρ = - αψ βφ φ2 -ψ φ - βφ 4M 4 ψ V 2 - M 4 ψ βφ , ( 15\n\n) p = - αψ βφ φ2 -ψ φ + βφ 4M 4 ψ V 2 + M 4 ψ βφ . ( 16\n\n)\n\nFrom Eq.( 1 ), one can see that a successful bounce requires:\n\nβφ 4M 4 ψ V 2 + M 4 ψ βφ = - αψ βφ φ2 -ψ φ < 0 . ( 17\n\n)\n\nWe will show below that (17) can be satisfied easily for our model. In the numerical study on the bouncing solution, we constrain the parameters α and β so that the model in (10) when expanding the derivative terms in the square root to the lowest order gives rise to a canonical kinetic term for the scalar field φ [16] , i.e., α + β > 0. In Fig. 4 and Fig. 5 we show the bounce solution for different potentials. In Fig. 4 , we take V (φ) = V 0 e -λφ 2 /M 2 with λ being a dimensionless parameter. One can see from this figure the scale factor initially decreases, then passes through its minimum and increases. Moreover, away from the bouncing point in the expanding phase, the EoS of the scalar field crosses w = -1 and approaches w = -0.6, which gives rise to a possible inflationary phase after the bouncing. In the numerical calculation we take the energy scale M to be 10 -2 M pl , and the hubble parameter is normalized with M 2 M pl . In Fig. 5 , we consider the model with potential V (φ) = V 0 /φ and then show another example of the bouncing solution 1 . Here the energy scale M is chosen to be 10 -2 M pl as well. One can see from this figure the clear picture of the bouncing, however the detailed evolution of the universe differs from the one shown in Fig. 4 . After entering the expanding phase, the EoS w crosses the cosmological constant boundary and approaches w = 1 3 , which is equivalent to the EoS of the radiation."},{"section_type":"DISCUSSION","section_title":"III. CONCLUSION AND DISCUSSIONS","text":"In this paper we have studied the possibility of obtaining a non-singular bounce in the presence of the Quintom matter. In the literature there have been a lot of efforts in constructing the bouncing universe, for instance, the Pre Big Bang scenario [21] , and the Ekpyrotic scenario [22] . In Refs. [23, 24, 25] and [26, 27] the authors have considered models with the modifications of gravity with the high order terms. In general these models modify the 4-dimensional 1 A Born-Infeld lagrangian with this potential provides a scaling solution, see Ref. [20] .\n\n-1 0 1 2 3 4 5 6 7 -5 -4 -3 -2 -1 0 1 0 1 2 3 -6 -4 -2 0 2 4 6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 6 w t H t a t FIG. 4: The plots of the evolution of the EoS w, the hubble parameter H and the scale factor a. Here in the numerical calculation we take the potential V (φ) = V0e -λφ 2 , α = -0.2, β = 2, λ = 2, V0 = 5, and the initial values are φ = 1, φ = 3, H = -1, and ψ = -80.\n\n1.0 1.5 2.0 2.5 3.0 -5 -4 -3 -2 -1 0 1 -3 -2 -1 0 1 2 3 -5 0 5 10 15 20 25 -2 -1 0 1 0 1 2 3 4 5 6 7 w t w=1/3 H t a t FIG. 5: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose the potential as V (φ) = V 0 φ , α = -0.2, β = 2, V0 = 0.7, and for the initial conditions φ = 10, φ = -3, H = -1, ψ = -40.\n\nEinstein gravity. However, the models we consider for bounce universe in this paper are restricted to be within the standard 4-dimensional FRW framework. Recently two papers [28, 29] have studied the possibilities of having a bounce universe with the ghost condensate. In the original formulation the ghost condensate [30] will not be able to give EoS crossing w = -1. The authors of these papers [28, 29] have considered a generalized model of ghost condensate [31] and shown the bouncing solutions. In this paper we have studied the general issue of obtaining a bouncing universe with the Quintom matter. Our results show that a universe in the presence of the Quintom matter will avoid the problem of the Big Bang singularity. Explicitly for the analytical and numerical studies we have considered three models: the phenomenological model, 7 the two-field model and the string-inspired Quintom model. The latter one is a generalization of the idea in Ref. [11] by introducing higher derivative terms to realize the EoS crossing w = -1. In this regard, this model for the bounce solution has the similarity with a recent paper [32] where the authors presented a bouncing solution with non-local SFT [12] ."},{"section_type":"OTHER","section_title":"Acknowledgments","text":"We thank Jie Liu, Jian-Xin Lu, Anupam Mazumdar, Shinji Tsujikawa, Jun-Qing Xia, and Gong-Bo Zhao, for discussions. This work is supported in part by National Natural Science Foundation of China under Grant Nos. 90303004, 10533010 and 10675136 and by the Chinese Academy of Science under Grant No. KJCX3-SYW-N2.\n\n[1] B. Feng, X. Wang and X. Zhang, Phys. Lett. B607, 35 (2005). [2] For example, see G.-B. Zhao, J.-Q. Xia, H. Li, C. Tao, J. M. Virey, Z.-H. Zhu and X. Zhang, arXiv:astro-ph/0612728. [3] R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003) [4] J.-Q. Xia, Y.-F. Cai, T.-T. Qiu, G.-B. Zhao and X.-M. Zhang, arXiv:astro-ph/0703202. [5] G.-B. Zhao, J.-Q. Xia, M. Li, B. Feng, and X. Zhang, Phys. Rev. D72, 123515 (2005). [6] R. R. Caldwell, M. Doran, Phys. Rev. D72, 043527 (2005). [7] A. Vikman, Phys. Rev. D71, 023515 (2005). [8] W. Hu, Phys. Rev. D71, 047301 (2005). [9] M. Kunz, and D. Sapone, Phys. Rev. D74, 123503 (2006), arXiv:astro-ph/0609040. [10] Z. Guo, Y. Piao, X. Zhang, and Y.-Z. Zhang, Phys. Lett. B608, 177 (2005). [11] M. Li, B. Feng, and X. Zhang, JCAP 0512, 002 (2005). [12] I. Y. Aref'eva, A. S. Koshelev and S. Y. Vernov, Phys. Rev. D 72, 064017 (2005); I. Y. Aref'eva and A. S. Koshelev, JHEP 0702, 041 (2007) [13] C. Armendariz-Picon, JCAP 0407, 007 (2004); H. Wei, and R.-G. Cai, Phys. Rev. D73, 083002 (2006). [14] R.-G. Cai, H.-S. Zhang, and A. Wang, Commun. Theor. Phys. 44, 948 (2005); P. S. Apostolopoulos, and N. Tetradis, Phys. Rev. D74, 064021 (2006); H.-S. Zhang, and Z.-H. Zhu, Phys. Rev. D75, 023510 (2007); A. Sheykhi, B. Wang, and N. Riazi, e-Print: arXiv:0704.0666 [hep-th]. [15] B. Feng, M. Li, Y. Piao, and X. Zhang, Phys. Lett. B634, 101 (2006); X.-F. Zhang, H. Li, Y.-S. Piao, and X. M. Zhang, Mod. Phys. Lett. A21, 231 (2006); H. Li, B. Feng, J.-Q. Xia, and X. Zhang, Phys. Rev. D73, 103503 (2006); X.-F. Zhang, and T.-T. Qiu, Phys. Lett. B642, 187 (2006); H. M. Sadjadi, and M. Alimohammadi, Phys. Rev. D74, 043506 (2006); Z.-K. Guo, Y.-S. Piao, X. Zhang, and Y.-Z. Zhang, astro-ph/0608165; Y.-F. Cai, H. Li, Y.-S. Piao, and X. Zhang, Phys. Lett. B646, 141 (2007), arXiv:gr-qc/0609039; W. Zhao, and Y. Zhang, Phys. Rev. D73, 123509 (2006); M. R. Setare, Phys. Lett. B641, 130 (2006); E. O. Kahya, and V. K. Onemli, gr-qc/0612026; X. Zhang, and F.-Q. Wu, Phys. Rev. D72, 043524 (2005); B. Wang, J. Zang, C.-Y. Lin, E. Abdalla, and S. Micheletti, e-Print: astro-ph/0607126; B. Wang, Y.-G. Gong, and E. Abdalla, Phys. Lett. B624, 141 (2005). [16] Y.-F. Cai, M.-Z. Li, J.-X. Lu, Y.-S. Piao, T.-T. Qiu and X.-M. Zhang, arXiv:hep-th/0701016. [17] A. A. Gerasimov and S. L. Shatashvili, JHEP 0010, 034 (2000). [18] D. Kutasov, M. Marino and G. W. Moore, JHEP 0010, 045 (2000); D. Kutasov, M. Marino and G. W. Moore, arXiv:hep-th/0010108. [19] P. Mukhopadhyay and A. Sen, JHEP 0211, 047 (2002). [20] M. R. Garousi, M. Sami and S. Tsujikawa, Phys. Rev. D 71, 083005 (2005). [21] G. Veneziano, Phys. Lett. B265, 287 (1991); M. Gasperini, and G. Veneziano, Astropart. Phys. 1, 317 (1993). [22] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Phys. Rev. D64, 123522 (2001); J. Khoury, B. A. Ovrut, N.\n\nSeiberg, P. J. Steinhardt, and N. Turok, Phys. Rev. D65, 086007 (2002). [23] R. Brustein, and R. Madden, Phys. Rev. D57, 712 (1998). [24] C. Cartier, E. J. Copeland, and R. Madden, JHEP 0001, 035 (2000). [25] S. Tsujikawa, R. Brandenberger, and F. Finelli, Phys. Rev. D66, 083513 (2002). [26] T. Biswas, A. Mazumdar, W. Siegel, JCAP 0603, 009 (2006). [27] T. Biswas, R. Brandenberger, A. Mazumdar, W. Siegel, arXiv:hep-th/0610274. [28] P. Creminelli and L. Senatore, arXiv:hep-th/0702165. [29] E. I. Buchbinder, J. Khoury and B. A. Ovrut, arXiv:hep-th/0702154. [30] N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, JHEP 0405, 074 (2004). [31] S. Tsujikawa, Phys. Rev. D72, 083512 (2005). [32] I. Y. Aref'eva, L. V. Joukovskaya, and S. Y. Vernov, arXiv:hep-th/0701184."}],"string":"[\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"The bouncing universe provides a possible solution to the Big Bang singularity problem. In this paper we study the bouncing solution in the universe dominated by the Quintom matter with an equation of state (EoS) crossing the cosmological constant boundary. We will show explicitly the analytical and numerical bouncing solutions in three types of models for the Quintom matter with an phenomenological EoS, the two scalar fields and a scalar field with a modified Born-Infeld action.\"\n },\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"A bouncing universe with an initial contraction to a non-vanishing minimal radius, then subsequent an expanding phase provides a possible solution to the singularity problem of the standard Big Bang cosmology. For a successful bounce, it can be shown that within the framework of the standard 4-dimensional Friedmann-Robertson-Walker (FRW) cosmology with Einstein gravity the null energy condition (NEC) is violated for a period of time around the bouncing point. Moreover, for the universe entering into the hot Big Bang era after the bouncing, the EoS of the matter content w in the universe must transit from w < -1 to w > -1.\\n\\nThe Quintom model [1], proposed to understand the behavior of dark energy with an EoS of w > -1 in the past and w < -1 at present, has been supported by the observational data [2] . Quintom is a dynamical model of dark energy. It differs from the cosmological constant, Quintessence, Phantom, K-essence and so on in the determination of the cosmological evolution. A salient feature of the Quintom model is that its EoS can smoothly cross over w = -1.\\n\\nIn the recent years there has been a lot of proposals for the Quintom-like models in the literature. In this paper we study the bouncing solution in the universe dominated by the Quintom matter and working with three specific models we will show explicitly the analytical and numerical solutions of the bounce.\\n\\nWe will start with a detailed examination on the necessary conditions required for a successful bounce. During the contracting phase, the scale factor a(t) is decreasing, i.e., ȧ(t) < 0, and in the expanding phase we have ȧ(t) > 0. At the bouncing point, ȧ(t) = 0, and around this point ä(t) > 0 for a period of time. Equivalently in the bouncing cosmology the hubble parameter H runs across zero from H < 0 to H > 0 and H = 0 at the bouncing point. A successful bounce requires around this point, Ḣ = -4πGρ(1 + w) > 0 . (1)\\n\\nFrom (1) one can see that w < -1 in a neighborhood of the bouncing point. After the bounce the universe needs to enter into the hot Big Bang era, otherwise the universe filled with the matter with an EoS w < -1 will reach the big rip singularity as what happens to the Phantom dark energy[3]. This requires the EoS of the matter to transit from w < -1 to w > -1.\\n\\nIn this paper, we study the bouncing solutions in the Quintom models. The paper is organized as follows. In section II, we present the analytical and numerical solutions for different types of models of the Quintom matter. Specifically we consider three models: i)a phenomenological Quintom fluid with a parameterized EoS crossing the cosmological constant boundary; ii)the two-field models of Quintom matter with one being the quintessence-like scalar and another the phantom-like scalar; iii) a single scalar with a Born-Infeld type action. III is the summary of the paper. 2 II. BOUNCING SOLUTION IN THE PRESENCE OF QUINTOM MATTER A. A phenomenological Quintom model\\n\\nWe start with a study on the possibility of obtaining the bouncing solution in a phenomenological Quintom matter described by the following EoS:\\n\\nw(t) = -r - s t 2 . ( 2\\n\\n)\\n\\nIn (2) r and s are parameters and we require that r < 1 and s > 0. One can see from (2) that w runs from negative infinity at t = 0 to the cosmological constant boundary at t = s 1-r and then crosses this boundary. Assuming that the universe is dominated by the matter with the EoS given by (2), we solve the Friedmann equation and obtain the corresponding evolution of hubble parameter H(t) and scale factor a(t) as follows,\\n\\nH(t) = 2 3 t (1 -r)t 2 + s , ( 3\\n\\n)\\n\\na(t) = (t 2 + s 1 -r ) 1 3(1-r) . ( 4\\n\\n)\\n\\nHere we choose t = 0 as the bouncing point and normalize a = 1 at this point. One can see that our solution provides a picture of the universe evolution with contracting for t < 0, and then bouncing at t = 0 to the expanding phase for t > 0. In Fig. 1 we plot the evolution of the EoS, the hubble parameter and the scale factor.\\n\\n-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 -10 -8 -6 -4 -2 0 2 4 6 8 10 0 1 2 3 4 5 6 7 8 9 10 11 12 t w H t a t FIG. 1: Plot of the evolution of the EoS w, the hubble parameter H and the scale factor a as a function of the cosmic time t.\\n\\nHere in the numerical calculation we have taken r = 0.6 and s = 1.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"One can see from\",\n \"text\": \"Fig. 1 that a non-singular bouncing happens at t = 0 with the hubble parameter H running across zero and a minimal non-vanishing scale factor a. At the bouncing point w approaches negative infinity. 3 B. Two-field Quintom model Having presented the bouncing solution with the phenomenological Quintom matter, we now study the bounce in the scalar field models of Quintom matter. However it is not easy to build a Quintom model theoretically. The No-Go theorem proven in Ref. [4] (also see Ref. [1, 5, 6, 7, 8, 9] ) forbids the traditional scalar field model with a lagrangian of general form L = L(φ, ∇ µ φ∇ µ φ) to have its EoS cross over the cosmological constant boundary. Therefore, to realize a viable Quintom field model in the framework of Einstein's gravity theory, it needs to introduce extra degree of freedom to the conventional theory with a single scalar field. The simplest Quintom model involves two scalars with one being the Quintessence-like and another the Phantom-like [1, 10] . This model has been studied in detail later on in the literature. In the recent years there have been a lot of activities in the theoretical study on Quintom-like models such as a single scalar with high-derivative [11, 12] , vector field [13] , extended theory of gravity [14] and so on, see e.g. [15] .\\n\\nIn this section we consider a two-field Quintom model with the action given by\\n\\nS = d 4 x √ -g 1 2 ∂ µ φ 1 ∂ µ φ 1 - 1 2 ∂ µ φ 2 ∂ µ φ 2 -V (φ 1 , φ 2 ) , ( 5\\n\\n)\\n\\nwhere the metric is in form of (+, -, -, -). Here the field φ 1 has a canonical kinetic term, but φ 2 is a ghost field. In the framework of FRW cosmology, we can easily obtain the energy density and the pressure of this model,\\n\\nρ = 1 2 φ2 1 - 1 2 φ2 2 + V , p = 1 2 φ2 1 - 1 2 φ2 2 -V , ( 6\\n\\n)\\n\\nand the Einstein equations are given by\\n\\nH 2 = 8πG 3 ( 1 2 φ2 1 - 1 2 φ2 2 + V ) , ( 7\\n\\n) φ1 + 3H φ1 + dV dφ 1 = 0 , ( 8\\n\\n) φ2 + 3H φ2 - dV dφ 2 = 0 . ( 9\\n\\n)\\n\\nFrom Eq. ( 1 ), we can see that a bouncing solution requires φ2 2 = φ2 1 + 2V when H crosses zero; and the Quintom behavior requires φ2 2 = φ2 1 when w crosses -1. These constraints can be easily satisfied in the parameter space of this model.\\n\\nIn Fig. 2 and Fig. 3 , we show the bouncing solution for two different type of potentials. In Fig. 2 we take\\n\\nV (φ 1 , φ 2 ) = V 1 e -λ1 φ 2 1 M 2 + V 2 e -λ2 φ 2 2 M 2 .\\n\\nIn the numerical calculation we normalize the dimensional parameters such as V 1 , V 2 , φ 1 and φ 2 by a mass scale M which we take specifically to be 10 -2 M pl . And the hubble parameter is normalized with M 2 M pl . One can see from this figure the non-singular behavior of the Hubble parameter and the scale factor for a bounce. The EoS w crosses over the w = -1 and approaches to negative infinity at the bouncing point. And due to the oscillatory behavior of the field φ 1 in the evolution, the EoS w is also oscillating around bouncing point.\\n\\nIn Fig. 3 , we take\\n\\nV (φ) = 1 2 m 2 φ 1 2 + V 0 φ 2 -2 .\\n\\nThis model also provides a bouncing solution, however the detailed evolution of the universe differs from the one shown in Fig. 2 . Fig. 3 shows that the EoS of the Quintom matter will approach w = 1 asymptotically.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. A single scalar with high-derivative terms\",\n \"text\": \"In this section we consider a class of Quintom models described by an effective lagrangian with higher derivative operators. Starting with a canonical scalar field with the lagrangian φ) . This type of models has been considered as a candidate for dark energy, however as shown by the No-Go theorem it does not give w crossing -1. As an effective theory we know that the lagrangian should include more operators, especially if these operators involve the term ✷φ, as pointed in Ref. [11] it will give rise to an EoS across w = -1. A connection of this type of Quintom theory to the string theory has been considered in Ref. [16] and [12] . In this paper we take the string-inspired model in [16] for the detailed study on the bouncing solution, where the action is given by\\n\\nL = 1 2 ∂ µ φ∂ µ φ -V (\\n\\nS = d 4 x √ -g -V (φ) 1 -α ′ ∇ µ φ∇ µ φ + β ′ φ✷φ . ( 10\\n\\n) 4 -30 -20 -10 0 10 20 30 -5 -4 -3 -2 -1 0 -30 -20 -10 0 10 20 30 -15 -12 -9 -6 -3 0 3 6 9 12 15 -1.0 -0.5 0.0 0.5 1.0 0 1 2 3 4 5 6 7 8 9 10 t w t H t a FIG. 2: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose V (φ1, φ2) = V1e -λ 1 φ 2 1 M 2 + V2e -λ 2 φ 2 2 M 2 with parameters: V1 = 15, V2 = 1, λ1 = -1.0, λ2 = 1.0, and for the initial conditions φ1 = 0.5, φ1 = 0.1, φ2 = 0.3, φ2 = 4.\\n\\n0.6 0.7 0.8 0.9 1.0 1.1 -5 -4 -3 -2 -1 0 1 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 -4 -2 0 2 4 6 8 10 12 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0 1 2 3 4 5 6 7 8 w t w=1 H t a t FIG. 3: The same plots as Fig. 2 with different potential and model parameters and for the initial conditions φ1 = 2, φ1 = 3, φ2 = 1, φ2 = 2. This is a generalized version of \\\"Born-Infeld\\\" action [17, 18] with the introduction of the β ′ term. To the lowest order, the Box-operator term φ✷φ is equivalent to the term ∇ µ φ∇ µ φ when the tachyon is on the top of its potential. However when the tachyon rolls down from the top of the potential, these two terms exhibit different dynamical behavior. The two parameters α ′ and β ′ in (10) could be arbitrary in the case of the background flux being turned on [19] . One interesting feature of this model is that it provides the possibility of its EoS w running across the cosmological constant boundary. In the analytical and numerical studies below to make two parameters (α ′ , β ′ ) dimensionless, it is convenient to redefine α = α ′ M 4 and β = β ′ M 4 where M is an energy scale of the effective theory of tachyon.\\n\\nV (φ) = 1 2 m 2 φ1 2 + V0φ2 -2 , m = 2, V0 = 0.4,\\n\\n5 From (10) we obtain the equation of motion for the scalar field φ:\\n\\nβ 2 ✷( V φ f ) + α∇ µ ( V ∇ µ φ f ) + M 4 V φ f + βV 2f ✷φ = 0 , ( 11\\n\\n)\\n\\nwhere f = 1 -α ′ ∇ µ φ∇ µ φ + β ′ φ✷φ and V φ = dV /dφ. Correspondingly, the stress energy tensor of the model is given by\\n\\nT µν = g µν [V f - β 2M 4 ∇ ρ ( φV f ∇ ρ φ)] + α M 4 V f ∇ µ φ∇ ν φ + β 2M 4 ∇ µ ( φV f )∇ ν φ + β 2M 4 ∇ ν ( φV f )∇ µ φ . ( 12\\n\\n)\\n\\nTechnically, it is very useful to define a parameter ψ ≡ ∂L ∂✷φ = -βφV 2M 4 f to solve (11) and (12) . In the framework of a flat FRW universe filled with a homogenous scalar field φ, we have the equations of motion in form of\\n\\nφ + 3H φ = βφ 4M 4 ψ 2 V 2 - M 4 βφ + α βφ φ2 , ( 13\\n\\n) ψ + 3H ψ = (2α + β)( M 4 ψ β 2 φ 2 - V 2 4M 4 ψ ) - βφ 2M 4 ψ V V φ -(2α -β) αψ β 2 φ 2 φ2 - 2α βφ ψ φ . ( 14\\n\\n)\\n\\nMoreover, the energy density and the pressure of this field can be written as\\n\\nρ = - αψ βφ φ2 -ψ φ - βφ 4M 4 ψ V 2 - M 4 ψ βφ , ( 15\\n\\n) p = - αψ βφ φ2 -ψ φ + βφ 4M 4 ψ V 2 + M 4 ψ βφ . ( 16\\n\\n)\\n\\nFrom Eq.( 1 ), one can see that a successful bounce requires:\\n\\nβφ 4M 4 ψ V 2 + M 4 ψ βφ = - αψ βφ φ2 -ψ φ < 0 . ( 17\\n\\n)\\n\\nWe will show below that (17) can be satisfied easily for our model. In the numerical study on the bouncing solution, we constrain the parameters α and β so that the model in (10) when expanding the derivative terms in the square root to the lowest order gives rise to a canonical kinetic term for the scalar field φ [16] , i.e., α + β > 0. In Fig. 4 and Fig. 5 we show the bounce solution for different potentials. In Fig. 4 , we take V (φ) = V 0 e -λφ 2 /M 2 with λ being a dimensionless parameter. One can see from this figure the scale factor initially decreases, then passes through its minimum and increases. Moreover, away from the bouncing point in the expanding phase, the EoS of the scalar field crosses w = -1 and approaches w = -0.6, which gives rise to a possible inflationary phase after the bouncing. In the numerical calculation we take the energy scale M to be 10 -2 M pl , and the hubble parameter is normalized with M 2 M pl . In Fig. 5 , we consider the model with potential V (φ) = V 0 /φ and then show another example of the bouncing solution 1 . Here the energy scale M is chosen to be 10 -2 M pl as well. One can see from this figure the clear picture of the bouncing, however the detailed evolution of the universe differs from the one shown in Fig. 4 . After entering the expanding phase, the EoS w crosses the cosmological constant boundary and approaches w = 1 3 , which is equivalent to the EoS of the radiation.\"\n },\n {\n \"section_type\": \"DISCUSSION\",\n \"section_title\": \"III. CONCLUSION AND DISCUSSIONS\",\n \"text\": \"In this paper we have studied the possibility of obtaining a non-singular bounce in the presence of the Quintom matter. In the literature there have been a lot of efforts in constructing the bouncing universe, for instance, the Pre Big Bang scenario [21] , and the Ekpyrotic scenario [22] . In Refs. [23, 24, 25] and [26, 27] the authors have considered models with the modifications of gravity with the high order terms. In general these models modify the 4-dimensional 1 A Born-Infeld lagrangian with this potential provides a scaling solution, see Ref. [20] .\\n\\n-1 0 1 2 3 4 5 6 7 -5 -4 -3 -2 -1 0 1 0 1 2 3 -6 -4 -2 0 2 4 6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 0 1 2 3 4 5 6 w t H t a t FIG. 4: The plots of the evolution of the EoS w, the hubble parameter H and the scale factor a. Here in the numerical calculation we take the potential V (φ) = V0e -λφ 2 , α = -0.2, β = 2, λ = 2, V0 = 5, and the initial values are φ = 1, φ = 3, H = -1, and ψ = -80.\\n\\n1.0 1.5 2.0 2.5 3.0 -5 -4 -3 -2 -1 0 1 -3 -2 -1 0 1 2 3 -5 0 5 10 15 20 25 -2 -1 0 1 0 1 2 3 4 5 6 7 w t w=1/3 H t a t FIG. 5: The plots of the evolutions of the EoS w, hubble parameter H and the scale factor a. In the numerical calculation we choose the potential as V (φ) = V 0 φ , α = -0.2, β = 2, V0 = 0.7, and for the initial conditions φ = 10, φ = -3, H = -1, ψ = -40.\\n\\nEinstein gravity. However, the models we consider for bounce universe in this paper are restricted to be within the standard 4-dimensional FRW framework. Recently two papers [28, 29] have studied the possibilities of having a bounce universe with the ghost condensate. In the original formulation the ghost condensate [30] will not be able to give EoS crossing w = -1. The authors of these papers [28, 29] have considered a generalized model of ghost condensate [31] and shown the bouncing solutions. In this paper we have studied the general issue of obtaining a bouncing universe with the Quintom matter. Our results show that a universe in the presence of the Quintom matter will avoid the problem of the Big Bang singularity. Explicitly for the analytical and numerical studies we have considered three models: the phenomenological model, 7 the two-field model and the string-inspired Quintom model. The latter one is a generalization of the idea in Ref. [11] by introducing higher derivative terms to realize the EoS crossing w = -1. In this regard, this model for the bounce solution has the similarity with a recent paper [32] where the authors presented a bouncing solution with non-local SFT [12] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments\",\n \"text\": \"We thank Jie Liu, Jian-Xin Lu, Anupam Mazumdar, Shinji Tsujikawa, Jun-Qing Xia, and Gong-Bo Zhao, for discussions. This work is supported in part by National Natural Science Foundation of China under Grant Nos. 90303004, 10533010 and 10675136 and by the Chinese Academy of Science under Grant No. KJCX3-SYW-N2.\\n\\n[1] B. Feng, X. Wang and X. Zhang, Phys. Lett. B607, 35 (2005). [2] For example, see G.-B. Zhao, J.-Q. Xia, H. Li, C. Tao, J. M. Virey, Z.-H. Zhu and X. Zhang, arXiv:astro-ph/0612728. [3] R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003) [4] J.-Q. Xia, Y.-F. Cai, T.-T. Qiu, G.-B. Zhao and X.-M. Zhang, arXiv:astro-ph/0703202. [5] G.-B. Zhao, J.-Q. Xia, M. Li, B. Feng, and X. Zhang, Phys. Rev. D72, 123515 (2005). [6] R. R. Caldwell, M. Doran, Phys. Rev. D72, 043527 (2005). [7] A. Vikman, Phys. Rev. D71, 023515 (2005). [8] W. Hu, Phys. Rev. D71, 047301 (2005). [9] M. Kunz, and D. Sapone, Phys. Rev. D74, 123503 (2006), arXiv:astro-ph/0609040. [10] Z. Guo, Y. Piao, X. Zhang, and Y.-Z. Zhang, Phys. Lett. B608, 177 (2005). [11] M. Li, B. Feng, and X. Zhang, JCAP 0512, 002 (2005). [12] I. Y. Aref'eva, A. S. Koshelev and S. Y. Vernov, Phys. Rev. D 72, 064017 (2005); I. Y. Aref'eva and A. S. Koshelev, JHEP 0702, 041 (2007) [13] C. Armendariz-Picon, JCAP 0407, 007 (2004); H. Wei, and R.-G. Cai, Phys. Rev. D73, 083002 (2006). [14] R.-G. Cai, H.-S. Zhang, and A. Wang, Commun. Theor. Phys. 44, 948 (2005); P. S. Apostolopoulos, and N. Tetradis, Phys. Rev. D74, 064021 (2006); H.-S. Zhang, and Z.-H. Zhu, Phys. Rev. D75, 023510 (2007); A. Sheykhi, B. Wang, and N. Riazi, e-Print: arXiv:0704.0666 [hep-th]. [15] B. Feng, M. Li, Y. Piao, and X. Zhang, Phys. Lett. B634, 101 (2006); X.-F. Zhang, H. Li, Y.-S. Piao, and X. M. Zhang, Mod. Phys. Lett. A21, 231 (2006); H. Li, B. Feng, J.-Q. Xia, and X. Zhang, Phys. Rev. D73, 103503 (2006); X.-F. Zhang, and T.-T. Qiu, Phys. Lett. B642, 187 (2006); H. M. Sadjadi, and M. Alimohammadi, Phys. Rev. D74, 043506 (2006); Z.-K. Guo, Y.-S. Piao, X. Zhang, and Y.-Z. Zhang, astro-ph/0608165; Y.-F. Cai, H. Li, Y.-S. Piao, and X. Zhang, Phys. Lett. B646, 141 (2007), arXiv:gr-qc/0609039; W. Zhao, and Y. Zhang, Phys. Rev. D73, 123509 (2006); M. R. Setare, Phys. Lett. B641, 130 (2006); E. O. Kahya, and V. K. Onemli, gr-qc/0612026; X. Zhang, and F.-Q. Wu, Phys. Rev. D72, 043524 (2005); B. Wang, J. Zang, C.-Y. Lin, E. Abdalla, and S. Micheletti, e-Print: astro-ph/0607126; B. Wang, Y.-G. Gong, and E. Abdalla, Phys. Lett. B624, 141 (2005). [16] Y.-F. Cai, M.-Z. Li, J.-X. Lu, Y.-S. Piao, T.-T. Qiu and X.-M. Zhang, arXiv:hep-th/0701016. [17] A. A. Gerasimov and S. L. Shatashvili, JHEP 0010, 034 (2000). [18] D. Kutasov, M. Marino and G. W. Moore, JHEP 0010, 045 (2000); D. Kutasov, M. Marino and G. W. Moore, arXiv:hep-th/0010108. [19] P. Mukhopadhyay and A. Sen, JHEP 0211, 047 (2002). [20] M. R. Garousi, M. Sami and S. Tsujikawa, Phys. Rev. D 71, 083005 (2005). [21] G. Veneziano, Phys. Lett. B265, 287 (1991); M. Gasperini, and G. Veneziano, Astropart. Phys. 1, 317 (1993). [22] J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Phys. Rev. D64, 123522 (2001); J. Khoury, B. A. Ovrut, N.\\n\\nSeiberg, P. J. Steinhardt, and N. Turok, Phys. Rev. D65, 086007 (2002). [23] R. Brustein, and R. Madden, Phys. Rev. D57, 712 (1998). [24] C. Cartier, E. J. Copeland, and R. Madden, JHEP 0001, 035 (2000). [25] S. Tsujikawa, R. Brandenberger, and F. Finelli, Phys. Rev. D66, 083513 (2002). [26] T. Biswas, A. Mazumdar, W. Siegel, JCAP 0603, 009 (2006). [27] T. Biswas, R. Brandenberger, A. Mazumdar, W. Siegel, arXiv:hep-th/0610274. [28] P. Creminelli and L. Senatore, arXiv:hep-th/0702165. [29] E. I. Buchbinder, J. Khoury and B. A. Ovrut, arXiv:hep-th/0702154. [30] N. Arkani-Hamed, H. C. Cheng, M. A. Luty and S. Mukohyama, JHEP 0405, 074 (2004). [31] S. Tsujikawa, Phys. Rev. D72, 083512 (2005). [32] I. Y. Aref'eva, L. V. Joukovskaya, and S. Y. Vernov, arXiv:hep-th/0701184.\"\n }\n]","isConversational":false}}},{"rowIdx":195,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1092"},"arxiv_id":{"kind":"string","value":"0704.1092"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1063/1.2746128"},"content_hash":{"kind":"string","value":"b7695653f6647e8e619ed5ff510d9714b7a8657922337075d92123288adfc2c0"},"title":{"kind":"string","value":"Simplifying additivity problems using direct sum constructions"},"abstract":{"kind":"string","value":"We study the additivity problems for the classical capacity of quantum\nchannels, the minimal output entropy and its convex closure. We show for each\nof them that additivity for arbitrary pairs of channels holds iff it holds for\narbitrary equal pairs, which in turn can be taken to be unital. In a similar\nsense, weak additivity is shown to imply strong additivity for any convex\nentanglement monotone. The implications are obtained by considering direct sums\nof channels (or states) for which we show how to obtain several information\ntheoretic quantities from their values on the summands. This provides a simple\nand general tool for lifting additivity results."},"authors":{"kind":"list like","value":["Motohisa Fukuda","Michael M. Wolf"],"string":"[\n \"Motohisa Fukuda\",\n \"Michael M. Wolf\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["quant-ph"],"string":"[\n \"quant-ph\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"quant-ph"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"A central question in classical and quantum information theory is, how much information can be transmitted through a given noisy channel. For classical channels the maximal asymptotically achievable rate-the capacity-was derived in the seminal work of Shannon [1] . For quantum channels, however, the matter is complicated by the existence of entanglement and the possibility of exploiting it in the encoding to protect information against decoherence. If one excludes this possibility, a capacity formula for the transmission of classical information through quantum channels was proven by Holevo [2] and Schumacher and Westmoreland [3] (HSW). Since then, considerable effort was devoted to the question whether (or in which cases) entangled inputs can lead to rates beyond the HSW capacity. This issue-the additivity problem for the HSW capacity-is still undecided, although for several classes of channels additivity has been shown to be true, i.e., entanglement does not seem to help in any case (see, e.g., [4, 5, 6, 7] and references therein). Instead, other additivity problems appeared which are similar in spirit but concern very different quantities like the minimal output entropy and the entanglement of formation, an entanglement measure for bipartite states for which in addition strong super-additivity has been conjectured.\n\nA major conceptional insight was then gained in [8, 9, 10, 11] where it was shown that all these additivity problems are globally equivalent in the sense that if additivity holds for one of these quantities in general, then it does so for all of them. Here 'in general' means that it has to be true for arbitrary pairs of channels (or states), a condition we will call strong additivity.\n\nIn this work we present a further conceptional simplification of these and related additivity problems. We show that strong additivity is implied by weak additivity, meaning additivity for arbitrary pairs of equal channels or states. Moreover, based on [12] we argue that it suffices to consider pairs of identical unital channels only. This observation may be a small step on a notorious path but it might guide future research as it for instance underlines recent attempts to understand the asymptotic structure of tensor powers of unital channels [13] . Moreover, one may think of other additivity questions than the ones stated above for which our techniques could be of use. In particular, we think of regularized quanti-ties (like quantum capacities (cf. [14, 15] ) or certain entanglement measures) for which weak additivity holds by definition.\n\nOur main tool is the use of direct sums of channels or states. For the latter case similar constructions appeared in [16, 17, 18] . We begin with a discussion of direct sum channels. This will contain more than what is needed for the subsequent additivity results as we think that these tools might be of independent interest.\n\nWe consider direct sums of channels, i.e., completely positive and trace preserving maps of the form T = ⊕ i T i , where each T i is a channel in its own right. Our aim in this section is to express information theoretic functionals of T in terms of their values for the T i 's. The definition of the quantities appearing in the following proposition will be given in the proof. Proposition 1 (Direct sums) Consider a direct sum T = ⊕ n i=1 T i , n ∈ N of arbitrary finite dimensional channels. Then 1. Minimal output α-Renyi entropy (α ≥ 1):\n\nS min,α T = min i S min,α (T i ) , ( 1\n\n)\n\n2. Coherent information:\n\nJ T = max i J(T i ) , ( 2\n\n)\n\n3. Mutual information:\n\nI T = max {λi} S({λ i }) + i λ i I(T i ), ( 3\n\n) = log i 2 I(Ti) , ( 4\n\n)\n\nwhere {λ i } is a probability distribution and S({λ i }) its entropy.\n\nχ T = max {λi} S({λ i }) + i λ i χ(T i ) ( 5\n\n) = log i 2 χ(Ti) . ( 6\n\n)\n\n2 Remark: Let us briefly comment on the interpretation of the above formulas. Concerning the HSW capacity, classical information can either be sent through the channels T i or it can be encoded in the choice of blocks i = 1, . . . , n. Eq.( 5 ) shows exactly the competition between these two ways of communicating classical information. For the quantum mutual information, which gives the entanglement assisted capacity [19], we obtain the same interpretation (note that the '2' comes from the fact that we take log in base 2). The coherent information is related (via regularization) to the quantum capacity [20] . In this case encoding information in the choice of blocks is not possible-this would be purely classical as all the coherences get lost. Similarly, for the minimal output entropies the minimum is obtained by putting all the weight into the least noisy channel.\n\nProof. 1. The α-Renyi entropy is defined as\n\nS α (ρ) = 1 1 -α log tr[ρ α ] = α 1 -α log ρ α , ( 7\n\n)\n\nfor 0 ≤ α ≤ ∞. Here • p is the Schatten p-norm.\n\nWhen α = 1 the functional is defined by its limit which is the minimal output entropy S min (T ) = inf ρ S(T (ρ)) with S(ρ) = -trρ log ρ the von Neumann entropy. Let us consider this case first. As the direct sum ⊕ i T i erases the off-diagonal blocks so that all possible outputs can be obtained upon block-diagonal inputs we can restrict to ρ = ⊕ i ρi . Here ρi is not necessarily normalized so that the weights tr[ρ i ] =: λ i form a probability distribution. Writing ρ i := ρi /λ i and using the concavity of von Neumann entropy we get\n\nS ⊕ i T i (λ i ρ i ) ≥ i λ i S(T i (ρ i )). ( 8\n\n)\n\nThis leads to Eq.( 1 ) when α = 1. For α > 1 the minimization of S α (T (ρ)) amounts to a maximization of T (ρ) α and the result follows from convexity of • α in a similar way.\n\n2. The coherent information is defined as\n\nJ(T ) = sup ρ S T (ρ) -S T ⊗ id(Ψ) , ( 9\n\n)\n\nwhere Ψ is a purification of ρ such that ρ = tr B Ψ. Since T and T ⊗ id erase the off-diagonal blocks we can replace ρ and Ψ by their diagonal blocks: 2 ) in a similar way as above.\n\n⊕ i λ i ρ i and ⊕ i λ i Ψ i . Here, Ψ i is an extension of ρ i . Since the conditional entropy S(ρ AB )-S(ρ A ) is concave in ρ AB [21] considering a convex decomposition of each Ψ i into pure states shows Eq.(\n\n3. The mutual information defined as\n\nI(T ) = sup ρ S(ρ) + S T (ρ) -S T ⊗ id(Ψ) ( 10\n\n)\n\nis concave in ρ so the maximum will be achieved by a block\n\ndiagonal ρ = ⊕ i λ i ρ i for T = ⊕ i T i . To see this, let V = ⊕ j exp{2πij/n}I j and average V k ρV * k over k = 1, . . . , n. Take a purification Ψ of ρ = ⊕ i λ i ρ i , and\n\nthen replace Ψ by its diagonal blocks: ⊕ i λ i Ψ i as before. However, each Ψ i is a purification of ρ i in this case. Indeed, suppose ρ i = j p ij |ij ij|, where {|ij } j is an orthonormal basis in the ith subspace. Then, Ψ is\n\nijkl λ i λ k p ij p kl |ij kl| ⊗ |ij kl|,\n\nand its ith diagonal block λ i Ψ i is\n\nλ i jl √ p ij p il |ij il| ⊗ |ij il| = λ i |Ψ i Ψ i |. Here, |Ψ i = j √ p ij |ij ⊗ |ij .\n\nExploiting this together with S(λρ) = λ(S(ρ) -log λ) then gives Eq.(3). Eq.(4) follows then from determining the optimal λ i via Lagrange multipliers in the following way. The maximization problem of\n\nS({λ i }) + λ i c i for a probability distribution {λ i } amounts then to maximizing S({λ i }) + i λ i c i + Λ λ i -1 , ( 11\n\n)\n\nwhere Λ is the Lagrange multiplier. Taking partial derivatives we obtain for extremal { λi }:\n\n-log λi - 1 ln 2 + c i + Λ = 0 ∀i ( 12\n\n) i λi -1 = 0. ( 13\n\n)\n\nHence (12) shows c i -log λi is a constant, say, C for ∀i, and by (13) we get C = log i 2 ci . Therefore\n\nS({ λi }) + i λi c i = i λi C = log i 2 ci . ( 14\n\n)\n\nAs this is lower bounded by min i [I(T i )] it must be the maximum.\n\n4. The HSW capacity χ is given by\n\nχ(T ) = sup ρ S(T (ρ)) -H T (ρ), ( 15\n\n) where H T (ρ) = inf P p k ρ k =ρ k p k S(T (ρ k )). ( 16\n\n)\n\nHere, H T (ρ) is the convex closure of the output entropy; {p k } is a probability distribution and ρ k are density matrices. The r.h.s. of Eq.( 15 ) for a fixed average input state ρ is a constraint HSW capacity which we will denote by χ(T, ρ). Since the inputs can again be assumed to be block-diagonal we have:\n\nχ (⊕ i T i , ⊕ i λ i ρ i ) = S(⊕ i λ i T i (ρ i )) - i λ i H Ti (ρ i ) = S({λ i }) + i λ i χ(T i , ρ i ). ( 17\n\n)\n\nThe first equality is explained by the fact that since the von Neumann entropy is concave there is an optimal decomposition of ⊕ i λ i ρ i for which each state has its support in one of the diagonal blocks. The second equality comes from\n\nS(λ i ρ i ) = λ i (S(ρ i ) -log λ i ).\n\nTaking the supremum over all states {ρ i } then leads to Eq.( 5 ). Again, Eq.(6) is obtained by using Lagrange multipliers as above. In fact, for unital channels (6) has been obtained in [30] .\n\nLet us now turn to the additivity conjectures and exploit Prop.1 in order to show that in several cases weak additivity (for equal channels or states) implies strong additivity (i.e., for different ones). Proposition 2 (Reduction for channels) The following (in-) equalities hold for arbitrary pairs of different channels T 1 and T 2 iff they hold for arbitrary equal pairs T 1 = T 2 .\n\n1. S min,α (T 1 ⊗ T 2 ) = S min,α (T 1 ) + S min,α (T 2 ), for any α ≥ 1. 2. χ(T 1 ⊗ T 2 ) = χ(T 1 ) + χ(T 2 ). 3. H T1⊗T2 (ρ) ≥ H T1 (ρ 1 ) + H T2 (ρ 2 ) for all states ρ with respective subsystems ρ 1 , ρ 2 . 4. H T1⊗T2 (ρ 1 ⊗ ρ 2 ) = H T1 (ρ 1 )+ H T2 (ρ 2 ) for all product states ρ 1 ⊗ ρ 2 .\n\nRemark: The conjectured equality in 1. is the additivity of the minimal output entropy when α = 1 [22], and it becomes the multiplicativity of maximal output p-norms for p = α > 1. This was conjectured to be true for all α ∈ [1, +∞] before a counterexample was found [23] ruling out all values α > 4.79. The equation in 2. is the conjectured additivity of the HSW capacity, which gives the classical capacity as long as entangled states are not allowed to be used in the encoding [2, 3] . The additivity would show that the HSW capacity itself is the unconstrained classical capacity of quantum channels. The conjectures 3. and 4. are called strong superadditivity and additivity of the convex closure of the output entropy. When T 1 , T 2 are partial traces they become strong superadditivity and additivity of entanglement of formation, respectively, which we discuss in greater detail below.\n\nWe note that Prop.2 remains valid in the case where 'arbitrary channels' refers to a restricted set of channels which is closed under direct sums and tensor products.\n\nProof. 1. Let σ 1 and σ 2 be optimal output states for T 1 and T 2 respectively. Then, form the following two channels:\n\nT ′ 1 (ρ) = T 1 (ρ) ⊗ σ 2 , T ′ 2 (ρ) = σ 1 ⊗ T 2 (ρ). ( 18\n\n)\n\nIt is not difficult to see that T 1 ⊗ T 2 and T ′ 1 ⊗ T ′ 2 share the additivity property. Hence we can assume that T 1 and T 2 have the same optimal output: S min,α (T 1 ) = S min,α (T 2 ). If we apply first weak additivity and then Prop. 1.1. we obtain:\n\nS min,α (((T 1 ⊕ T 2 ) ⊗ (T 1 ⊕ T 2 ))) (19) = 2S min,α (T 1 ⊕ T 2 ) = S min,α (T 1 ) + S min,α (T 2 ). (20)\n\nOn the other hand, if we first apply Prop. 1.1. and then weak additivity, we obtain that (19) and thus (20) is upper bounded by S min,α (T 1 ⊗ T 2 ). The converse inequality is trivial.\n\n2. Consider χ T 1 ⊕ T 2 ⊗2 = 2χ T 1 ⊕ T 2 = 2 log 2 χ(T1) + 2 χ(T2)\n\n= log 2 2χ(T1) + 2 2χ(T2) + 2 χ(T1)+χ(T2)+1 .\n\nThis follows from first applying weak additivity and then the proposition 1.4. On the other hand, applying them in reverse order we have χ T 1 ⊕ T 2 ⊗2 = log 2 χ(T1⊗T1) + 2 χ(T2⊗T2) + 2 • 2 χ(T1⊗T2) = log 2 2χ(T1) + 2 2χ(T2) + 2 χ(T1⊗T2)+1 .\n\nTogether they prove the claimed equality.\n\nFor 3. we obtain by weak superadditivity,\n\nH T1⊗T2 (ρ) = H (T1⊕T2)⊗(T1⊕T2) (0 ⊕ ρ ⊕ 0 ⊕ 0) ≥ H T1⊕T2 (ρ 1 ⊕ 0) + H T1⊕T2 (0 ⊕ ρ 2 ) = H T1 (ρ 1 ) + H T2 (ρ 2 ). ( 21\n\n)\n\nHere, ρ 1 , ρ 2 are reduced states of ρ. This proves 3. and the statement 4. follows in a similar way when replacing ρ by a product state.\n\nProposition 3 (Unital channels) Proving one of the conjectures in proposition 2 for all pairs of identical unital channels would show the conjecture is true for arbitrary channels.\n\nProof. In [12] a unital channel T is constructed for a given channel T so that these two channels T and T share the following additivity properties: additivity of minimal output α-Renyi entropy, and strong superadditivity and additivity of the convex closure of the output entropy. Hence these conjectures can be restricted to products T1 ⊗ T2 for all channels T 1 , T 2 .\n\nAs for the HSW, we have the same reduction but for a different reason (See the remark below). Finally, for the above two unital channels T1 , T2 we can construct the direct sum T1 ⊕ T2 which is again a unital channel. Then the result follows from the proof of proposition 2.\n\nRemark: We explain local relation between minimal output entropy and the HSW capacity, which was implicitly written but not clear in [12] . Since the unital extension T sort of mixes up outputs of T we have the following formula.\n\nχ( T1 ⊗ T2 ) = log d 1 d 2 -S min ( T1 ⊗ T2 ), ( 22\n\n)\n\nwhere d 1 , d 2 are the dimensions of the output spaces of T1 and T2 respectively. Hence the additivity of HSW capacity is equivalent to the additivity of the minimal output entropy for products of those extensions T1 ⊗ T2 by Eq.( 22 ). Hence the additivity conjecture of the HSW capacity can also be restricted to products T1 ⊗ T2 for all channels T 1 , T 2 by using global equivalence [8, 9, 10, 11] .\n\nFinally, we will discuss additivity issues for entanglement measures. The one already mentioned is the entanglement of formation which was introduced in [24] . Since then the following conjectures have been considered:\n\nE F (ρ) ≥ E F (ρ 1 ) + E F (ρ 2 ) (23) E F (ρ 1 ⊗ ρ 2 ) = E F (ρ 1 ) + E F (ρ 2 ). ( 24\n\n)\n\n4 In fact, both are again globally equivalent to the additivity of the HSW capacity and the minimal output entropy. Moreover, additivity would imply that E F equals an important operationally defined entanglement measure, the entanglement cost\n\nE c , since E c (ρ) = lim n→∞ E F (ρ ⊗n ) [25].\n\nThe entanglement of formation E F (ρ) is the convex closure of output entropy H T (ρ) when T is a partial trace.\n\nFollowing a similar strategy as above we will now show that strong additivity in the sense of Eq.(24) is again implied by weak additivity (i.e., Eq.(24) with ρ 1 = ρ 2 ). In fact, this will not only hold for E F but for any convex entanglement monotone [24, 26, 27] . The main reason behind is that every such functional satisfies [28]:\n\nf (⊕ i λ i ρ i ) = i λ i f (ρ i ), ( 25\n\n)\n\nwhere {λ i } is a probability distribution and ρ i are states as before.\n\nProposition 4 (Convex entanglement monotones) [29] Suppose f is a convex entanglement monotone which is weakly additive, i.e.,\n\nf (ρ 1 ⊗ ρ 2 ) = f (ρ 1 ) + f (ρ 2 ) for all ρ 1 = ρ 2 .\n\nThen f is strongly additive in the sense that this holds also for all ρ 1 = ρ 2 .\n\nProof. Let ρ = 1 2 (ρ 1 ⊕ ρ 2 ). Then f (ρ ⊗ ρ) = 2f (ρ) = f (ρ 1 ) + f (ρ 2 ). (26)\n\nHere, we applied the weak additivity and then (25). Applying them in reverse order we get\n\nf (ρ ⊗ ρ) = 1 4 2 i,j=1 f (ρ i ⊗ ρ j ) = 1 2 (f (ρ 1 ) + f (ρ 2 ) + f (ρ 1 ⊗ ρ 2 )). ( 27\n\n)\n\nUsing similar ideas, it has recently been shown that for regularized entanglement measures like E c or the asymptotic relative entropy of entanglement, monotonicity (i.e., essentially Eq.(25)) and strong additivity are equivalent [18] . Acknowledgement M.F. would like to thank his supervisor Y.M.Suhov for constant encouragement and numerous discussions. M.W. thanks K.G. Vollbrecht for discussions and J.I. Cirac for support. Both authors thank M. B. Ruskai for bringing [30] to their attention.\n\n[1] C. E. Shannon, \" A Mathematical Theory of Communication\", Bell System Technical Journal, 27 379E23 and 623E56 (1948). [2] A. S. Holevo \"The capacity of the quantum channel with general signal states\", IEEE Trans. Info. Theory, 44, 269-273, (1998). [3] B. Schumacher and M. D. Westmoreland, \"Sending classical information via noisy quantum channels\", Phys. Rev. A, 56, 131-138, (1997). [4] M.M. Wolf, J. Eisert, \"Classical information capacity of a class of quantum channels \", New J. Phys. 7, 93 (2005); quant-ph/0412133 [5] C. King, \"The capacity of the quantum depolarizing channel \", IEEE Trans. Inf. Theo. 49, 221 (2003); quant-ph/0204172 [6] C. King, \"Additivity for unital qubit channels\", J. Math. Phys. 4641-4653, (2002) . [7] P.W. Shor, \"Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels\", J. Math. Phys. 4334 (2002) ; quant-ph/0201149 [8] P.W. Shor, \"Equivalence of Additivity Questions in Quantum Information Theory\", Comm. Math. Phys., 246, Issue 3, 453-472 (2004); quant-ph/0305035. [9] A.A. Pomeransky, \"Strong superadditivity of the entanglement of formation follows from its additivity\", Phys. Rev. A 68, 032317 (2003); quant-ph/0305056. [10] K.M.R. Audenaert and S.L. Braunstein, \"On Strong Subadditivity of the Entanglement of Formation\", Comm. Math. Phys. No 3, 443-452, (2004) ; quant-ph/0303045. [11] K. Matsumoto, T. Shimono, and A. Winter, \"Remarks on additivity of the Holevo channel capacity and of the entanglement of formation \", Commun. Math. Phys. 246, 437 (2004); quant-ph/0206148. [12] M. Fukuda, \"Simplification of additivity conjecture in quantum information theory\", quant-ph/0608010. [13] J.A. Smolin, F. Verstraete, A. Winter, \"Entanglement of assistance and multipartite state distillation\", Phys. Rev. A 72, 052317 (2005); quant-ph/0505038. [14] M. Wolf, D. Perez-Garcia, \"Quantum Capacities of Channels with small Environment \", Phys. Rev. A 75, 012303 (2007); quant-ph/0607070. [15] G. Smith, J.A. Smolin, A. Winter, \"The quantum capacity with symmetric side channels \", quant-ph/0607039. [16] P.W. Shor, J.A. Smolin, B.M. Terhal, \"Nonadditivity of Bipartite Distillable Entanglement follows from Conjecture on Bound Entangled Werner States \", Phys. Rev. Lett. 86, 2681 (2001); quant-ph/0010054. [17] K.G.H. Vollbrecht, R.F. Werner, M.M. Wolf, \"On the irreversibility of entanglement distillation \", Phys. Rev. A 69, 062304 (2004); quant-ph/0301072. [18] F.G.S.L. Brandao, M. Horodecki, M.B. Plenio, S. Virmani, \"Remarks on the equivalence of full additivity and monotonicity for the entanglement cost\", quant-ph/0702136. [19] C. H. Bennett, P.W. Shor, J. A. Smolin, and A. V.\n\nThapliyal,\"Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem\", IEEE Trans. Inf. Theory 48, 2637 (2002); quant-ph/0106052. [20] P.W. Shor, \"The quantum channel capacity and coherent information\", lecture notes, MSRI Workshop on Quantum Computation (2002); I. Devetak, IEEE Trans. Inf. Th. 51, 44 (2005); S. Lloyd, Phys. Rev. A 55, 1613 (1997). [21] M.B. Ruskai, \"Inequalities for Quantum Entropy: A Review with Conditions for Equality \", J. Math. Phys. 43, 4358 (2002); erratum 46, 019901 (2005); quant-ph/0205064. [22] C. King and M. B. Ruskai, \"Minimal entropy of states emerging from noisy quantum channels\", IEEE Trans. Info. Theory, 47, 192-209 (2001); quant-ph/9911079. [23] R.F. Werner and A.S. Holevo, \"Counterexample to an ad-5 ditivity conjecture for output purity of quantum channels\", quant-ph/0203003. [24] C. H. Bennet, D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, \"Mixed-state entanglement and quantum error correction\", Phys. Rev. A, 54, 3824-3851, (1996) ; quant-ph/9604024. [25] P.M. Hayden, M. Horodecki, B.M. Terhal, \"The asymptotic entanglement cost of preparing a quantum state \", J. Phys. A: Math. Gen. 34, 6891 (2001); quant-ph/0008134. [26] G. Vidal, J. Mod. Opt., 47, 355 (2000); quant-ph/9807077. [27] V. Vedral and M. B. Plenio, Phys. Rev. A, 57, 1619 (1998); quant-ph/9707035. [28] M. Horodecki, \"Simplifying monotonicity conditions for entanglement measures\", Open Syst. Inf. Dyn., 12, 231 (2005), quant-ph/0412210. [29] This result is implicit in the recent work [18] . However, it has already been proven and been communicated by one of us at the ERATO conference, Kyoto 2003 (unpublished). [30] E. Stormer, \"A reduction theorem for capacity of positive maps\", quant-ph/0510040."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"A central question in classical and quantum information theory is, how much information can be transmitted through a given noisy channel. For classical channels the maximal asymptotically achievable rate-the capacity-was derived in the seminal work of Shannon [1] . For quantum channels, however, the matter is complicated by the existence of entanglement and the possibility of exploiting it in the encoding to protect information against decoherence. If one excludes this possibility, a capacity formula for the transmission of classical information through quantum channels was proven by Holevo [2] and Schumacher and Westmoreland [3] (HSW). Since then, considerable effort was devoted to the question whether (or in which cases) entangled inputs can lead to rates beyond the HSW capacity. This issue-the additivity problem for the HSW capacity-is still undecided, although for several classes of channels additivity has been shown to be true, i.e., entanglement does not seem to help in any case (see, e.g., [4, 5, 6, 7] and references therein). Instead, other additivity problems appeared which are similar in spirit but concern very different quantities like the minimal output entropy and the entanglement of formation, an entanglement measure for bipartite states for which in addition strong super-additivity has been conjectured.\n\nA major conceptional insight was then gained in [8, 9, 10, 11] where it was shown that all these additivity problems are globally equivalent in the sense that if additivity holds for one of these quantities in general, then it does so for all of them. Here 'in general' means that it has to be true for arbitrary pairs of channels (or states), a condition we will call strong additivity.\n\nIn this work we present a further conceptional simplification of these and related additivity problems. We show that strong additivity is implied by weak additivity, meaning additivity for arbitrary pairs of equal channels or states. Moreover, based on [12] we argue that it suffices to consider pairs of identical unital channels only. This observation may be a small step on a notorious path but it might guide future research as it for instance underlines recent attempts to understand the asymptotic structure of tensor powers of unital channels [13] . Moreover, one may think of other additivity questions than the ones stated above for which our techniques could be of use. In particular, we think of regularized quanti-ties (like quantum capacities (cf. [14, 15] ) or certain entanglement measures) for which weak additivity holds by definition.\n\nOur main tool is the use of direct sums of channels or states. For the latter case similar constructions appeared in [16, 17, 18] . We begin with a discussion of direct sum channels. This will contain more than what is needed for the subsequent additivity results as we think that these tools might be of independent interest."},{"section_type":"OTHER","section_title":"II. DIRECT SUMS OF QUANTUM CHANNELS","text":"We consider direct sums of channels, i.e., completely positive and trace preserving maps of the form T = ⊕ i T i , where each T i is a channel in its own right. Our aim in this section is to express information theoretic functionals of T in terms of their values for the T i 's. The definition of the quantities appearing in the following proposition will be given in the proof. Proposition 1 (Direct sums) Consider a direct sum T = ⊕ n i=1 T i , n ∈ N of arbitrary finite dimensional channels. Then 1. Minimal output α-Renyi entropy (α ≥ 1):\n\nS min,α T = min i S min,α (T i ) , ( 1\n\n)\n\n2. Coherent information:\n\nJ T = max i J(T i ) , ( 2\n\n)\n\n3. Mutual information:\n\nI T = max {λi} S({λ i }) + i λ i I(T i ), ( 3\n\n) = log i 2 I(Ti) , ( 4\n\n)\n\nwhere {λ i } is a probability distribution and S({λ i }) its entropy."},{"section_type":"OTHER","section_title":"HSW capacity:","text":"χ T = max {λi} S({λ i }) + i λ i χ(T i ) ( 5\n\n) = log i 2 χ(Ti) . ( 6\n\n)\n\n2 Remark: Let us briefly comment on the interpretation of the above formulas. Concerning the HSW capacity, classical information can either be sent through the channels T i or it can be encoded in the choice of blocks i = 1, . . . , n. Eq.( 5 ) shows exactly the competition between these two ways of communicating classical information. For the quantum mutual information, which gives the entanglement assisted capacity [19], we obtain the same interpretation (note that the '2' comes from the fact that we take log in base 2). The coherent information is related (via regularization) to the quantum capacity [20] . In this case encoding information in the choice of blocks is not possible-this would be purely classical as all the coherences get lost. Similarly, for the minimal output entropies the minimum is obtained by putting all the weight into the least noisy channel.\n\nProof. 1. The α-Renyi entropy is defined as\n\nS α (ρ) = 1 1 -α log tr[ρ α ] = α 1 -α log ρ α , ( 7\n\n)\n\nfor 0 ≤ α ≤ ∞. Here • p is the Schatten p-norm.\n\nWhen α = 1 the functional is defined by its limit which is the minimal output entropy S min (T ) = inf ρ S(T (ρ)) with S(ρ) = -trρ log ρ the von Neumann entropy. Let us consider this case first. As the direct sum ⊕ i T i erases the off-diagonal blocks so that all possible outputs can be obtained upon block-diagonal inputs we can restrict to ρ = ⊕ i ρi . Here ρi is not necessarily normalized so that the weights tr[ρ i ] =: λ i form a probability distribution. Writing ρ i := ρi /λ i and using the concavity of von Neumann entropy we get\n\nS ⊕ i T i (λ i ρ i ) ≥ i λ i S(T i (ρ i )). ( 8\n\n)\n\nThis leads to Eq.( 1 ) when α = 1. For α > 1 the minimization of S α (T (ρ)) amounts to a maximization of T (ρ) α and the result follows from convexity of • α in a similar way.\n\n2. The coherent information is defined as\n\nJ(T ) = sup ρ S T (ρ) -S T ⊗ id(Ψ) , ( 9\n\n)\n\nwhere Ψ is a purification of ρ such that ρ = tr B Ψ. Since T and T ⊗ id erase the off-diagonal blocks we can replace ρ and Ψ by their diagonal blocks: 2 ) in a similar way as above.\n\n⊕ i λ i ρ i and ⊕ i λ i Ψ i . Here, Ψ i is an extension of ρ i . Since the conditional entropy S(ρ AB )-S(ρ A ) is concave in ρ AB [21] considering a convex decomposition of each Ψ i into pure states shows Eq.(\n\n3. The mutual information defined as\n\nI(T ) = sup ρ S(ρ) + S T (ρ) -S T ⊗ id(Ψ) ( 10\n\n)\n\nis concave in ρ so the maximum will be achieved by a block\n\ndiagonal ρ = ⊕ i λ i ρ i for T = ⊕ i T i . To see this, let V = ⊕ j exp{2πij/n}I j and average V k ρV * k over k = 1, . . . , n. Take a purification Ψ of ρ = ⊕ i λ i ρ i , and\n\nthen replace Ψ by its diagonal blocks: ⊕ i λ i Ψ i as before. However, each Ψ i is a purification of ρ i in this case. Indeed, suppose ρ i = j p ij |ij ij|, where {|ij } j is an orthonormal basis in the ith subspace. Then, Ψ is\n\nijkl λ i λ k p ij p kl |ij kl| ⊗ |ij kl|,\n\nand its ith diagonal block λ i Ψ i is\n\nλ i jl √ p ij p il |ij il| ⊗ |ij il| = λ i |Ψ i Ψ i |. Here, |Ψ i = j √ p ij |ij ⊗ |ij .\n\nExploiting this together with S(λρ) = λ(S(ρ) -log λ) then gives Eq.(3). Eq.(4) follows then from determining the optimal λ i via Lagrange multipliers in the following way. The maximization problem of\n\nS({λ i }) + λ i c i for a probability distribution {λ i } amounts then to maximizing S({λ i }) + i λ i c i + Λ λ i -1 , ( 11\n\n)\n\nwhere Λ is the Lagrange multiplier. Taking partial derivatives we obtain for extremal { λi }:\n\n-log λi - 1 ln 2 + c i + Λ = 0 ∀i ( 12\n\n) i λi -1 = 0. ( 13\n\n)\n\nHence (12) shows c i -log λi is a constant, say, C for ∀i, and by (13) we get C = log i 2 ci . Therefore\n\nS({ λi }) + i λi c i = i λi C = log i 2 ci . ( 14\n\n)\n\nAs this is lower bounded by min i [I(T i )] it must be the maximum.\n\n4. The HSW capacity χ is given by\n\nχ(T ) = sup ρ S(T (ρ)) -H T (ρ), ( 15\n\n) where H T (ρ) = inf P p k ρ k =ρ k p k S(T (ρ k )). ( 16\n\n)\n\nHere, H T (ρ) is the convex closure of the output entropy; {p k } is a probability distribution and ρ k are density matrices. The r.h.s. of Eq.( 15 ) for a fixed average input state ρ is a constraint HSW capacity which we will denote by χ(T, ρ). Since the inputs can again be assumed to be block-diagonal we have:\n\nχ (⊕ i T i , ⊕ i λ i ρ i ) = S(⊕ i λ i T i (ρ i )) - i λ i H Ti (ρ i ) = S({λ i }) + i λ i χ(T i , ρ i ). ( 17\n\n)\n\nThe first equality is explained by the fact that since the von Neumann entropy is concave there is an optimal decomposition of ⊕ i λ i ρ i for which each state has its support in one of the diagonal blocks. The second equality comes from\n\nS(λ i ρ i ) = λ i (S(ρ i ) -log λ i ).\n\nTaking the supremum over all states {ρ i } then leads to Eq.( 5 ). Again, Eq.(6) is obtained by using Lagrange multipliers as above. In fact, for unital channels (6) has been obtained in [30] ."},{"section_type":"OTHER","section_title":"III. SIMPLIFYING ADDITIVITY PROBLEMS","text":"Let us now turn to the additivity conjectures and exploit Prop.1 in order to show that in several cases weak additivity (for equal channels or states) implies strong additivity (i.e., for different ones). Proposition 2 (Reduction for channels) The following (in-) equalities hold for arbitrary pairs of different channels T 1 and T 2 iff they hold for arbitrary equal pairs T 1 = T 2 .\n\n1. S min,α (T 1 ⊗ T 2 ) = S min,α (T 1 ) + S min,α (T 2 ), for any α ≥ 1. 2. χ(T 1 ⊗ T 2 ) = χ(T 1 ) + χ(T 2 ). 3. H T1⊗T2 (ρ) ≥ H T1 (ρ 1 ) + H T2 (ρ 2 ) for all states ρ with respective subsystems ρ 1 , ρ 2 . 4. H T1⊗T2 (ρ 1 ⊗ ρ 2 ) = H T1 (ρ 1 )+ H T2 (ρ 2 ) for all product states ρ 1 ⊗ ρ 2 .\n\nRemark: The conjectured equality in 1. is the additivity of the minimal output entropy when α = 1 [22], and it becomes the multiplicativity of maximal output p-norms for p = α > 1. This was conjectured to be true for all α ∈ [1, +∞] before a counterexample was found [23] ruling out all values α > 4.79. The equation in 2. is the conjectured additivity of the HSW capacity, which gives the classical capacity as long as entangled states are not allowed to be used in the encoding [2, 3] . The additivity would show that the HSW capacity itself is the unconstrained classical capacity of quantum channels. The conjectures 3. and 4. are called strong superadditivity and additivity of the convex closure of the output entropy. When T 1 , T 2 are partial traces they become strong superadditivity and additivity of entanglement of formation, respectively, which we discuss in greater detail below.\n\nWe note that Prop.2 remains valid in the case where 'arbitrary channels' refers to a restricted set of channels which is closed under direct sums and tensor products.\n\nProof. 1. Let σ 1 and σ 2 be optimal output states for T 1 and T 2 respectively. Then, form the following two channels:\n\nT ′ 1 (ρ) = T 1 (ρ) ⊗ σ 2 , T ′ 2 (ρ) = σ 1 ⊗ T 2 (ρ). ( 18\n\n)\n\nIt is not difficult to see that T 1 ⊗ T 2 and T ′ 1 ⊗ T ′ 2 share the additivity property. Hence we can assume that T 1 and T 2 have the same optimal output: S min,α (T 1 ) = S min,α (T 2 ). If we apply first weak additivity and then Prop. 1.1. we obtain:\n\nS min,α (((T 1 ⊕ T 2 ) ⊗ (T 1 ⊕ T 2 ))) (19) = 2S min,α (T 1 ⊕ T 2 ) = S min,α (T 1 ) + S min,α (T 2 ). (20)\n\nOn the other hand, if we first apply Prop. 1.1. and then weak additivity, we obtain that (19) and thus (20) is upper bounded by S min,α (T 1 ⊗ T 2 ). The converse inequality is trivial.\n\n2. Consider χ T 1 ⊕ T 2 ⊗2 = 2χ T 1 ⊕ T 2 = 2 log 2 χ(T1) + 2 χ(T2)\n\n= log 2 2χ(T1) + 2 2χ(T2) + 2 χ(T1)+χ(T2)+1 .\n\nThis follows from first applying weak additivity and then the proposition 1.4. On the other hand, applying them in reverse order we have χ T 1 ⊕ T 2 ⊗2 = log 2 χ(T1⊗T1) + 2 χ(T2⊗T2) + 2 • 2 χ(T1⊗T2) = log 2 2χ(T1) + 2 2χ(T2) + 2 χ(T1⊗T2)+1 .\n\nTogether they prove the claimed equality.\n\nFor 3. we obtain by weak superadditivity,\n\nH T1⊗T2 (ρ) = H (T1⊕T2)⊗(T1⊕T2) (0 ⊕ ρ ⊕ 0 ⊕ 0) ≥ H T1⊕T2 (ρ 1 ⊕ 0) + H T1⊕T2 (0 ⊕ ρ 2 ) = H T1 (ρ 1 ) + H T2 (ρ 2 ). ( 21\n\n)\n\nHere, ρ 1 , ρ 2 are reduced states of ρ. This proves 3. and the statement 4. follows in a similar way when replacing ρ by a product state.\n\nProposition 3 (Unital channels) Proving one of the conjectures in proposition 2 for all pairs of identical unital channels would show the conjecture is true for arbitrary channels.\n\nProof. In [12] a unital channel T is constructed for a given channel T so that these two channels T and T share the following additivity properties: additivity of minimal output α-Renyi entropy, and strong superadditivity and additivity of the convex closure of the output entropy. Hence these conjectures can be restricted to products T1 ⊗ T2 for all channels T 1 , T 2 .\n\nAs for the HSW, we have the same reduction but for a different reason (See the remark below). Finally, for the above two unital channels T1 , T2 we can construct the direct sum T1 ⊕ T2 which is again a unital channel. Then the result follows from the proof of proposition 2.\n\nRemark: We explain local relation between minimal output entropy and the HSW capacity, which was implicitly written but not clear in [12] . Since the unital extension T sort of mixes up outputs of T we have the following formula.\n\nχ( T1 ⊗ T2 ) = log d 1 d 2 -S min ( T1 ⊗ T2 ), ( 22\n\n)\n\nwhere d 1 , d 2 are the dimensions of the output spaces of T1 and T2 respectively. Hence the additivity of HSW capacity is equivalent to the additivity of the minimal output entropy for products of those extensions T1 ⊗ T2 by Eq.( 22 ). Hence the additivity conjecture of the HSW capacity can also be restricted to products T1 ⊗ T2 for all channels T 1 , T 2 by using global equivalence [8, 9, 10, 11] .\n\nFinally, we will discuss additivity issues for entanglement measures. The one already mentioned is the entanglement of formation which was introduced in [24] . Since then the following conjectures have been considered:\n\nE F (ρ) ≥ E F (ρ 1 ) + E F (ρ 2 ) (23) E F (ρ 1 ⊗ ρ 2 ) = E F (ρ 1 ) + E F (ρ 2 ). ( 24\n\n)\n\n4 In fact, both are again globally equivalent to the additivity of the HSW capacity and the minimal output entropy. Moreover, additivity would imply that E F equals an important operationally defined entanglement measure, the entanglement cost\n\nE c , since E c (ρ) = lim n→∞ E F (ρ ⊗n ) [25].\n\nThe entanglement of formation E F (ρ) is the convex closure of output entropy H T (ρ) when T is a partial trace.\n\nFollowing a similar strategy as above we will now show that strong additivity in the sense of Eq.(24) is again implied by weak additivity (i.e., Eq.(24) with ρ 1 = ρ 2 ). In fact, this will not only hold for E F but for any convex entanglement monotone [24, 26, 27] . The main reason behind is that every such functional satisfies [28]:\n\nf (⊕ i λ i ρ i ) = i λ i f (ρ i ), ( 25\n\n)\n\nwhere {λ i } is a probability distribution and ρ i are states as before.\n\nProposition 4 (Convex entanglement monotones) [29] Suppose f is a convex entanglement monotone which is weakly additive, i.e.,\n\nf (ρ 1 ⊗ ρ 2 ) = f (ρ 1 ) + f (ρ 2 ) for all ρ 1 = ρ 2 .\n\nThen f is strongly additive in the sense that this holds also for all ρ 1 = ρ 2 .\n\nProof. Let ρ = 1 2 (ρ 1 ⊕ ρ 2 ). Then f (ρ ⊗ ρ) = 2f (ρ) = f (ρ 1 ) + f (ρ 2 ). (26)\n\nHere, we applied the weak additivity and then (25). Applying them in reverse order we get\n\nf (ρ ⊗ ρ) = 1 4 2 i,j=1 f (ρ i ⊗ ρ j ) = 1 2 (f (ρ 1 ) + f (ρ 2 ) + f (ρ 1 ⊗ ρ 2 )). ( 27\n\n)\n\nUsing similar ideas, it has recently been shown that for regularized entanglement measures like E c or the asymptotic relative entropy of entanglement, monotonicity (i.e., essentially Eq.(25)) and strong additivity are equivalent [18] . Acknowledgement M.F. would like to thank his supervisor Y.M.Suhov for constant encouragement and numerous discussions. M.W. thanks K.G. Vollbrecht for discussions and J.I. Cirac for support. Both authors thank M. B. Ruskai for bringing [30] to their attention.\n\n[1] C. E. Shannon, \" A Mathematical Theory of Communication\", Bell System Technical Journal, 27 379E23 and 623E56 (1948). [2] A. S. Holevo \"The capacity of the quantum channel with general signal states\", IEEE Trans. Info. Theory, 44, 269-273, (1998). [3] B. Schumacher and M. D. Westmoreland, \"Sending classical information via noisy quantum channels\", Phys. Rev. A, 56, 131-138, (1997). [4] M.M. Wolf, J. Eisert, \"Classical information capacity of a class of quantum channels \", New J. Phys. 7, 93 (2005); quant-ph/0412133 [5] C. King, \"The capacity of the quantum depolarizing channel \", IEEE Trans. Inf. Theo. 49, 221 (2003); quant-ph/0204172 [6] C. King, \"Additivity for unital qubit channels\", J. Math. Phys. 4641-4653, (2002) . [7] P.W. Shor, \"Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels\", J. Math. Phys. 4334 (2002) ; quant-ph/0201149 [8] P.W. Shor, \"Equivalence of Additivity Questions in Quantum Information Theory\", Comm. Math. Phys., 246, Issue 3, 453-472 (2004); quant-ph/0305035. [9] A.A. Pomeransky, \"Strong superadditivity of the entanglement of formation follows from its additivity\", Phys. Rev. A 68, 032317 (2003); quant-ph/0305056. [10] K.M.R. Audenaert and S.L. Braunstein, \"On Strong Subadditivity of the Entanglement of Formation\", Comm. Math. Phys. No 3, 443-452, (2004) ; quant-ph/0303045. [11] K. Matsumoto, T. Shimono, and A. Winter, \"Remarks on additivity of the Holevo channel capacity and of the entanglement of formation \", Commun. Math. Phys. 246, 437 (2004); quant-ph/0206148. [12] M. Fukuda, \"Simplification of additivity conjecture in quantum information theory\", quant-ph/0608010. [13] J.A. Smolin, F. Verstraete, A. Winter, \"Entanglement of assistance and multipartite state distillation\", Phys. Rev. A 72, 052317 (2005); quant-ph/0505038. [14] M. Wolf, D. Perez-Garcia, \"Quantum Capacities of Channels with small Environment \", Phys. Rev. A 75, 012303 (2007); quant-ph/0607070. [15] G. Smith, J.A. Smolin, A. Winter, \"The quantum capacity with symmetric side channels \", quant-ph/0607039. [16] P.W. Shor, J.A. Smolin, B.M. Terhal, \"Nonadditivity of Bipartite Distillable Entanglement follows from Conjecture on Bound Entangled Werner States \", Phys. Rev. Lett. 86, 2681 (2001); quant-ph/0010054. [17] K.G.H. Vollbrecht, R.F. Werner, M.M. Wolf, \"On the irreversibility of entanglement distillation \", Phys. Rev. A 69, 062304 (2004); quant-ph/0301072. [18] F.G.S.L. Brandao, M. Horodecki, M.B. Plenio, S. Virmani, \"Remarks on the equivalence of full additivity and monotonicity for the entanglement cost\", quant-ph/0702136. [19] C. H. Bennett, P.W. Shor, J. A. Smolin, and A. V.\n\nThapliyal,\"Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem\", IEEE Trans. Inf. Theory 48, 2637 (2002); quant-ph/0106052. [20] P.W. Shor, \"The quantum channel capacity and coherent information\", lecture notes, MSRI Workshop on Quantum Computation (2002); I. Devetak, IEEE Trans. Inf. Th. 51, 44 (2005); S. Lloyd, Phys. Rev. A 55, 1613 (1997). [21] M.B. Ruskai, \"Inequalities for Quantum Entropy: A Review with Conditions for Equality \", J. Math. Phys. 43, 4358 (2002); erratum 46, 019901 (2005); quant-ph/0205064. [22] C. King and M. B. Ruskai, \"Minimal entropy of states emerging from noisy quantum channels\", IEEE Trans. Info. Theory, 47, 192-209 (2001); quant-ph/9911079. [23] R.F. Werner and A.S. Holevo, \"Counterexample to an ad-5 ditivity conjecture for output purity of quantum channels\", quant-ph/0203003. [24] C. H. Bennet, D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, \"Mixed-state entanglement and quantum error correction\", Phys. Rev. A, 54, 3824-3851, (1996) ; quant-ph/9604024. [25] P.M. Hayden, M. Horodecki, B.M. Terhal, \"The asymptotic entanglement cost of preparing a quantum state \", J. Phys. A: Math. Gen. 34, 6891 (2001); quant-ph/0008134. [26] G. Vidal, J. Mod. Opt., 47, 355 (2000); quant-ph/9807077. [27] V. Vedral and M. B. Plenio, Phys. Rev. A, 57, 1619 (1998); quant-ph/9707035. [28] M. Horodecki, \"Simplifying monotonicity conditions for entanglement measures\", Open Syst. Inf. Dyn., 12, 231 (2005), quant-ph/0412210. [29] This result is implicit in the recent work [18] . However, it has already been proven and been communicated by one of us at the ERATO conference, Kyoto 2003 (unpublished). [30] E. Stormer, \"A reduction theorem for capacity of positive maps\", quant-ph/0510040."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"A central question in classical and quantum information theory is, how much information can be transmitted through a given noisy channel. For classical channels the maximal asymptotically achievable rate-the capacity-was derived in the seminal work of Shannon [1] . For quantum channels, however, the matter is complicated by the existence of entanglement and the possibility of exploiting it in the encoding to protect information against decoherence. If one excludes this possibility, a capacity formula for the transmission of classical information through quantum channels was proven by Holevo [2] and Schumacher and Westmoreland [3] (HSW). Since then, considerable effort was devoted to the question whether (or in which cases) entangled inputs can lead to rates beyond the HSW capacity. This issue-the additivity problem for the HSW capacity-is still undecided, although for several classes of channels additivity has been shown to be true, i.e., entanglement does not seem to help in any case (see, e.g., [4, 5, 6, 7] and references therein). Instead, other additivity problems appeared which are similar in spirit but concern very different quantities like the minimal output entropy and the entanglement of formation, an entanglement measure for bipartite states for which in addition strong super-additivity has been conjectured.\\n\\nA major conceptional insight was then gained in [8, 9, 10, 11] where it was shown that all these additivity problems are globally equivalent in the sense that if additivity holds for one of these quantities in general, then it does so for all of them. Here 'in general' means that it has to be true for arbitrary pairs of channels (or states), a condition we will call strong additivity.\\n\\nIn this work we present a further conceptional simplification of these and related additivity problems. We show that strong additivity is implied by weak additivity, meaning additivity for arbitrary pairs of equal channels or states. Moreover, based on [12] we argue that it suffices to consider pairs of identical unital channels only. This observation may be a small step on a notorious path but it might guide future research as it for instance underlines recent attempts to understand the asymptotic structure of tensor powers of unital channels [13] . Moreover, one may think of other additivity questions than the ones stated above for which our techniques could be of use. In particular, we think of regularized quanti-ties (like quantum capacities (cf. [14, 15] ) or certain entanglement measures) for which weak additivity holds by definition.\\n\\nOur main tool is the use of direct sums of channels or states. For the latter case similar constructions appeared in [16, 17, 18] . We begin with a discussion of direct sum channels. This will contain more than what is needed for the subsequent additivity results as we think that these tools might be of independent interest.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. DIRECT SUMS OF QUANTUM CHANNELS\",\n \"text\": \"We consider direct sums of channels, i.e., completely positive and trace preserving maps of the form T = ⊕ i T i , where each T i is a channel in its own right. Our aim in this section is to express information theoretic functionals of T in terms of their values for the T i 's. The definition of the quantities appearing in the following proposition will be given in the proof. Proposition 1 (Direct sums) Consider a direct sum T = ⊕ n i=1 T i , n ∈ N of arbitrary finite dimensional channels. Then 1. Minimal output α-Renyi entropy (α ≥ 1):\\n\\nS min,α T = min i S min,α (T i ) , ( 1\\n\\n)\\n\\n2. Coherent information:\\n\\nJ T = max i J(T i ) , ( 2\\n\\n)\\n\\n3. Mutual information:\\n\\nI T = max {λi} S({λ i }) + i λ i I(T i ), ( 3\\n\\n) = log i 2 I(Ti) , ( 4\\n\\n)\\n\\nwhere {λ i } is a probability distribution and S({λ i }) its entropy.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"HSW capacity:\",\n \"text\": \"χ T = max {λi} S({λ i }) + i λ i χ(T i ) ( 5\\n\\n) = log i 2 χ(Ti) . ( 6\\n\\n)\\n\\n2 Remark: Let us briefly comment on the interpretation of the above formulas. Concerning the HSW capacity, classical information can either be sent through the channels T i or it can be encoded in the choice of blocks i = 1, . . . , n. Eq.( 5 ) shows exactly the competition between these two ways of communicating classical information. For the quantum mutual information, which gives the entanglement assisted capacity [19], we obtain the same interpretation (note that the '2' comes from the fact that we take log in base 2). The coherent information is related (via regularization) to the quantum capacity [20] . In this case encoding information in the choice of blocks is not possible-this would be purely classical as all the coherences get lost. Similarly, for the minimal output entropies the minimum is obtained by putting all the weight into the least noisy channel.\\n\\nProof. 1. The α-Renyi entropy is defined as\\n\\nS α (ρ) = 1 1 -α log tr[ρ α ] = α 1 -α log ρ α , ( 7\\n\\n)\\n\\nfor 0 ≤ α ≤ ∞. Here • p is the Schatten p-norm.\\n\\nWhen α = 1 the functional is defined by its limit which is the minimal output entropy S min (T ) = inf ρ S(T (ρ)) with S(ρ) = -trρ log ρ the von Neumann entropy. Let us consider this case first. As the direct sum ⊕ i T i erases the off-diagonal blocks so that all possible outputs can be obtained upon block-diagonal inputs we can restrict to ρ = ⊕ i ρi . Here ρi is not necessarily normalized so that the weights tr[ρ i ] =: λ i form a probability distribution. Writing ρ i := ρi /λ i and using the concavity of von Neumann entropy we get\\n\\nS ⊕ i T i (λ i ρ i ) ≥ i λ i S(T i (ρ i )). ( 8\\n\\n)\\n\\nThis leads to Eq.( 1 ) when α = 1. For α > 1 the minimization of S α (T (ρ)) amounts to a maximization of T (ρ) α and the result follows from convexity of • α in a similar way.\\n\\n2. The coherent information is defined as\\n\\nJ(T ) = sup ρ S T (ρ) -S T ⊗ id(Ψ) , ( 9\\n\\n)\\n\\nwhere Ψ is a purification of ρ such that ρ = tr B Ψ. Since T and T ⊗ id erase the off-diagonal blocks we can replace ρ and Ψ by their diagonal blocks: 2 ) in a similar way as above.\\n\\n⊕ i λ i ρ i and ⊕ i λ i Ψ i . Here, Ψ i is an extension of ρ i . Since the conditional entropy S(ρ AB )-S(ρ A ) is concave in ρ AB [21] considering a convex decomposition of each Ψ i into pure states shows Eq.(\\n\\n3. The mutual information defined as\\n\\nI(T ) = sup ρ S(ρ) + S T (ρ) -S T ⊗ id(Ψ) ( 10\\n\\n)\\n\\nis concave in ρ so the maximum will be achieved by a block\\n\\ndiagonal ρ = ⊕ i λ i ρ i for T = ⊕ i T i . To see this, let V = ⊕ j exp{2πij/n}I j and average V k ρV * k over k = 1, . . . , n. Take a purification Ψ of ρ = ⊕ i λ i ρ i , and\\n\\nthen replace Ψ by its diagonal blocks: ⊕ i λ i Ψ i as before. However, each Ψ i is a purification of ρ i in this case. Indeed, suppose ρ i = j p ij |ij ij|, where {|ij } j is an orthonormal basis in the ith subspace. Then, Ψ is\\n\\nijkl λ i λ k p ij p kl |ij kl| ⊗ |ij kl|,\\n\\nand its ith diagonal block λ i Ψ i is\\n\\nλ i jl √ p ij p il |ij il| ⊗ |ij il| = λ i |Ψ i Ψ i |. Here, |Ψ i = j √ p ij |ij ⊗ |ij .\\n\\nExploiting this together with S(λρ) = λ(S(ρ) -log λ) then gives Eq.(3). Eq.(4) follows then from determining the optimal λ i via Lagrange multipliers in the following way. The maximization problem of\\n\\nS({λ i }) + λ i c i for a probability distribution {λ i } amounts then to maximizing S({λ i }) + i λ i c i + Λ λ i -1 , ( 11\\n\\n)\\n\\nwhere Λ is the Lagrange multiplier. Taking partial derivatives we obtain for extremal { λi }:\\n\\n-log λi - 1 ln 2 + c i + Λ = 0 ∀i ( 12\\n\\n) i λi -1 = 0. ( 13\\n\\n)\\n\\nHence (12) shows c i -log λi is a constant, say, C for ∀i, and by (13) we get C = log i 2 ci . Therefore\\n\\nS({ λi }) + i λi c i = i λi C = log i 2 ci . ( 14\\n\\n)\\n\\nAs this is lower bounded by min i [I(T i )] it must be the maximum.\\n\\n4. The HSW capacity χ is given by\\n\\nχ(T ) = sup ρ S(T (ρ)) -H T (ρ), ( 15\\n\\n) where H T (ρ) = inf P p k ρ k =ρ k p k S(T (ρ k )). ( 16\\n\\n)\\n\\nHere, H T (ρ) is the convex closure of the output entropy; {p k } is a probability distribution and ρ k are density matrices. The r.h.s. of Eq.( 15 ) for a fixed average input state ρ is a constraint HSW capacity which we will denote by χ(T, ρ). Since the inputs can again be assumed to be block-diagonal we have:\\n\\nχ (⊕ i T i , ⊕ i λ i ρ i ) = S(⊕ i λ i T i (ρ i )) - i λ i H Ti (ρ i ) = S({λ i }) + i λ i χ(T i , ρ i ). ( 17\\n\\n)\\n\\nThe first equality is explained by the fact that since the von Neumann entropy is concave there is an optimal decomposition of ⊕ i λ i ρ i for which each state has its support in one of the diagonal blocks. The second equality comes from\\n\\nS(λ i ρ i ) = λ i (S(ρ i ) -log λ i ).\\n\\nTaking the supremum over all states {ρ i } then leads to Eq.( 5 ). Again, Eq.(6) is obtained by using Lagrange multipliers as above. In fact, for unital channels (6) has been obtained in [30] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. SIMPLIFYING ADDITIVITY PROBLEMS\",\n \"text\": \"Let us now turn to the additivity conjectures and exploit Prop.1 in order to show that in several cases weak additivity (for equal channels or states) implies strong additivity (i.e., for different ones). Proposition 2 (Reduction for channels) The following (in-) equalities hold for arbitrary pairs of different channels T 1 and T 2 iff they hold for arbitrary equal pairs T 1 = T 2 .\\n\\n1. S min,α (T 1 ⊗ T 2 ) = S min,α (T 1 ) + S min,α (T 2 ), for any α ≥ 1. 2. χ(T 1 ⊗ T 2 ) = χ(T 1 ) + χ(T 2 ). 3. H T1⊗T2 (ρ) ≥ H T1 (ρ 1 ) + H T2 (ρ 2 ) for all states ρ with respective subsystems ρ 1 , ρ 2 . 4. H T1⊗T2 (ρ 1 ⊗ ρ 2 ) = H T1 (ρ 1 )+ H T2 (ρ 2 ) for all product states ρ 1 ⊗ ρ 2 .\\n\\nRemark: The conjectured equality in 1. is the additivity of the minimal output entropy when α = 1 [22], and it becomes the multiplicativity of maximal output p-norms for p = α > 1. This was conjectured to be true for all α ∈ [1, +∞] before a counterexample was found [23] ruling out all values α > 4.79. The equation in 2. is the conjectured additivity of the HSW capacity, which gives the classical capacity as long as entangled states are not allowed to be used in the encoding [2, 3] . The additivity would show that the HSW capacity itself is the unconstrained classical capacity of quantum channels. The conjectures 3. and 4. are called strong superadditivity and additivity of the convex closure of the output entropy. When T 1 , T 2 are partial traces they become strong superadditivity and additivity of entanglement of formation, respectively, which we discuss in greater detail below.\\n\\nWe note that Prop.2 remains valid in the case where 'arbitrary channels' refers to a restricted set of channels which is closed under direct sums and tensor products.\\n\\nProof. 1. Let σ 1 and σ 2 be optimal output states for T 1 and T 2 respectively. Then, form the following two channels:\\n\\nT ′ 1 (ρ) = T 1 (ρ) ⊗ σ 2 , T ′ 2 (ρ) = σ 1 ⊗ T 2 (ρ). ( 18\\n\\n)\\n\\nIt is not difficult to see that T 1 ⊗ T 2 and T ′ 1 ⊗ T ′ 2 share the additivity property. Hence we can assume that T 1 and T 2 have the same optimal output: S min,α (T 1 ) = S min,α (T 2 ). If we apply first weak additivity and then Prop. 1.1. we obtain:\\n\\nS min,α (((T 1 ⊕ T 2 ) ⊗ (T 1 ⊕ T 2 ))) (19) = 2S min,α (T 1 ⊕ T 2 ) = S min,α (T 1 ) + S min,α (T 2 ). (20)\\n\\nOn the other hand, if we first apply Prop. 1.1. and then weak additivity, we obtain that (19) and thus (20) is upper bounded by S min,α (T 1 ⊗ T 2 ). The converse inequality is trivial.\\n\\n2. Consider χ T 1 ⊕ T 2 ⊗2 = 2χ T 1 ⊕ T 2 = 2 log 2 χ(T1) + 2 χ(T2)\\n\\n= log 2 2χ(T1) + 2 2χ(T2) + 2 χ(T1)+χ(T2)+1 .\\n\\nThis follows from first applying weak additivity and then the proposition 1.4. On the other hand, applying them in reverse order we have χ T 1 ⊕ T 2 ⊗2 = log 2 χ(T1⊗T1) + 2 χ(T2⊗T2) + 2 • 2 χ(T1⊗T2) = log 2 2χ(T1) + 2 2χ(T2) + 2 χ(T1⊗T2)+1 .\\n\\nTogether they prove the claimed equality.\\n\\nFor 3. we obtain by weak superadditivity,\\n\\nH T1⊗T2 (ρ) = H (T1⊕T2)⊗(T1⊕T2) (0 ⊕ ρ ⊕ 0 ⊕ 0) ≥ H T1⊕T2 (ρ 1 ⊕ 0) + H T1⊕T2 (0 ⊕ ρ 2 ) = H T1 (ρ 1 ) + H T2 (ρ 2 ). ( 21\\n\\n)\\n\\nHere, ρ 1 , ρ 2 are reduced states of ρ. This proves 3. and the statement 4. follows in a similar way when replacing ρ by a product state.\\n\\nProposition 3 (Unital channels) Proving one of the conjectures in proposition 2 for all pairs of identical unital channels would show the conjecture is true for arbitrary channels.\\n\\nProof. In [12] a unital channel T is constructed for a given channel T so that these two channels T and T share the following additivity properties: additivity of minimal output α-Renyi entropy, and strong superadditivity and additivity of the convex closure of the output entropy. Hence these conjectures can be restricted to products T1 ⊗ T2 for all channels T 1 , T 2 .\\n\\nAs for the HSW, we have the same reduction but for a different reason (See the remark below). Finally, for the above two unital channels T1 , T2 we can construct the direct sum T1 ⊕ T2 which is again a unital channel. Then the result follows from the proof of proposition 2.\\n\\nRemark: We explain local relation between minimal output entropy and the HSW capacity, which was implicitly written but not clear in [12] . Since the unital extension T sort of mixes up outputs of T we have the following formula.\\n\\nχ( T1 ⊗ T2 ) = log d 1 d 2 -S min ( T1 ⊗ T2 ), ( 22\\n\\n)\\n\\nwhere d 1 , d 2 are the dimensions of the output spaces of T1 and T2 respectively. Hence the additivity of HSW capacity is equivalent to the additivity of the minimal output entropy for products of those extensions T1 ⊗ T2 by Eq.( 22 ). Hence the additivity conjecture of the HSW capacity can also be restricted to products T1 ⊗ T2 for all channels T 1 , T 2 by using global equivalence [8, 9, 10, 11] .\\n\\nFinally, we will discuss additivity issues for entanglement measures. The one already mentioned is the entanglement of formation which was introduced in [24] . Since then the following conjectures have been considered:\\n\\nE F (ρ) ≥ E F (ρ 1 ) + E F (ρ 2 ) (23) E F (ρ 1 ⊗ ρ 2 ) = E F (ρ 1 ) + E F (ρ 2 ). ( 24\\n\\n)\\n\\n4 In fact, both are again globally equivalent to the additivity of the HSW capacity and the minimal output entropy. Moreover, additivity would imply that E F equals an important operationally defined entanglement measure, the entanglement cost\\n\\nE c , since E c (ρ) = lim n→∞ E F (ρ ⊗n ) [25].\\n\\nThe entanglement of formation E F (ρ) is the convex closure of output entropy H T (ρ) when T is a partial trace.\\n\\nFollowing a similar strategy as above we will now show that strong additivity in the sense of Eq.(24) is again implied by weak additivity (i.e., Eq.(24) with ρ 1 = ρ 2 ). In fact, this will not only hold for E F but for any convex entanglement monotone [24, 26, 27] . The main reason behind is that every such functional satisfies [28]:\\n\\nf (⊕ i λ i ρ i ) = i λ i f (ρ i ), ( 25\\n\\n)\\n\\nwhere {λ i } is a probability distribution and ρ i are states as before.\\n\\nProposition 4 (Convex entanglement monotones) [29] Suppose f is a convex entanglement monotone which is weakly additive, i.e.,\\n\\nf (ρ 1 ⊗ ρ 2 ) = f (ρ 1 ) + f (ρ 2 ) for all ρ 1 = ρ 2 .\\n\\nThen f is strongly additive in the sense that this holds also for all ρ 1 = ρ 2 .\\n\\nProof. Let ρ = 1 2 (ρ 1 ⊕ ρ 2 ). Then f (ρ ⊗ ρ) = 2f (ρ) = f (ρ 1 ) + f (ρ 2 ). (26)\\n\\nHere, we applied the weak additivity and then (25). Applying them in reverse order we get\\n\\nf (ρ ⊗ ρ) = 1 4 2 i,j=1 f (ρ i ⊗ ρ j ) = 1 2 (f (ρ 1 ) + f (ρ 2 ) + f (ρ 1 ⊗ ρ 2 )). ( 27\\n\\n)\\n\\nUsing similar ideas, it has recently been shown that for regularized entanglement measures like E c or the asymptotic relative entropy of entanglement, monotonicity (i.e., essentially Eq.(25)) and strong additivity are equivalent [18] . Acknowledgement M.F. would like to thank his supervisor Y.M.Suhov for constant encouragement and numerous discussions. M.W. thanks K.G. Vollbrecht for discussions and J.I. Cirac for support. Both authors thank M. B. Ruskai for bringing [30] to their attention.\\n\\n[1] C. E. Shannon, \\\" A Mathematical Theory of Communication\\\", Bell System Technical Journal, 27 379E23 and 623E56 (1948). [2] A. S. Holevo \\\"The capacity of the quantum channel with general signal states\\\", IEEE Trans. Info. Theory, 44, 269-273, (1998). [3] B. Schumacher and M. D. Westmoreland, \\\"Sending classical information via noisy quantum channels\\\", Phys. Rev. A, 56, 131-138, (1997). [4] M.M. Wolf, J. Eisert, \\\"Classical information capacity of a class of quantum channels \\\", New J. Phys. 7, 93 (2005); quant-ph/0412133 [5] C. King, \\\"The capacity of the quantum depolarizing channel \\\", IEEE Trans. Inf. Theo. 49, 221 (2003); quant-ph/0204172 [6] C. King, \\\"Additivity for unital qubit channels\\\", J. Math. Phys. 4641-4653, (2002) . [7] P.W. Shor, \\\"Additivity of the Classical Capacity of Entanglement-Breaking Quantum Channels\\\", J. Math. Phys. 4334 (2002) ; quant-ph/0201149 [8] P.W. Shor, \\\"Equivalence of Additivity Questions in Quantum Information Theory\\\", Comm. Math. Phys., 246, Issue 3, 453-472 (2004); quant-ph/0305035. [9] A.A. Pomeransky, \\\"Strong superadditivity of the entanglement of formation follows from its additivity\\\", Phys. Rev. A 68, 032317 (2003); quant-ph/0305056. [10] K.M.R. Audenaert and S.L. Braunstein, \\\"On Strong Subadditivity of the Entanglement of Formation\\\", Comm. Math. Phys. No 3, 443-452, (2004) ; quant-ph/0303045. [11] K. Matsumoto, T. Shimono, and A. Winter, \\\"Remarks on additivity of the Holevo channel capacity and of the entanglement of formation \\\", Commun. Math. Phys. 246, 437 (2004); quant-ph/0206148. [12] M. Fukuda, \\\"Simplification of additivity conjecture in quantum information theory\\\", quant-ph/0608010. [13] J.A. Smolin, F. Verstraete, A. Winter, \\\"Entanglement of assistance and multipartite state distillation\\\", Phys. Rev. A 72, 052317 (2005); quant-ph/0505038. [14] M. Wolf, D. Perez-Garcia, \\\"Quantum Capacities of Channels with small Environment \\\", Phys. Rev. A 75, 012303 (2007); quant-ph/0607070. [15] G. Smith, J.A. Smolin, A. Winter, \\\"The quantum capacity with symmetric side channels \\\", quant-ph/0607039. [16] P.W. Shor, J.A. Smolin, B.M. Terhal, \\\"Nonadditivity of Bipartite Distillable Entanglement follows from Conjecture on Bound Entangled Werner States \\\", Phys. Rev. Lett. 86, 2681 (2001); quant-ph/0010054. [17] K.G.H. Vollbrecht, R.F. Werner, M.M. Wolf, \\\"On the irreversibility of entanglement distillation \\\", Phys. Rev. A 69, 062304 (2004); quant-ph/0301072. [18] F.G.S.L. Brandao, M. Horodecki, M.B. Plenio, S. Virmani, \\\"Remarks on the equivalence of full additivity and monotonicity for the entanglement cost\\\", quant-ph/0702136. [19] C. H. Bennett, P.W. Shor, J. A. Smolin, and A. V.\\n\\nThapliyal,\\\"Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem\\\", IEEE Trans. Inf. Theory 48, 2637 (2002); quant-ph/0106052. [20] P.W. Shor, \\\"The quantum channel capacity and coherent information\\\", lecture notes, MSRI Workshop on Quantum Computation (2002); I. Devetak, IEEE Trans. Inf. Th. 51, 44 (2005); S. Lloyd, Phys. Rev. A 55, 1613 (1997). [21] M.B. Ruskai, \\\"Inequalities for Quantum Entropy: A Review with Conditions for Equality \\\", J. Math. Phys. 43, 4358 (2002); erratum 46, 019901 (2005); quant-ph/0205064. [22] C. King and M. B. Ruskai, \\\"Minimal entropy of states emerging from noisy quantum channels\\\", IEEE Trans. Info. Theory, 47, 192-209 (2001); quant-ph/9911079. [23] R.F. Werner and A.S. Holevo, \\\"Counterexample to an ad-5 ditivity conjecture for output purity of quantum channels\\\", quant-ph/0203003. [24] C. H. Bennet, D. P. DiVincenzo, J. A. Smolin, W. K. Wootters, \\\"Mixed-state entanglement and quantum error correction\\\", Phys. Rev. A, 54, 3824-3851, (1996) ; quant-ph/9604024. [25] P.M. Hayden, M. Horodecki, B.M. Terhal, \\\"The asymptotic entanglement cost of preparing a quantum state \\\", J. Phys. A: Math. Gen. 34, 6891 (2001); quant-ph/0008134. [26] G. Vidal, J. Mod. Opt., 47, 355 (2000); quant-ph/9807077. [27] V. Vedral and M. B. Plenio, Phys. Rev. A, 57, 1619 (1998); quant-ph/9707035. [28] M. Horodecki, \\\"Simplifying monotonicity conditions for entanglement measures\\\", Open Syst. Inf. Dyn., 12, 231 (2005), quant-ph/0412210. [29] This result is implicit in the recent work [18] . However, it has already been proven and been communicated by one of us at the ERATO conference, Kyoto 2003 (unpublished). [30] E. Stormer, \\\"A reduction theorem for capacity of positive maps\\\", quant-ph/0510040.\"\n }\n]","isConversational":false}}},{"rowIdx":196,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1096"},"arxiv_id":{"kind":"string","value":"0704.1096"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevA.76.053603"},"content_hash":{"kind":"string","value":"4fce9d3a936e06bb27860d451986f511743ca489f3216c4b6002564bb2e9de92"},"title":{"kind":"string","value":"Formation of a sonic horizon in isotropically expanding Bose-Einstein\n condensates"},"abstract":{"kind":"string","value":"We propose a simple experiment to create a sonic horizon in isotropically\ntrapped cold atoms within currently available experimental techniques.\nNumerical simulation of the Gross-Pitaevskii equation shows that the sonic\nhorizon should appear by making the condensate expand. The expansion is\ntriggered by changing the interaction which can be controlled by the Feshbach\nresonance in real experiments. The sonic horizon is shown to be quasi-static\nfor sufficiently strong interaction or large number of atoms. The\ncharacteristic temperature that is associated with particle emission from the\nhorizon, which corresponds to the Hawking temperature in an ideal situation, is\nestimated to be a few nK."},"authors":{"kind":"list like","value":["Yasunari Kurita","Takao Morinari"],"string":"[\n \"Yasunari Kurita\",\n \"Takao Morinari\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["cond-mat.other","gr-qc","hep-th"],"string":"[\n \"cond-mat.other\",\n \"gr-qc\",\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"cond-mat.other"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"For the exploration of cosmology and gravitational physics, it is necessary to have a deep understanding of quantum filed theory in curved spacetime: It is widely believed that everything except for the spacetime itself should originate from quantum fluctuations in the early Universe. Quantum effects on curved spacetime, such as the Hawking radiation, give us theoretical support for black hole thermodynamics. However, it is extremely hard to verify such quantum effects experimentally. For instance, the Hawking radiation is thermal radiation emitted from a dynamically formed stationary black hole [1] . However, the characteristic temperature of the thermal radiation, the Hawking temperature, is on the order of several tens of nanokelvins at most, which is much lower than the cosmic microwave background radiation temperature. So detecting thermal radiation from a real black hole is almost impossible.\n\nOne way to circumvent this difficulty is to make use of artificial black holes [2][3] . Unruh showed in his seminal paper [4] that excitations in a supersonic flow corresponds to a scalar field equation on a curved spacetime including a horizon. Since the phenomenon of the Hawking radiation can be separated from gravitational physics, it is possible to detect the corresponding phenomenon in a fluid system with sonic horizon [4] . The basic idea is to identify fluid flow with curved spacetime and excitation modes with fields on the curved spacetime. A black hole event horizon corresponds to a sonic horizon in a fluid. For the purpose of investigating the quantum effects, a quantum fluid should be considered. As such a quantum fluid, Bose-Einstein condensates (BEC) in trapped cold atoms [5, 6] are one of the most suitable systems [7]- [9] . A crucial advantage is that one can control scattering length between atoms by making use of the Feshbach resonance [10] . In fact, that experimental technique was used in observing jets and bursts in a collapsing condensate, which is called \"Bose-Novae\" [11] . An remarkable explanation of burst and jet phenomena in Bose-Novae was proposed in [12] [13] , based on quantum field theory of particle creation and structure formation in cosmological spacetime.\n\nIn order to verify the Hawking effect in fluid analogy, it is necessary to create a stationary sonic horizon because it is a phenomenon on a dynamically formed stationary black hole. Although several possibilities have been discussed so far [7, 8] , it seems difficult to realize exactly stationary sonic horizon in cold atoms. However, if one can make a quasi-static horizon for high frequency modes, particle emission from the horizon is also expected. In this paper, we numerically demonstrate that a quasistatic horizon is realized without introducing new experimental techniques beyond currently available ones. We consider an expanding BEC driven by a sudden change of the interaction. Numerically solving the Gross-Pitaevskii (GP) equation, we show that a quasi-static horizon appear.\n\nWe note that there have been made great efforts to create cosmological geometry using expanding BEC[14]- [21] . In these papers, the analogue models with specific cosmological metrics such as Friedmann-Robertson-Walker (FRW) metric or de Sitter metric were discussed and the effects of particle creation in these cosmological spacetimes were investigated. However, in this paper, we do not intend to obtain any cosmological analogue model with well-known analytic metric. But we try to obtain dynamically formed quasi-static sonic horizon. Furthermore, the sonic horizon should be formed in hydrodynamic regime of the condensate because the spacetime analogy is only valid in such regime. The appearance of a horizon due to expansion of a condensate was noticed in the previous works, for example in [15] , and its formation itself is not surprising. But it is non-trivial whether the condensate flow at the horizon is in the hydrodynamic regime, or not. In this paper, we show that the quasi-static sonic horizon will appear in the hydrodynamic regime of the condensate by changing the atomic interaction instantaneously.\n\nIn the coherent state path integral formulation, the action of bosons is given by\n\nS = dt d 3 r i φ∂ t φ - 2 2m ∇ φ • ∇φ -V ext φφ - 1 2 U 0 ( φφ) 2 , ( 1\n\n)\n\nwhere V ext is the confining potential and U 0 = 4π 2 a/m with a the s-wave scattering length. For φ and V ext , spatial and time dependences are implicit. The saddle point equation for this action leads to the GP equation:\n\ni ∂ t Ψ = - 2 2m ∇ 2 + V ext + U 0 |Ψ| 2 Ψ. ( 2\n\n)\n\nThis GP equation governs the dynamics of the condensate whose order parameter is given by Ψ. Now we consider hydrodynamical approximation. We denote the bosonic field φ as φ = √ ρ 0 + ρe i (ϕ0+ϕ) , where √ ρ 0 and ϕ 0 are the amplitude and the phase of Ψ, respectively. (Namely, Ψ = √ ρ 0 exp(iϕ 0 ).) The fields ρ and ϕ describe the non-condensate part of the bosonic field. If the density gradient is sufficiently smooth over the scale determined by the local healing length ξ(r, t) ≡ /(2mρ 0 U 0 ) 1/2 , or, in other words if the conditions,\n\n|ξ∇ρ 0 /ρ 0 | 2 ≪ 1 and |ξ∇ρ/ρ| 2 ≪ 1, ( 3\n\n)\n\nare satisfied, hydrodynamical approximation is justified. (The condition (3) shall be examined later.) Under the above condition, the equation for ρ is\n\nρ = -( φ + v 0 • ∇ϕ)/U 0 , ( 4\n\n)\n\nwhere v 0 = m ∇ϕ 0 is the background fluid velocity, and the effective action for ϕ is\n\nS eff = dt d 3 r 2 2U 0 ( φ + v 0 • ∇ϕ) 2 - ρ 0 U 0 m (∇ϕ) 2 .( 5\n\n)\n\nTaking variation with respect to ϕ, we find that the field equation for ϕ has the form of a propagating wave equation. Also, the equation for the field ϕ can be expressed as ∂ µ ( √ -gg µν ∂ ν )ϕ = 0, where g µν is the inverse of the following matrix:\n\ng µν ∝ -(c 2 s -v 0 2 ) -v 0 -v 0 1 , ( 6\n\n)\n\nwith c s = ρ 0 U 0 /m and g = detg µν . Thus, the equation is equivalent to an equation for a massless field on a curved spacetime determined by the metric (6) with c s the speed of \"light.\" Note that in order to interpret the quantity c s as a velocity, U 0 must be positive because, for negative U 0 , c s becomes pure imaginary. Hereafter we consider positive U 0 , which leads to an effective spacetime with Lorentzian signature.\n\nFor the excitation modes of ϕ whose frequencies, say ω, are much higher than the frequency ω BEC , which is associated with the condensate motion, the condensate will be quasi-static. (For moderate changes of the interaction, ω BEC turns out to be the trapping harmonic potential frequency ω ho , as shall be discussed below .) The analogy between fields on the curved spacetime and excitation modes on the fluid flow is meaningful only when the conditions (3) are satisfied. The latter condition in Eq.( 3 ) turns out to be ω 2 ≪ (c s /ξ) 2 , by using Eq.( 4 ). Thus, the frequency ω has an upper limit. The former condition in Eq.( 3 ) is satisfied in the regions far from the edge of the condensate. (In contrast, if one is very close to the edge, zero-point oscillations become dominant, and so the former condition in (3) is not satisfied.) If there exists intermediate region for ω of\n\nω BEC ≪ ω ≪ c s /ξ, ( 7\n\n)\n\nthen the hydrodynamical approximation is justified and the condensate is quasi-static for excitation modes. Note that those modes are associated with particle emission from the horizon if the hydrodynamical flow has a dynamically formed sonic horizon. The necessary condition for the existence of the intermediate region (7) is\n\nc s ξω BEC ≫ 1. ( 8\n\n)\n\nIn the following, we mainly consider condensate satisfying the above condition.\n\nNow we investigate sonic horizon formation in an expanding BEC trapped in isotropic harmonic potential, V ext = mω 2 ho r 2 /2, where r is the radial coordinate. Initially, we set the condensate in a ground state with an initial atomic interaction a i . At t = 0, the atomic interaction is changed suddenly from a i to a f (> a i ), which makes the condensate expand. Then, formation of sonic horizon can be expected. The reason is as follows: The sound velocity is proportional to square root of the condensate density and a decreasing function of r. In contrast, the fluid velocity is an increasing function of r and the condensate expands fast around its edge whereas v 0 (r = 0) = 0 due to the boundary condition. Therefore, at an intermediate radius, v 0 exceeds c s and the fluid flow is transonic. It has a surface satisfying c s = |v 0 | which is called a sonic horizon. We should note that the sonic horizon corresponds to a horizon in the analogue spacetime defined by the metric (6) . We also note that the subsonic region is around the center of the condensate and inside of the sonic horizon.\n\nIn general, if a fluid has a static sonic horizon and a proper quantum state for an excitation field is realized, then it is theoretically predicted that the horizon will emit thermal radiation of the quantum field. As will 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 25 30 t c a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 1: Times tc in each simulation are shown in units of ω -1 ho . The horizontal axis is the ratio of a f to ai.\n\nbe discussed in Appendix A, if the sonic horizon in the expanding condensate is quasi-static for the field ϕ, the horizon will emit thermal radiation into the center of the condensate. The temperature characterizing the thermal emission (Hawking temperature) is given by the following formula:\n\nT pc = 2πk B ∂ r (v 0 -c s )| rH , ( 9\n\n)\n\nwhere r H is the horizon radius and k B is the Boltzmann's constant[4] [24] . From the above expression, it is found that the Hawking temperature is determined by gradient of fluid and sound velocity at the horizon. Thus, it is important to investigate the velocity gradients at the horizon. In the derivation of the formula (9), it is assumed that the dynamically formed horizon is static, but in actual experiments, this assumption is not satisfied exactly. Therefore, the spectrum of the particle is not fully given by the single Planck's distribution function, but rather given by a superposition of the Planck's distribution functions with slightly different temperatures. Even if this is the case, the energy scale of the particle creation emitted from the dynamically formed horizon is on the order of T pc . We have simulated the expansion of the condensate by solving numerically (using the Crank-Nicolson scheme) the time-dependent GP equation. The initial groundstate wave function is obtained by solving the GP equation using the steepest descent method for an initial swave scattering length a i and the number of atoms N . We have computed c s and the radial velocity of the condensate via\n\nc s = (Ψ * Ψ)U 0 /m, (10) v 0 = [Ψ * ∂ r Ψ -(∂ r Ψ * ) Ψ] /(2mi|Ψ| 2 ), ( 11\n\n)\n\nand searched for parameter sets leading\n\nto |v 0 | > c s .\n\nIn the following, we assume that the condensate consists of N = 10 5 Rb atoms. (The values of atomic interaction given below are those in the case of N = 10 5 . If 0 2 4 6 8 10 0 2 4 6 8 10 12 14 r c s v 0 ξ FIG. 2: Sound velocity cs(solid line) and the fluid velocity v0(dashed line) versus r at t = 0.4 in the case of ai = 200a0 and a f = 5ai are shown in units of ( ω ho /m) 1/2 . The healing length ξ(dotted line) is shown as well in units of a ho = ( /mω ho ) 1/2 . N = 10 5 /n with an integer n, then a i and a f should be multiplied by n.) The initial atomic interaction is assumed to be a i = 50a 0 , 200a 0 and 800a 0 where a 0 = 0.53 × 10 -10 m is the Bohr radius. The following change of the atomic interaction has been simulated: 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30. Just after t = 0, the condensate begins to expand in the trapping potential and the expansion is accelerated for a while. At some time, say t = t c , the expansion turns to be decelerated. Figure 1 shows t c as a function of a f /a i . It is seen that t c does not depend on the initial strength of the interaction. For t := ω ho t < π/2, the condensate continues to expand, and at t ≃ π/2, the condensate starts to collapse. Therefore, we turn off the trapping potential at t = π/4 and make the condensate expand freely in order to keep the horizon for a while.\n\na f /a i =\n\nAs far as we have investigated, sonic horizon always appears in the sence of the surface where v 0 exceeds c s . As an example of a sonic horizon, we show Fig. 2 which is the snapshot at t = 0.4 in the case of a i = 200a 0 and a f = 5a i . We see that, around r = 7a ho , the fluid velocity exceeds the sound velocity, and the sonic horizon exists there. In this case, we find that, at t ≡ ω ho t = 0.11, the horizon appears. Fig. 3 shows the time dependence of the radius of the horizon, say r H , and the velocity gradient at the horizon\n\n∂ r (v 0 -c s )| rH .\n\nIf we keep the trapping potential for a long time, an oscillating behavior of the condensate is observed. The period of the oscillation is about π in units of ω -1 ho and we find ω BEC ≃ ω ho , within a moderate change of the interaction. This oscillation is just like an oscillation of a droplet confined in a harmonic potential. Therefore, the condition (8) can be rewritten as\n\nc s ξω ho ≫ 1. ( 12\n\n) 4\n\nFIG. 3: Time dependence of ∂r(v0 -cs)r H (solid line) in units of ω ho and the position of the sonic horizon rH (dashed line) in units of a ho = ( /mω ho ) 1/2 in the case of ai = 200a0 and a f = 5ai.\n\n0 100 200 300 400 500 0 5 10 15 20 25 30 c s /ξω ho a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 4: cs/ξω ho as a function of a f /ai is shown for each initial scattering length.\n\nNow, we are interested in sonic horizon where (12) is satisfied. The intermediate region (7) exists if, for example, the following inequality is satisfied:\n\nc s ξω ho ≥ 22.5. ( 13\n\n)\n\nFor this choice of the lower bound, there exists a region of ω satisfying both conditions of ω ≥ 10ω ho and (c s /ωξ) 2 ≥ 5. The condition (13) ensures hydrodynamic flow and quasi-static nature of the condensate. Fig. 4 shows c s /ξω ho as a function of a f /a i . We define horizon life time as the time interval during which the condition (13) continues to be satisfied at the horizon. The horizon life time is shown in Fig. 5 . As far as we have investigated, the condensate flow satisfying (13) at sonic horizon appears only when\n\na f ≥ 5a i for a i = 50a 0 , a f ≥ 4a i for a i = 200a 0 and a f ≥ 3a i for a i = 800a 0 .\n\nThe Hawking temperature at t = t c and t = 0.79 (just 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 30 a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 5: Horizon life time as a function of a f /ai is shown in units of ω -1 ho .\n\n0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 T pc a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 6: Hawking temperature at t = tc in units of nK. In the evaluation, we assume ω ho = 1400 Hz.\n\nafter turning off the trapping potential) are shown in Fig. 6 and Fig. 7, respectively. In the evaluation, we assume the frequency ω ho = 1400 Hz. The Hawking temperature at t = t c depends on the ratio a f /a i almost linearly. In contrast, for a f /a i ≥ 9, the Hawking temperature at t = 0.79 does not depend on the ratio so much. From the simulations, the temperature is expected to be a few nK. For this spherically symmetric trap, one may concern the three-body recombination loss of condensed atoms. Now, we check the effect of three-body losses for the given peak density. This effects may be taken into account by incorporating the imaginary term describing the inelastic process in the GP equation [22]\n\ni ∂ t Ψ = - 2 2m ∇ 2 + V ext + U 0 |Ψ| 2 Ψ - i 2 K 3 |Ψ| 4 Ψ,\n\nwhere K 3 denotes three-body recombination loss-rate coefficient. Then, the three-body loss is proportional to the 5 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 T pc a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 7: Hawking temperature at t = 0.79 in units of nK, when just after the trapping potential is turned off. In the evaluation, we assume ω ho = 1400 Hz.\n\ncube of the atomic density\n\n∂ ∂t |Ψ| 2 d 3 r = -K 3 |Ψ| 6 d 3 r,\n\nwhich implies that the three-body loss rate is given by\n\nR 3 ≡ K 3 |Ψ| 6 d 3 r/ |Ψ| 2 d 3 r.\n\nFor the value of K 3 , we assume K 3 = 2 × 10 -28 cm 6 /s, according to [23] . Of course, high atomic density causes many inelastic processes and gives high atomic loss rate. In our numerical simulation, the upper limit of the loss rate can be estimated by use of the peak density as R 3 ≤ 3×10s -1 , where the total atomic number was set to be N = |Ψ| 2 d 3 r = 10 5 . Then, the three-body loss can be ignored because we consider the time scale of ≤ 10 ms. In the above evaluation for Hawking temperature, horizon lifetime and R 3 , we have assumed that the trapping frequency is ω ho = 1400 Hz. Note that ω ho is the energy scale of the system. Therefore, a large value of ω ho is plausible to increase the characteristic temperature for the particle emission, though the time evolution process becomes rapid for large ω ho . If lower frequency is assumed, lower temperature, longer horizon lifetime and fewer three-body loss rate would be expected. As an example, Fig. 8 shows ω ho -dependence of the Hawking temperature and the horizon lifetime in the case of a i = 200a 0 and a f = 10a i .\n\nIn the above numerical simulations, we assume there is no dynamical instability. Now, we check whether there is dynamical instability or not, within Gaussian approximation. For that purpose, we study the Bogoliubov-de Gennes equations: the second quantized field equations 0 0.5 1 1.5 2 2.5 3 3.5 4 0 200 400 600 800 1000 1200 1400 ω ho T H horizon life time FIG. 8: ω ho -dependence of TH and horizon lifetime in the case of ai = 200a0 and a f = 10ai. The Hawking temperature is shown in units of nK and horizon lifetime is in units of ω -1 ho .\n\nfor the excitation fields δφ and δφ are given by\n\ni ∂ t δφ = - 2 2m ∇ 2 + V ext + 2U 0 |Ψ| 2 δφ + U 0 Ψ 2 δφ, -i ∂ t δφ = - 2 2m ∇ 2 + V ext + 2U 0 |Ψ| 2 δφ + U 0 Ψ 2 δφ.\n\nThe excitation spectrum is computed by performing the Bogoliubov transformation:\n\nδφ = α u α (r) b α e -iEαt/ -v α (r) b † α e iEαt/ ,( 14\n\n) δφ = α u * α (r) b † α e iEαt/ -v * α (r) b α e -iEαt/ .( 15\n\n)\n\nThe energy spectrum E α is calculated by diagonalizing the skew symmetric matrix, which is carried out by using a routine in LAPACK. For the parameter values taken above, we find that all eigenvalues do not have the imaginary parts within numerical errors. Therefore, within Gaussian approximation, there is no dynamical instability. In addition, we find that there is no level crossing.\n\nTo summarize, we have proposed an experiment to create a quasi-static sonic horizon using an expanding BEC. It has been shown that the dynamically formed quasi-static sonic horizon is in hydrodynamic regime as it should be to discuss analogy with curved spacetime in BEC. Under suitable choices of the interaction parameter and the confining potential, the characteristic temperature of the particle emission is expected to be a few nK for sufficiently strong confining potential. Large number of atoms or strong atomic interaction improves the quasi-static nature of the horizon. 6 Of course, other effect such as cosmological particle creation can occur in this expanding BEC setup, as discussed in [14]- [21] . In this paper, we have focused on how to make dynamically formed quasi-static sonic horizon in the hydrodynamic regime of the condensate flow. In order to investigate cosmological particle creation effect and other excitations arising from depletion, we need a different numerical simulation scheme. The result will be reported in a future publication.\n\nFurthermore, it is interesting to investigate numerically the behavior of negative frequency modes with positive norm which seem to be related to Hawking effect as was discussed in [26] [27] . This point shall be investigated in a future publication. Acknowledgments Y.K. thanks Hideki Ishihara, Ken-ichi Nakao, and Makoto Tsubota for useful discussions. The authors thank Michikazu Kobayashi and Takashi Uneyama for useful comments on numerical simulations. Y.K. was partially supported by the Yukawa memorial foundation. This work was also supported by the 21st Century COE \"Center for Diversity and Universality in Physics\" and \"Constitution of wide-angle mathematical basis focused on knots\" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. The numerical calculations were carried out on Altix3700 BX2 at YITP in Kyoto University.\n\nAPPENDIX A: PARTICLE CREATION PHENOMENON\n\nHere we focus on spherically symmetric quantum fluctuations by symmetry. At t ≤ 0, the fluid velocity v 0 = 0, and the metric of the initial static effective spacetime is\n\nds 2 ∝ -c 2 s dt 2 + dr 2 + r 2 dΩ 2 S 2 , ( A1\n\n)\n\nwhere dΩ 2 S 2 is the element of solid angle on the unit sphere S 2 . After the increase of the interaction, the effective spacetime evolves dynamically as the BEC starts to expand. Then, the sonic horizon is formed as was shown by the above numerical simulation. If the effective spacetime is static, we can introduce a following time coordinate: τ = t + v 0 dr/(c 2 s -v 2 0 ), and the effective spacetime metric becomes\n\nds 2 ∝ -(c 2 s -v 2 0 )dτ 2 + c 2 s dr 2 c 2 s -v 2 0 + r 2 dΩ 2 S 2 . ( A2\n\n)\n\nFrom this expression, it is found that the horizon is located at the surface where the condition c s = |v 0 | is satisfied. A new coordinate v is introduced as v ≡ τ + r * where r * ≡ c s dr/(c 2 s -v 2 0 ), which is a coordinate characterizing ingoing light-like (null) rays in the effective spacetime.\n\nWe assume here that the initial state of the quantum field ϕ is the vacuum state for the static observer in the initial effective spacetime. Under the time evolution of the effective spacetime caused by the expansion of the condensate, the creation and annihilation operators for the field ϕ also evolve, and particle creation occurs.\n\nNow we consider an observer who moves along his or her outgoing geodesic with proper time λ, crossing the horizon at λ = 0. Hereafter, we term the observer geodesic observer. If we assume that the horizon is located at r = r H , the proper time λ is related to the coordinate v there via λ ≈ -λ 0\n\ne -2c H α v , where c H ≡ c s (r H ), α ≡ 2c H ∂ r (v 0 -c s )| r=rH\n\nand λ 0 is a constant. The ingoing mode functions ϕ ω = e -iωv have λ-dependence near the horizon as\n\nϕ ω ≈ exp i 2c H ω α ln(-λ) . ( A3\n\n)\n\nInitially, the state is the vacuum for the static observer and therefore the geodesic observer would see no excitation at short distance, because there will be no much higher positive frequency excitations than those determined by the time scale of the dynamical expansion of the BEC. If we ignore the short distance cut-off determined by the healing length, or equivalently, if the latter condition in Eq.(3) is ignored, this λ-dependence of the ingoing mode functions implies that the particle creation from the horizon into the inside of the condensate has thermal spectrum with the temperature given by (9). Furthermore, even if the short distance cut-off is taken into account, it is known that the result does not change in principle [25] . Therefore, the particle emission from horizon will occur in the case of expanding condensate, even where the subsonic region is inside of the horizon.\n\n[1] S.W. Hawking, Nature 248, 30 (1974); S.W. Hawking, Commun. Math. Phys. 43, 199 (1975). [2] Artificial Black Holes, edited by M. Novello, M. Visser, and G. Volovik (World Scientific, 2002). [3] C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel.\n\n8, 12 (2005) [4] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E.A. Cornell, Science 269, 198 (1995). [6] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [7] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys.\n\nRev. Lett. 85, 4643 (2000) [8] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys. Rev. A 63, 023611 (2001) [9] C. Barcelo, S. Liberati and M. Visser, Class. Quantum Grav. 18, 1137 (2001). [10] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151 (1998); P. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, and B.J. Verhaar, Phys. Rev. Lett. 81, 69 (1998); J.L. Roberts, N.R. Claussen, J.P. Burke, C.H. Greene, E.A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81, 5109 (1998). [11] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Nature 412, 295 (2001). [12] E. A. Calzetta and B. L. Hu, Phys. Rev. A 68, 043625 (2003) [13] E. A. Calzetta and B. L. Hu, cond-mat/0208569 [14] C. Barcelo, S. Liberati and M. Visser, Int.J.Mod. Phys. D 12 1641 (2003). [15] C. Barcelo, S. Liberati and M. Visser, Phys. Rev. A 68 053613 (2003).\n\n[16] P. O. Fedichev and U. R. Fischer, Phys. Rev. Lett. 91, 240407 (2003). [17] P. O. Fedichev and U. R. Fischer, Phys. Rev. A. 69, 033602 (2004). [18] P. O. Fedichev and U. R. Fischer, Phys. Rev. D 69, 064021 (2004). [19] U. R. Fischer and R. Schützhold, Phys. Rev. A 70, 063615 (2004). [20] J. E. Lidsey, Class. Quantum Grav. 21, 777 (2004). [21] S. E. C. Weinfurtner, gr-qc/0404063. [22] Yu. Kagan, A.E. Muryshev, and G.V. Shlyapnikov, Phys.\n\nRev. Lett. 81 933 (1998). [23] H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002). [24] For derivation details of the particle creation, see, for example, T. Jacobson, gr-qc/0308048. [25] S. Corley and T. Jacobson, Phys. Rev. D 54, 1568 (1996). [26] U. Leonhardt, T. Kiss and P. Öhberg, J. Opt. B 5 S 42 (2003). [27] U. Leonhardt, T. Kiss and P. Öhberg, Phys. Rev. A 67, 033602 (2003)"},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"For the exploration of cosmology and gravitational physics, it is necessary to have a deep understanding of quantum filed theory in curved spacetime: It is widely believed that everything except for the spacetime itself should originate from quantum fluctuations in the early Universe. Quantum effects on curved spacetime, such as the Hawking radiation, give us theoretical support for black hole thermodynamics. However, it is extremely hard to verify such quantum effects experimentally. For instance, the Hawking radiation is thermal radiation emitted from a dynamically formed stationary black hole [1] . However, the characteristic temperature of the thermal radiation, the Hawking temperature, is on the order of several tens of nanokelvins at most, which is much lower than the cosmic microwave background radiation temperature. So detecting thermal radiation from a real black hole is almost impossible.\n\nOne way to circumvent this difficulty is to make use of artificial black holes [2][3] . Unruh showed in his seminal paper [4] that excitations in a supersonic flow corresponds to a scalar field equation on a curved spacetime including a horizon. Since the phenomenon of the Hawking radiation can be separated from gravitational physics, it is possible to detect the corresponding phenomenon in a fluid system with sonic horizon [4] . The basic idea is to identify fluid flow with curved spacetime and excitation modes with fields on the curved spacetime. A black hole event horizon corresponds to a sonic horizon in a fluid. For the purpose of investigating the quantum effects, a quantum fluid should be considered. As such a quantum fluid, Bose-Einstein condensates (BEC) in trapped cold atoms [5, 6] are one of the most suitable systems [7]- [9] . A crucial advantage is that one can control scattering length between atoms by making use of the Feshbach resonance [10] . In fact, that experimental technique was used in observing jets and bursts in a collapsing condensate, which is called \"Bose-Novae\" [11] . An remarkable explanation of burst and jet phenomena in Bose-Novae was proposed in [12] [13] , based on quantum field theory of particle creation and structure formation in cosmological spacetime.\n\nIn order to verify the Hawking effect in fluid analogy, it is necessary to create a stationary sonic horizon because it is a phenomenon on a dynamically formed stationary black hole. Although several possibilities have been discussed so far [7, 8] , it seems difficult to realize exactly stationary sonic horizon in cold atoms. However, if one can make a quasi-static horizon for high frequency modes, particle emission from the horizon is also expected. In this paper, we numerically demonstrate that a quasistatic horizon is realized without introducing new experimental techniques beyond currently available ones. We consider an expanding BEC driven by a sudden change of the interaction. Numerically solving the Gross-Pitaevskii (GP) equation, we show that a quasi-static horizon appear.\n\nWe note that there have been made great efforts to create cosmological geometry using expanding BEC[14]- [21] . In these papers, the analogue models with specific cosmological metrics such as Friedmann-Robertson-Walker (FRW) metric or de Sitter metric were discussed and the effects of particle creation in these cosmological spacetimes were investigated. However, in this paper, we do not intend to obtain any cosmological analogue model with well-known analytic metric. But we try to obtain dynamically formed quasi-static sonic horizon. Furthermore, the sonic horizon should be formed in hydrodynamic regime of the condensate because the spacetime analogy is only valid in such regime. The appearance of a horizon due to expansion of a condensate was noticed in the previous works, for example in [15] , and its formation itself is not surprising. But it is non-trivial whether the condensate flow at the horizon is in the hydrodynamic regime, or not. In this paper, we show that the quasi-static sonic horizon will appear in the hydrodynamic regime of the condensate by changing the atomic interaction instantaneously."},{"section_type":"OTHER","section_title":"II. ANALOGUE SPACETIME IN BEC","text":"In the coherent state path integral formulation, the action of bosons is given by\n\nS = dt d 3 r i φ∂ t φ - 2 2m ∇ φ • ∇φ -V ext φφ - 1 2 U 0 ( φφ) 2 , ( 1\n\n)\n\nwhere V ext is the confining potential and U 0 = 4π 2 a/m with a the s-wave scattering length. For φ and V ext , spatial and time dependences are implicit. The saddle point equation for this action leads to the GP equation:\n\ni ∂ t Ψ = - 2 2m ∇ 2 + V ext + U 0 |Ψ| 2 Ψ. ( 2\n\n)\n\nThis GP equation governs the dynamics of the condensate whose order parameter is given by Ψ. Now we consider hydrodynamical approximation. We denote the bosonic field φ as φ = √ ρ 0 + ρe i (ϕ0+ϕ) , where √ ρ 0 and ϕ 0 are the amplitude and the phase of Ψ, respectively. (Namely, Ψ = √ ρ 0 exp(iϕ 0 ).) The fields ρ and ϕ describe the non-condensate part of the bosonic field. If the density gradient is sufficiently smooth over the scale determined by the local healing length ξ(r, t) ≡ /(2mρ 0 U 0 ) 1/2 , or, in other words if the conditions,\n\n|ξ∇ρ 0 /ρ 0 | 2 ≪ 1 and |ξ∇ρ/ρ| 2 ≪ 1, ( 3\n\n)\n\nare satisfied, hydrodynamical approximation is justified. (The condition (3) shall be examined later.) Under the above condition, the equation for ρ is\n\nρ = -( φ + v 0 • ∇ϕ)/U 0 , ( 4\n\n)\n\nwhere v 0 = m ∇ϕ 0 is the background fluid velocity, and the effective action for ϕ is\n\nS eff = dt d 3 r 2 2U 0 ( φ + v 0 • ∇ϕ) 2 - ρ 0 U 0 m (∇ϕ) 2 .( 5\n\n)\n\nTaking variation with respect to ϕ, we find that the field equation for ϕ has the form of a propagating wave equation. Also, the equation for the field ϕ can be expressed as ∂ µ ( √ -gg µν ∂ ν )ϕ = 0, where g µν is the inverse of the following matrix:\n\ng µν ∝ -(c 2 s -v 0 2 ) -v 0 -v 0 1 , ( 6\n\n)\n\nwith c s = ρ 0 U 0 /m and g = detg µν . Thus, the equation is equivalent to an equation for a massless field on a curved spacetime determined by the metric (6) with c s the speed of \"light.\" Note that in order to interpret the quantity c s as a velocity, U 0 must be positive because, for negative U 0 , c s becomes pure imaginary. Hereafter we consider positive U 0 , which leads to an effective spacetime with Lorentzian signature.\n\nFor the excitation modes of ϕ whose frequencies, say ω, are much higher than the frequency ω BEC , which is associated with the condensate motion, the condensate will be quasi-static. (For moderate changes of the interaction, ω BEC turns out to be the trapping harmonic potential frequency ω ho , as shall be discussed below .) The analogy between fields on the curved spacetime and excitation modes on the fluid flow is meaningful only when the conditions (3) are satisfied. The latter condition in Eq.( 3 ) turns out to be ω 2 ≪ (c s /ξ) 2 , by using Eq.( 4 ). Thus, the frequency ω has an upper limit. The former condition in Eq.( 3 ) is satisfied in the regions far from the edge of the condensate. (In contrast, if one is very close to the edge, zero-point oscillations become dominant, and so the former condition in (3) is not satisfied.) If there exists intermediate region for ω of\n\nω BEC ≪ ω ≪ c s /ξ, ( 7\n\n)\n\nthen the hydrodynamical approximation is justified and the condensate is quasi-static for excitation modes. Note that those modes are associated with particle emission from the horizon if the hydrodynamical flow has a dynamically formed sonic horizon. The necessary condition for the existence of the intermediate region (7) is\n\nc s ξω BEC ≫ 1. ( 8\n\n)\n\nIn the following, we mainly consider condensate satisfying the above condition."},{"section_type":"OTHER","section_title":"III. FORMATION OF SONIC HORIZON","text":"Now we investigate sonic horizon formation in an expanding BEC trapped in isotropic harmonic potential, V ext = mω 2 ho r 2 /2, where r is the radial coordinate. Initially, we set the condensate in a ground state with an initial atomic interaction a i . At t = 0, the atomic interaction is changed suddenly from a i to a f (> a i ), which makes the condensate expand. Then, formation of sonic horizon can be expected. The reason is as follows: The sound velocity is proportional to square root of the condensate density and a decreasing function of r. In contrast, the fluid velocity is an increasing function of r and the condensate expands fast around its edge whereas v 0 (r = 0) = 0 due to the boundary condition. Therefore, at an intermediate radius, v 0 exceeds c s and the fluid flow is transonic. It has a surface satisfying c s = |v 0 | which is called a sonic horizon. We should note that the sonic horizon corresponds to a horizon in the analogue spacetime defined by the metric (6) . We also note that the subsonic region is around the center of the condensate and inside of the sonic horizon.\n\nIn general, if a fluid has a static sonic horizon and a proper quantum state for an excitation field is realized, then it is theoretically predicted that the horizon will emit thermal radiation of the quantum field. As will 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 25 30 t c a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 1: Times tc in each simulation are shown in units of ω -1 ho . The horizontal axis is the ratio of a f to ai.\n\nbe discussed in Appendix A, if the sonic horizon in the expanding condensate is quasi-static for the field ϕ, the horizon will emit thermal radiation into the center of the condensate. The temperature characterizing the thermal emission (Hawking temperature) is given by the following formula:\n\nT pc = 2πk B ∂ r (v 0 -c s )| rH , ( 9\n\n)\n\nwhere r H is the horizon radius and k B is the Boltzmann's constant[4] [24] . From the above expression, it is found that the Hawking temperature is determined by gradient of fluid and sound velocity at the horizon. Thus, it is important to investigate the velocity gradients at the horizon. In the derivation of the formula (9), it is assumed that the dynamically formed horizon is static, but in actual experiments, this assumption is not satisfied exactly. Therefore, the spectrum of the particle is not fully given by the single Planck's distribution function, but rather given by a superposition of the Planck's distribution functions with slightly different temperatures. Even if this is the case, the energy scale of the particle creation emitted from the dynamically formed horizon is on the order of T pc . We have simulated the expansion of the condensate by solving numerically (using the Crank-Nicolson scheme) the time-dependent GP equation. The initial groundstate wave function is obtained by solving the GP equation using the steepest descent method for an initial swave scattering length a i and the number of atoms N . We have computed c s and the radial velocity of the condensate via\n\nc s = (Ψ * Ψ)U 0 /m, (10) v 0 = [Ψ * ∂ r Ψ -(∂ r Ψ * ) Ψ] /(2mi|Ψ| 2 ), ( 11\n\n)\n\nand searched for parameter sets leading\n\nto |v 0 | > c s .\n\nIn the following, we assume that the condensate consists of N = 10 5 Rb atoms. (The values of atomic interaction given below are those in the case of N = 10 5 . If 0 2 4 6 8 10 0 2 4 6 8 10 12 14 r c s v 0 ξ FIG. 2: Sound velocity cs(solid line) and the fluid velocity v0(dashed line) versus r at t = 0.4 in the case of ai = 200a0 and a f = 5ai are shown in units of ( ω ho /m) 1/2 . The healing length ξ(dotted line) is shown as well in units of a ho = ( /mω ho ) 1/2 . N = 10 5 /n with an integer n, then a i and a f should be multiplied by n.) The initial atomic interaction is assumed to be a i = 50a 0 , 200a 0 and 800a 0 where a 0 = 0.53 × 10 -10 m is the Bohr radius. The following change of the atomic interaction has been simulated: 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30. Just after t = 0, the condensate begins to expand in the trapping potential and the expansion is accelerated for a while. At some time, say t = t c , the expansion turns to be decelerated. Figure 1 shows t c as a function of a f /a i . It is seen that t c does not depend on the initial strength of the interaction. For t := ω ho t < π/2, the condensate continues to expand, and at t ≃ π/2, the condensate starts to collapse. Therefore, we turn off the trapping potential at t = π/4 and make the condensate expand freely in order to keep the horizon for a while.\n\na f /a i =\n\nAs far as we have investigated, sonic horizon always appears in the sence of the surface where v 0 exceeds c s . As an example of a sonic horizon, we show Fig. 2 which is the snapshot at t = 0.4 in the case of a i = 200a 0 and a f = 5a i . We see that, around r = 7a ho , the fluid velocity exceeds the sound velocity, and the sonic horizon exists there. In this case, we find that, at t ≡ ω ho t = 0.11, the horizon appears. Fig. 3 shows the time dependence of the radius of the horizon, say r H , and the velocity gradient at the horizon\n\n∂ r (v 0 -c s )| rH .\n\nIf we keep the trapping potential for a long time, an oscillating behavior of the condensate is observed. The period of the oscillation is about π in units of ω -1 ho and we find ω BEC ≃ ω ho , within a moderate change of the interaction. This oscillation is just like an oscillation of a droplet confined in a harmonic potential. Therefore, the condition (8) can be rewritten as\n\nc s ξω ho ≫ 1. ( 12\n\n) 4\n\nFIG. 3: Time dependence of ∂r(v0 -cs)r H (solid line) in units of ω ho and the position of the sonic horizon rH (dashed line) in units of a ho = ( /mω ho ) 1/2 in the case of ai = 200a0 and a f = 5ai.\n\n0 100 200 300 400 500 0 5 10 15 20 25 30 c s /ξω ho a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 4: cs/ξω ho as a function of a f /ai is shown for each initial scattering length.\n\nNow, we are interested in sonic horizon where (12) is satisfied. The intermediate region (7) exists if, for example, the following inequality is satisfied:\n\nc s ξω ho ≥ 22.5. ( 13\n\n)\n\nFor this choice of the lower bound, there exists a region of ω satisfying both conditions of ω ≥ 10ω ho and (c s /ωξ) 2 ≥ 5. The condition (13) ensures hydrodynamic flow and quasi-static nature of the condensate. Fig. 4 shows c s /ξω ho as a function of a f /a i . We define horizon life time as the time interval during which the condition (13) continues to be satisfied at the horizon. The horizon life time is shown in Fig. 5 . As far as we have investigated, the condensate flow satisfying (13) at sonic horizon appears only when\n\na f ≥ 5a i for a i = 50a 0 , a f ≥ 4a i for a i = 200a 0 and a f ≥ 3a i for a i = 800a 0 .\n\nThe Hawking temperature at t = t c and t = 0.79 (just 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 30 a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 5: Horizon life time as a function of a f /ai is shown in units of ω -1 ho .\n\n0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 T pc a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 6: Hawking temperature at t = tc in units of nK. In the evaluation, we assume ω ho = 1400 Hz.\n\nafter turning off the trapping potential) are shown in Fig. 6 and Fig. 7, respectively. In the evaluation, we assume the frequency ω ho = 1400 Hz. The Hawking temperature at t = t c depends on the ratio a f /a i almost linearly. In contrast, for a f /a i ≥ 9, the Hawking temperature at t = 0.79 does not depend on the ratio so much. From the simulations, the temperature is expected to be a few nK. For this spherically symmetric trap, one may concern the three-body recombination loss of condensed atoms. Now, we check the effect of three-body losses for the given peak density. This effects may be taken into account by incorporating the imaginary term describing the inelastic process in the GP equation [22]\n\ni ∂ t Ψ = - 2 2m ∇ 2 + V ext + U 0 |Ψ| 2 Ψ - i 2 K 3 |Ψ| 4 Ψ,\n\nwhere K 3 denotes three-body recombination loss-rate coefficient. Then, the three-body loss is proportional to the 5 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 T pc a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 7: Hawking temperature at t = 0.79 in units of nK, when just after the trapping potential is turned off. In the evaluation, we assume ω ho = 1400 Hz.\n\ncube of the atomic density\n\n∂ ∂t |Ψ| 2 d 3 r = -K 3 |Ψ| 6 d 3 r,\n\nwhich implies that the three-body loss rate is given by\n\nR 3 ≡ K 3 |Ψ| 6 d 3 r/ |Ψ| 2 d 3 r.\n\nFor the value of K 3 , we assume K 3 = 2 × 10 -28 cm 6 /s, according to [23] . Of course, high atomic density causes many inelastic processes and gives high atomic loss rate. In our numerical simulation, the upper limit of the loss rate can be estimated by use of the peak density as R 3 ≤ 3×10s -1 , where the total atomic number was set to be N = |Ψ| 2 d 3 r = 10 5 . Then, the three-body loss can be ignored because we consider the time scale of ≤ 10 ms. In the above evaluation for Hawking temperature, horizon lifetime and R 3 , we have assumed that the trapping frequency is ω ho = 1400 Hz. Note that ω ho is the energy scale of the system. Therefore, a large value of ω ho is plausible to increase the characteristic temperature for the particle emission, though the time evolution process becomes rapid for large ω ho . If lower frequency is assumed, lower temperature, longer horizon lifetime and fewer three-body loss rate would be expected. As an example, Fig. 8 shows ω ho -dependence of the Hawking temperature and the horizon lifetime in the case of a i = 200a 0 and a f = 10a i ."},{"section_type":"OTHER","section_title":"IV. BOGOLIUBOV SPECTRUM","text":"In the above numerical simulations, we assume there is no dynamical instability. Now, we check whether there is dynamical instability or not, within Gaussian approximation. For that purpose, we study the Bogoliubov-de Gennes equations: the second quantized field equations 0 0.5 1 1.5 2 2.5 3 3.5 4 0 200 400 600 800 1000 1200 1400 ω ho T H horizon life time FIG. 8: ω ho -dependence of TH and horizon lifetime in the case of ai = 200a0 and a f = 10ai. The Hawking temperature is shown in units of nK and horizon lifetime is in units of ω -1 ho .\n\nfor the excitation fields δφ and δφ are given by\n\ni ∂ t δφ = - 2 2m ∇ 2 + V ext + 2U 0 |Ψ| 2 δφ + U 0 Ψ 2 δφ, -i ∂ t δφ = - 2 2m ∇ 2 + V ext + 2U 0 |Ψ| 2 δφ + U 0 Ψ 2 δφ.\n\nThe excitation spectrum is computed by performing the Bogoliubov transformation:\n\nδφ = α u α (r) b α e -iEαt/ -v α (r) b † α e iEαt/ ,( 14\n\n) δφ = α u * α (r) b † α e iEαt/ -v * α (r) b α e -iEαt/ .( 15\n\n)\n\nThe energy spectrum E α is calculated by diagonalizing the skew symmetric matrix, which is carried out by using a routine in LAPACK. For the parameter values taken above, we find that all eigenvalues do not have the imaginary parts within numerical errors. Therefore, within Gaussian approximation, there is no dynamical instability. In addition, we find that there is no level crossing."},{"section_type":"OTHER","section_title":"V. SUMMARY","text":"To summarize, we have proposed an experiment to create a quasi-static sonic horizon using an expanding BEC. It has been shown that the dynamically formed quasi-static sonic horizon is in hydrodynamic regime as it should be to discuss analogy with curved spacetime in BEC. Under suitable choices of the interaction parameter and the confining potential, the characteristic temperature of the particle emission is expected to be a few nK for sufficiently strong confining potential. Large number of atoms or strong atomic interaction improves the quasi-static nature of the horizon. 6 Of course, other effect such as cosmological particle creation can occur in this expanding BEC setup, as discussed in [14]- [21] . In this paper, we have focused on how to make dynamically formed quasi-static sonic horizon in the hydrodynamic regime of the condensate flow. In order to investigate cosmological particle creation effect and other excitations arising from depletion, we need a different numerical simulation scheme. The result will be reported in a future publication.\n\nFurthermore, it is interesting to investigate numerically the behavior of negative frequency modes with positive norm which seem to be related to Hawking effect as was discussed in [26] [27] . This point shall be investigated in a future publication. Acknowledgments Y.K. thanks Hideki Ishihara, Ken-ichi Nakao, and Makoto Tsubota for useful discussions. The authors thank Michikazu Kobayashi and Takashi Uneyama for useful comments on numerical simulations. Y.K. was partially supported by the Yukawa memorial foundation. This work was also supported by the 21st Century COE \"Center for Diversity and Universality in Physics\" and \"Constitution of wide-angle mathematical basis focused on knots\" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. The numerical calculations were carried out on Altix3700 BX2 at YITP in Kyoto University.\n\nAPPENDIX A: PARTICLE CREATION PHENOMENON\n\nHere we focus on spherically symmetric quantum fluctuations by symmetry. At t ≤ 0, the fluid velocity v 0 = 0, and the metric of the initial static effective spacetime is\n\nds 2 ∝ -c 2 s dt 2 + dr 2 + r 2 dΩ 2 S 2 , ( A1\n\n)\n\nwhere dΩ 2 S 2 is the element of solid angle on the unit sphere S 2 . After the increase of the interaction, the effective spacetime evolves dynamically as the BEC starts to expand. Then, the sonic horizon is formed as was shown by the above numerical simulation. If the effective spacetime is static, we can introduce a following time coordinate: τ = t + v 0 dr/(c 2 s -v 2 0 ), and the effective spacetime metric becomes\n\nds 2 ∝ -(c 2 s -v 2 0 )dτ 2 + c 2 s dr 2 c 2 s -v 2 0 + r 2 dΩ 2 S 2 . ( A2\n\n)\n\nFrom this expression, it is found that the horizon is located at the surface where the condition c s = |v 0 | is satisfied. A new coordinate v is introduced as v ≡ τ + r * where r * ≡ c s dr/(c 2 s -v 2 0 ), which is a coordinate characterizing ingoing light-like (null) rays in the effective spacetime.\n\nWe assume here that the initial state of the quantum field ϕ is the vacuum state for the static observer in the initial effective spacetime. Under the time evolution of the effective spacetime caused by the expansion of the condensate, the creation and annihilation operators for the field ϕ also evolve, and particle creation occurs.\n\nNow we consider an observer who moves along his or her outgoing geodesic with proper time λ, crossing the horizon at λ = 0. Hereafter, we term the observer geodesic observer. If we assume that the horizon is located at r = r H , the proper time λ is related to the coordinate v there via λ ≈ -λ 0\n\ne -2c H α v , where c H ≡ c s (r H ), α ≡ 2c H ∂ r (v 0 -c s )| r=rH\n\nand λ 0 is a constant. The ingoing mode functions ϕ ω = e -iωv have λ-dependence near the horizon as\n\nϕ ω ≈ exp i 2c H ω α ln(-λ) . ( A3\n\n)\n\nInitially, the state is the vacuum for the static observer and therefore the geodesic observer would see no excitation at short distance, because there will be no much higher positive frequency excitations than those determined by the time scale of the dynamical expansion of the BEC. If we ignore the short distance cut-off determined by the healing length, or equivalently, if the latter condition in Eq.(3) is ignored, this λ-dependence of the ingoing mode functions implies that the particle creation from the horizon into the inside of the condensate has thermal spectrum with the temperature given by (9). Furthermore, even if the short distance cut-off is taken into account, it is known that the result does not change in principle [25] . Therefore, the particle emission from horizon will occur in the case of expanding condensate, even where the subsonic region is inside of the horizon.\n\n[1] S.W. Hawking, Nature 248, 30 (1974); S.W. Hawking, Commun. Math. Phys. 43, 199 (1975). [2] Artificial Black Holes, edited by M. Novello, M. Visser, and G. Volovik (World Scientific, 2002). [3] C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel.\n\n8, 12 (2005) [4] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E.A. Cornell, Science 269, 198 (1995). [6] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [7] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys.\n\nRev. Lett. 85, 4643 (2000) [8] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys. Rev. A 63, 023611 (2001) [9] C. Barcelo, S. Liberati and M. Visser, Class. Quantum Grav. 18, 1137 (2001). [10] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151 (1998); P. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, and B.J. Verhaar, Phys. Rev. Lett. 81, 69 (1998); J.L. Roberts, N.R. Claussen, J.P. Burke, C.H. Greene, E.A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81, 5109 (1998). [11] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Nature 412, 295 (2001). [12] E. A. Calzetta and B. L. Hu, Phys. Rev. A 68, 043625 (2003) [13] E. A. Calzetta and B. L. Hu, cond-mat/0208569 [14] C. Barcelo, S. Liberati and M. Visser, Int.J.Mod. Phys. D 12 1641 (2003). [15] C. Barcelo, S. Liberati and M. Visser, Phys. Rev. A 68 053613 (2003).\n\n[16] P. O. Fedichev and U. R. Fischer, Phys. Rev. Lett. 91, 240407 (2003). [17] P. O. Fedichev and U. R. Fischer, Phys. Rev. A. 69, 033602 (2004). [18] P. O. Fedichev and U. R. Fischer, Phys. Rev. D 69, 064021 (2004). [19] U. R. Fischer and R. Schützhold, Phys. Rev. A 70, 063615 (2004). [20] J. E. Lidsey, Class. Quantum Grav. 21, 777 (2004). [21] S. E. C. Weinfurtner, gr-qc/0404063. [22] Yu. Kagan, A.E. Muryshev, and G.V. Shlyapnikov, Phys.\n\nRev. Lett. 81 933 (1998). [23] H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002). [24] For derivation details of the particle creation, see, for example, T. Jacobson, gr-qc/0308048. [25] S. Corley and T. Jacobson, Phys. Rev. D 54, 1568 (1996). [26] U. Leonhardt, T. Kiss and P. Öhberg, J. Opt. B 5 S 42 (2003). [27] U. Leonhardt, T. Kiss and P. Öhberg, Phys. Rev. A 67, 033602 (2003)"}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"For the exploration of cosmology and gravitational physics, it is necessary to have a deep understanding of quantum filed theory in curved spacetime: It is widely believed that everything except for the spacetime itself should originate from quantum fluctuations in the early Universe. Quantum effects on curved spacetime, such as the Hawking radiation, give us theoretical support for black hole thermodynamics. However, it is extremely hard to verify such quantum effects experimentally. For instance, the Hawking radiation is thermal radiation emitted from a dynamically formed stationary black hole [1] . However, the characteristic temperature of the thermal radiation, the Hawking temperature, is on the order of several tens of nanokelvins at most, which is much lower than the cosmic microwave background radiation temperature. So detecting thermal radiation from a real black hole is almost impossible.\\n\\nOne way to circumvent this difficulty is to make use of artificial black holes [2][3] . Unruh showed in his seminal paper [4] that excitations in a supersonic flow corresponds to a scalar field equation on a curved spacetime including a horizon. Since the phenomenon of the Hawking radiation can be separated from gravitational physics, it is possible to detect the corresponding phenomenon in a fluid system with sonic horizon [4] . The basic idea is to identify fluid flow with curved spacetime and excitation modes with fields on the curved spacetime. A black hole event horizon corresponds to a sonic horizon in a fluid. For the purpose of investigating the quantum effects, a quantum fluid should be considered. As such a quantum fluid, Bose-Einstein condensates (BEC) in trapped cold atoms [5, 6] are one of the most suitable systems [7]- [9] . A crucial advantage is that one can control scattering length between atoms by making use of the Feshbach resonance [10] . In fact, that experimental technique was used in observing jets and bursts in a collapsing condensate, which is called \\\"Bose-Novae\\\" [11] . An remarkable explanation of burst and jet phenomena in Bose-Novae was proposed in [12] [13] , based on quantum field theory of particle creation and structure formation in cosmological spacetime.\\n\\nIn order to verify the Hawking effect in fluid analogy, it is necessary to create a stationary sonic horizon because it is a phenomenon on a dynamically formed stationary black hole. Although several possibilities have been discussed so far [7, 8] , it seems difficult to realize exactly stationary sonic horizon in cold atoms. However, if one can make a quasi-static horizon for high frequency modes, particle emission from the horizon is also expected. In this paper, we numerically demonstrate that a quasistatic horizon is realized without introducing new experimental techniques beyond currently available ones. We consider an expanding BEC driven by a sudden change of the interaction. Numerically solving the Gross-Pitaevskii (GP) equation, we show that a quasi-static horizon appear.\\n\\nWe note that there have been made great efforts to create cosmological geometry using expanding BEC[14]- [21] . In these papers, the analogue models with specific cosmological metrics such as Friedmann-Robertson-Walker (FRW) metric or de Sitter metric were discussed and the effects of particle creation in these cosmological spacetimes were investigated. However, in this paper, we do not intend to obtain any cosmological analogue model with well-known analytic metric. But we try to obtain dynamically formed quasi-static sonic horizon. Furthermore, the sonic horizon should be formed in hydrodynamic regime of the condensate because the spacetime analogy is only valid in such regime. The appearance of a horizon due to expansion of a condensate was noticed in the previous works, for example in [15] , and its formation itself is not surprising. But it is non-trivial whether the condensate flow at the horizon is in the hydrodynamic regime, or not. In this paper, we show that the quasi-static sonic horizon will appear in the hydrodynamic regime of the condensate by changing the atomic interaction instantaneously.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"II. ANALOGUE SPACETIME IN BEC\",\n \"text\": \"In the coherent state path integral formulation, the action of bosons is given by\\n\\nS = dt d 3 r i φ∂ t φ - 2 2m ∇ φ • ∇φ -V ext φφ - 1 2 U 0 ( φφ) 2 , ( 1\\n\\n)\\n\\nwhere V ext is the confining potential and U 0 = 4π 2 a/m with a the s-wave scattering length. For φ and V ext , spatial and time dependences are implicit. The saddle point equation for this action leads to the GP equation:\\n\\ni ∂ t Ψ = - 2 2m ∇ 2 + V ext + U 0 |Ψ| 2 Ψ. ( 2\\n\\n)\\n\\nThis GP equation governs the dynamics of the condensate whose order parameter is given by Ψ. Now we consider hydrodynamical approximation. We denote the bosonic field φ as φ = √ ρ 0 + ρe i (ϕ0+ϕ) , where √ ρ 0 and ϕ 0 are the amplitude and the phase of Ψ, respectively. (Namely, Ψ = √ ρ 0 exp(iϕ 0 ).) The fields ρ and ϕ describe the non-condensate part of the bosonic field. If the density gradient is sufficiently smooth over the scale determined by the local healing length ξ(r, t) ≡ /(2mρ 0 U 0 ) 1/2 , or, in other words if the conditions,\\n\\n|ξ∇ρ 0 /ρ 0 | 2 ≪ 1 and |ξ∇ρ/ρ| 2 ≪ 1, ( 3\\n\\n)\\n\\nare satisfied, hydrodynamical approximation is justified. (The condition (3) shall be examined later.) Under the above condition, the equation for ρ is\\n\\nρ = -( φ + v 0 • ∇ϕ)/U 0 , ( 4\\n\\n)\\n\\nwhere v 0 = m ∇ϕ 0 is the background fluid velocity, and the effective action for ϕ is\\n\\nS eff = dt d 3 r 2 2U 0 ( φ + v 0 • ∇ϕ) 2 - ρ 0 U 0 m (∇ϕ) 2 .( 5\\n\\n)\\n\\nTaking variation with respect to ϕ, we find that the field equation for ϕ has the form of a propagating wave equation. Also, the equation for the field ϕ can be expressed as ∂ µ ( √ -gg µν ∂ ν )ϕ = 0, where g µν is the inverse of the following matrix:\\n\\ng µν ∝ -(c 2 s -v 0 2 ) -v 0 -v 0 1 , ( 6\\n\\n)\\n\\nwith c s = ρ 0 U 0 /m and g = detg µν . Thus, the equation is equivalent to an equation for a massless field on a curved spacetime determined by the metric (6) with c s the speed of \\\"light.\\\" Note that in order to interpret the quantity c s as a velocity, U 0 must be positive because, for negative U 0 , c s becomes pure imaginary. Hereafter we consider positive U 0 , which leads to an effective spacetime with Lorentzian signature.\\n\\nFor the excitation modes of ϕ whose frequencies, say ω, are much higher than the frequency ω BEC , which is associated with the condensate motion, the condensate will be quasi-static. (For moderate changes of the interaction, ω BEC turns out to be the trapping harmonic potential frequency ω ho , as shall be discussed below .) The analogy between fields on the curved spacetime and excitation modes on the fluid flow is meaningful only when the conditions (3) are satisfied. The latter condition in Eq.( 3 ) turns out to be ω 2 ≪ (c s /ξ) 2 , by using Eq.( 4 ). Thus, the frequency ω has an upper limit. The former condition in Eq.( 3 ) is satisfied in the regions far from the edge of the condensate. (In contrast, if one is very close to the edge, zero-point oscillations become dominant, and so the former condition in (3) is not satisfied.) If there exists intermediate region for ω of\\n\\nω BEC ≪ ω ≪ c s /ξ, ( 7\\n\\n)\\n\\nthen the hydrodynamical approximation is justified and the condensate is quasi-static for excitation modes. Note that those modes are associated with particle emission from the horizon if the hydrodynamical flow has a dynamically formed sonic horizon. The necessary condition for the existence of the intermediate region (7) is\\n\\nc s ξω BEC ≫ 1. ( 8\\n\\n)\\n\\nIn the following, we mainly consider condensate satisfying the above condition.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"III. FORMATION OF SONIC HORIZON\",\n \"text\": \"Now we investigate sonic horizon formation in an expanding BEC trapped in isotropic harmonic potential, V ext = mω 2 ho r 2 /2, where r is the radial coordinate. Initially, we set the condensate in a ground state with an initial atomic interaction a i . At t = 0, the atomic interaction is changed suddenly from a i to a f (> a i ), which makes the condensate expand. Then, formation of sonic horizon can be expected. The reason is as follows: The sound velocity is proportional to square root of the condensate density and a decreasing function of r. In contrast, the fluid velocity is an increasing function of r and the condensate expands fast around its edge whereas v 0 (r = 0) = 0 due to the boundary condition. Therefore, at an intermediate radius, v 0 exceeds c s and the fluid flow is transonic. It has a surface satisfying c s = |v 0 | which is called a sonic horizon. We should note that the sonic horizon corresponds to a horizon in the analogue spacetime defined by the metric (6) . We also note that the subsonic region is around the center of the condensate and inside of the sonic horizon.\\n\\nIn general, if a fluid has a static sonic horizon and a proper quantum state for an excitation field is realized, then it is theoretically predicted that the horizon will emit thermal radiation of the quantum field. As will 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0 5 10 15 20 25 30 t c a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 1: Times tc in each simulation are shown in units of ω -1 ho . The horizontal axis is the ratio of a f to ai.\\n\\nbe discussed in Appendix A, if the sonic horizon in the expanding condensate is quasi-static for the field ϕ, the horizon will emit thermal radiation into the center of the condensate. The temperature characterizing the thermal emission (Hawking temperature) is given by the following formula:\\n\\nT pc = 2πk B ∂ r (v 0 -c s )| rH , ( 9\\n\\n)\\n\\nwhere r H is the horizon radius and k B is the Boltzmann's constant[4] [24] . From the above expression, it is found that the Hawking temperature is determined by gradient of fluid and sound velocity at the horizon. Thus, it is important to investigate the velocity gradients at the horizon. In the derivation of the formula (9), it is assumed that the dynamically formed horizon is static, but in actual experiments, this assumption is not satisfied exactly. Therefore, the spectrum of the particle is not fully given by the single Planck's distribution function, but rather given by a superposition of the Planck's distribution functions with slightly different temperatures. Even if this is the case, the energy scale of the particle creation emitted from the dynamically formed horizon is on the order of T pc . We have simulated the expansion of the condensate by solving numerically (using the Crank-Nicolson scheme) the time-dependent GP equation. The initial groundstate wave function is obtained by solving the GP equation using the steepest descent method for an initial swave scattering length a i and the number of atoms N . We have computed c s and the radial velocity of the condensate via\\n\\nc s = (Ψ * Ψ)U 0 /m, (10) v 0 = [Ψ * ∂ r Ψ -(∂ r Ψ * ) Ψ] /(2mi|Ψ| 2 ), ( 11\\n\\n)\\n\\nand searched for parameter sets leading\\n\\nto |v 0 | > c s .\\n\\nIn the following, we assume that the condensate consists of N = 10 5 Rb atoms. (The values of atomic interaction given below are those in the case of N = 10 5 . If 0 2 4 6 8 10 0 2 4 6 8 10 12 14 r c s v 0 ξ FIG. 2: Sound velocity cs(solid line) and the fluid velocity v0(dashed line) versus r at t = 0.4 in the case of ai = 200a0 and a f = 5ai are shown in units of ( ω ho /m) 1/2 . The healing length ξ(dotted line) is shown as well in units of a ho = ( /mω ho ) 1/2 . N = 10 5 /n with an integer n, then a i and a f should be multiplied by n.) The initial atomic interaction is assumed to be a i = 50a 0 , 200a 0 and 800a 0 where a 0 = 0.53 × 10 -10 m is the Bohr radius. The following change of the atomic interaction has been simulated: 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20, 25, 30. Just after t = 0, the condensate begins to expand in the trapping potential and the expansion is accelerated for a while. At some time, say t = t c , the expansion turns to be decelerated. Figure 1 shows t c as a function of a f /a i . It is seen that t c does not depend on the initial strength of the interaction. For t := ω ho t < π/2, the condensate continues to expand, and at t ≃ π/2, the condensate starts to collapse. Therefore, we turn off the trapping potential at t = π/4 and make the condensate expand freely in order to keep the horizon for a while.\\n\\na f /a i =\\n\\nAs far as we have investigated, sonic horizon always appears in the sence of the surface where v 0 exceeds c s . As an example of a sonic horizon, we show Fig. 2 which is the snapshot at t = 0.4 in the case of a i = 200a 0 and a f = 5a i . We see that, around r = 7a ho , the fluid velocity exceeds the sound velocity, and the sonic horizon exists there. In this case, we find that, at t ≡ ω ho t = 0.11, the horizon appears. Fig. 3 shows the time dependence of the radius of the horizon, say r H , and the velocity gradient at the horizon\\n\\n∂ r (v 0 -c s )| rH .\\n\\nIf we keep the trapping potential for a long time, an oscillating behavior of the condensate is observed. The period of the oscillation is about π in units of ω -1 ho and we find ω BEC ≃ ω ho , within a moderate change of the interaction. This oscillation is just like an oscillation of a droplet confined in a harmonic potential. Therefore, the condition (8) can be rewritten as\\n\\nc s ξω ho ≫ 1. ( 12\\n\\n) 4\\n\\nFIG. 3: Time dependence of ∂r(v0 -cs)r H (solid line) in units of ω ho and the position of the sonic horizon rH (dashed line) in units of a ho = ( /mω ho ) 1/2 in the case of ai = 200a0 and a f = 5ai.\\n\\n0 100 200 300 400 500 0 5 10 15 20 25 30 c s /ξω ho a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 4: cs/ξω ho as a function of a f /ai is shown for each initial scattering length.\\n\\nNow, we are interested in sonic horizon where (12) is satisfied. The intermediate region (7) exists if, for example, the following inequality is satisfied:\\n\\nc s ξω ho ≥ 22.5. ( 13\\n\\n)\\n\\nFor this choice of the lower bound, there exists a region of ω satisfying both conditions of ω ≥ 10ω ho and (c s /ωξ) 2 ≥ 5. The condition (13) ensures hydrodynamic flow and quasi-static nature of the condensate. Fig. 4 shows c s /ξω ho as a function of a f /a i . We define horizon life time as the time interval during which the condition (13) continues to be satisfied at the horizon. The horizon life time is shown in Fig. 5 . As far as we have investigated, the condensate flow satisfying (13) at sonic horizon appears only when\\n\\na f ≥ 5a i for a i = 50a 0 , a f ≥ 4a i for a i = 200a 0 and a f ≥ 3a i for a i = 800a 0 .\\n\\nThe Hawking temperature at t = t c and t = 0.79 (just 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 25 30 a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 5: Horizon life time as a function of a f /ai is shown in units of ω -1 ho .\\n\\n0 1 2 3 4 5 6 7 8 0 5 10 15 20 25 30 T pc a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 6: Hawking temperature at t = tc in units of nK. In the evaluation, we assume ω ho = 1400 Hz.\\n\\nafter turning off the trapping potential) are shown in Fig. 6 and Fig. 7, respectively. In the evaluation, we assume the frequency ω ho = 1400 Hz. The Hawking temperature at t = t c depends on the ratio a f /a i almost linearly. In contrast, for a f /a i ≥ 9, the Hawking temperature at t = 0.79 does not depend on the ratio so much. From the simulations, the temperature is expected to be a few nK. For this spherically symmetric trap, one may concern the three-body recombination loss of condensed atoms. Now, we check the effect of three-body losses for the given peak density. This effects may be taken into account by incorporating the imaginary term describing the inelastic process in the GP equation [22]\\n\\ni ∂ t Ψ = - 2 2m ∇ 2 + V ext + U 0 |Ψ| 2 Ψ - i 2 K 3 |Ψ| 4 Ψ,\\n\\nwhere K 3 denotes three-body recombination loss-rate coefficient. Then, the three-body loss is proportional to the 5 0 0.5 1 1.5 2 2.5 3 0 5 10 15 20 25 30 T pc a f /a i a i =50a 0 a i =200a 0 a i =800a 0 FIG. 7: Hawking temperature at t = 0.79 in units of nK, when just after the trapping potential is turned off. In the evaluation, we assume ω ho = 1400 Hz.\\n\\ncube of the atomic density\\n\\n∂ ∂t |Ψ| 2 d 3 r = -K 3 |Ψ| 6 d 3 r,\\n\\nwhich implies that the three-body loss rate is given by\\n\\nR 3 ≡ K 3 |Ψ| 6 d 3 r/ |Ψ| 2 d 3 r.\\n\\nFor the value of K 3 , we assume K 3 = 2 × 10 -28 cm 6 /s, according to [23] . Of course, high atomic density causes many inelastic processes and gives high atomic loss rate. In our numerical simulation, the upper limit of the loss rate can be estimated by use of the peak density as R 3 ≤ 3×10s -1 , where the total atomic number was set to be N = |Ψ| 2 d 3 r = 10 5 . Then, the three-body loss can be ignored because we consider the time scale of ≤ 10 ms. In the above evaluation for Hawking temperature, horizon lifetime and R 3 , we have assumed that the trapping frequency is ω ho = 1400 Hz. Note that ω ho is the energy scale of the system. Therefore, a large value of ω ho is plausible to increase the characteristic temperature for the particle emission, though the time evolution process becomes rapid for large ω ho . If lower frequency is assumed, lower temperature, longer horizon lifetime and fewer three-body loss rate would be expected. As an example, Fig. 8 shows ω ho -dependence of the Hawking temperature and the horizon lifetime in the case of a i = 200a 0 and a f = 10a i .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"IV. BOGOLIUBOV SPECTRUM\",\n \"text\": \"In the above numerical simulations, we assume there is no dynamical instability. Now, we check whether there is dynamical instability or not, within Gaussian approximation. For that purpose, we study the Bogoliubov-de Gennes equations: the second quantized field equations 0 0.5 1 1.5 2 2.5 3 3.5 4 0 200 400 600 800 1000 1200 1400 ω ho T H horizon life time FIG. 8: ω ho -dependence of TH and horizon lifetime in the case of ai = 200a0 and a f = 10ai. The Hawking temperature is shown in units of nK and horizon lifetime is in units of ω -1 ho .\\n\\nfor the excitation fields δφ and δφ are given by\\n\\ni ∂ t δφ = - 2 2m ∇ 2 + V ext + 2U 0 |Ψ| 2 δφ + U 0 Ψ 2 δφ, -i ∂ t δφ = - 2 2m ∇ 2 + V ext + 2U 0 |Ψ| 2 δφ + U 0 Ψ 2 δφ.\\n\\nThe excitation spectrum is computed by performing the Bogoliubov transformation:\\n\\nδφ = α u α (r) b α e -iEαt/ -v α (r) b † α e iEαt/ ,( 14\\n\\n) δφ = α u * α (r) b † α e iEαt/ -v * α (r) b α e -iEαt/ .( 15\\n\\n)\\n\\nThe energy spectrum E α is calculated by diagonalizing the skew symmetric matrix, which is carried out by using a routine in LAPACK. For the parameter values taken above, we find that all eigenvalues do not have the imaginary parts within numerical errors. Therefore, within Gaussian approximation, there is no dynamical instability. In addition, we find that there is no level crossing.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"V. SUMMARY\",\n \"text\": \"To summarize, we have proposed an experiment to create a quasi-static sonic horizon using an expanding BEC. It has been shown that the dynamically formed quasi-static sonic horizon is in hydrodynamic regime as it should be to discuss analogy with curved spacetime in BEC. Under suitable choices of the interaction parameter and the confining potential, the characteristic temperature of the particle emission is expected to be a few nK for sufficiently strong confining potential. Large number of atoms or strong atomic interaction improves the quasi-static nature of the horizon. 6 Of course, other effect such as cosmological particle creation can occur in this expanding BEC setup, as discussed in [14]- [21] . In this paper, we have focused on how to make dynamically formed quasi-static sonic horizon in the hydrodynamic regime of the condensate flow. In order to investigate cosmological particle creation effect and other excitations arising from depletion, we need a different numerical simulation scheme. The result will be reported in a future publication.\\n\\nFurthermore, it is interesting to investigate numerically the behavior of negative frequency modes with positive norm which seem to be related to Hawking effect as was discussed in [26] [27] . This point shall be investigated in a future publication. Acknowledgments Y.K. thanks Hideki Ishihara, Ken-ichi Nakao, and Makoto Tsubota for useful discussions. The authors thank Michikazu Kobayashi and Takashi Uneyama for useful comments on numerical simulations. Y.K. was partially supported by the Yukawa memorial foundation. This work was also supported by the 21st Century COE \\\"Center for Diversity and Universality in Physics\\\" and \\\"Constitution of wide-angle mathematical basis focused on knots\\\" from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. The numerical calculations were carried out on Altix3700 BX2 at YITP in Kyoto University.\\n\\nAPPENDIX A: PARTICLE CREATION PHENOMENON\\n\\nHere we focus on spherically symmetric quantum fluctuations by symmetry. At t ≤ 0, the fluid velocity v 0 = 0, and the metric of the initial static effective spacetime is\\n\\nds 2 ∝ -c 2 s dt 2 + dr 2 + r 2 dΩ 2 S 2 , ( A1\\n\\n)\\n\\nwhere dΩ 2 S 2 is the element of solid angle on the unit sphere S 2 . After the increase of the interaction, the effective spacetime evolves dynamically as the BEC starts to expand. Then, the sonic horizon is formed as was shown by the above numerical simulation. If the effective spacetime is static, we can introduce a following time coordinate: τ = t + v 0 dr/(c 2 s -v 2 0 ), and the effective spacetime metric becomes\\n\\nds 2 ∝ -(c 2 s -v 2 0 )dτ 2 + c 2 s dr 2 c 2 s -v 2 0 + r 2 dΩ 2 S 2 . ( A2\\n\\n)\\n\\nFrom this expression, it is found that the horizon is located at the surface where the condition c s = |v 0 | is satisfied. A new coordinate v is introduced as v ≡ τ + r * where r * ≡ c s dr/(c 2 s -v 2 0 ), which is a coordinate characterizing ingoing light-like (null) rays in the effective spacetime.\\n\\nWe assume here that the initial state of the quantum field ϕ is the vacuum state for the static observer in the initial effective spacetime. Under the time evolution of the effective spacetime caused by the expansion of the condensate, the creation and annihilation operators for the field ϕ also evolve, and particle creation occurs.\\n\\nNow we consider an observer who moves along his or her outgoing geodesic with proper time λ, crossing the horizon at λ = 0. Hereafter, we term the observer geodesic observer. If we assume that the horizon is located at r = r H , the proper time λ is related to the coordinate v there via λ ≈ -λ 0\\n\\ne -2c H α v , where c H ≡ c s (r H ), α ≡ 2c H ∂ r (v 0 -c s )| r=rH\\n\\nand λ 0 is a constant. The ingoing mode functions ϕ ω = e -iωv have λ-dependence near the horizon as\\n\\nϕ ω ≈ exp i 2c H ω α ln(-λ) . ( A3\\n\\n)\\n\\nInitially, the state is the vacuum for the static observer and therefore the geodesic observer would see no excitation at short distance, because there will be no much higher positive frequency excitations than those determined by the time scale of the dynamical expansion of the BEC. If we ignore the short distance cut-off determined by the healing length, or equivalently, if the latter condition in Eq.(3) is ignored, this λ-dependence of the ingoing mode functions implies that the particle creation from the horizon into the inside of the condensate has thermal spectrum with the temperature given by (9). Furthermore, even if the short distance cut-off is taken into account, it is known that the result does not change in principle [25] . Therefore, the particle emission from horizon will occur in the case of expanding condensate, even where the subsonic region is inside of the horizon.\\n\\n[1] S.W. Hawking, Nature 248, 30 (1974); S.W. Hawking, Commun. Math. Phys. 43, 199 (1975). [2] Artificial Black Holes, edited by M. Novello, M. Visser, and G. Volovik (World Scientific, 2002). [3] C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel.\\n\\n8, 12 (2005) [4] W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). [5] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E.A. Cornell, Science 269, 198 (1995). [6] K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [7] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys.\\n\\nRev. Lett. 85, 4643 (2000) [8] L. J. Garay, J. R. Anglin, J. I. Cirac and P. Zoller, Phys. Rev. A 63, 023611 (2001) [9] C. Barcelo, S. Liberati and M. Visser, Class. Quantum Grav. 18, 1137 (2001). [10] S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature 392, 151 (1998); P. Courteille, R.S. Freeland, D.J. Heinzen, F.A. van Abeelen, and B.J. Verhaar, Phys. Rev. Lett. 81, 69 (1998); J.L. Roberts, N.R. Claussen, J.P. Burke, C.H. Greene, E.A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81, 5109 (1998). [11] E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E. Wieman, Nature 412, 295 (2001). [12] E. A. Calzetta and B. L. Hu, Phys. Rev. A 68, 043625 (2003) [13] E. A. Calzetta and B. L. Hu, cond-mat/0208569 [14] C. Barcelo, S. Liberati and M. Visser, Int.J.Mod. Phys. D 12 1641 (2003). [15] C. Barcelo, S. Liberati and M. Visser, Phys. Rev. A 68 053613 (2003).\\n\\n[16] P. O. Fedichev and U. R. Fischer, Phys. Rev. Lett. 91, 240407 (2003). [17] P. O. Fedichev and U. R. Fischer, Phys. Rev. A. 69, 033602 (2004). [18] P. O. Fedichev and U. R. Fischer, Phys. Rev. D 69, 064021 (2004). [19] U. R. Fischer and R. Schützhold, Phys. Rev. A 70, 063615 (2004). [20] J. E. Lidsey, Class. Quantum Grav. 21, 777 (2004). [21] S. E. C. Weinfurtner, gr-qc/0404063. [22] Yu. Kagan, A.E. Muryshev, and G.V. Shlyapnikov, Phys.\\n\\nRev. Lett. 81 933 (1998). [23] H. Saito and M. Ueda, Phys. Rev. A 65, 033624 (2002). [24] For derivation details of the particle creation, see, for example, T. Jacobson, gr-qc/0308048. [25] S. Corley and T. Jacobson, Phys. Rev. D 54, 1568 (1996). [26] U. Leonhardt, T. Kiss and P. Öhberg, J. Opt. B 5 S 42 (2003). [27] U. Leonhardt, T. Kiss and P. Öhberg, Phys. Rev. A 67, 033602 (2003)\"\n }\n]","isConversational":false}}},{"rowIdx":197,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1104"},"arxiv_id":{"kind":"string","value":"0704.1104"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.047701"},"content_hash":{"kind":"string","value":"125b45f03427d978496db2f46aa6048dda50a3d8b80ada4086123ac0491cebaa"},"title":{"kind":"string","value":"On the induction of the four-dimensional Lorentz-breaking non-Abelian\n Chern-Simons action"},"abstract":{"kind":"string","value":"A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is\ngenerated as a one-loop perturbative correction via an appropriate\nLorentz-breaking coupling of the non-Abelian gauge field to the spinor field.\nThis term is shown to be regularization dependent but nevertheless it can be\nfound unambiguously in different regularization schemes at zero and finite\ntemperature."},"authors":{"kind":"list like","value":["M. Gomes","J. R. Nascimento","E. Passos","A. Yu. Petrov","A. J. da Silva"],"string":"[\n \"M. Gomes\",\n \"J. R. Nascimento\",\n \"E. Passos\",\n \"A. Yu. Petrov\",\n \"A. J. da Silva\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th"],"string":"[\n \"hep-th\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is generated as a one-loop perturbative correction via an appropriate Lorentz-breaking coupling of the non-Abelian gauge field to a spinor field. This term is shown to be regularization dependent but nevertheless it can be found unambiguously in different regularization schemes at zero and at finite temperature.\n\nDuring the last years, different aspects of the Lorentz symmetry breaking have been intensively studied [1] . One of the theoretical consequences of this effect is the birefringence of light in vacuum. After the formulation of the concept of noncommutativity of the space-time [2] , which implies in Lorentz symmetry breaking (see the discussion in [3] ), The interest in this subject has greatly increased . One of the implications of the Lorentz symmetry breaking is the possibility of introducing a lot of new couplings in the Standard Model [4] . These terms may arise from radiative corrections to some Lorentz-breaking field theories at zero [5, 6, 7, 8, 9, 10] and at finite temperature [11, 12, 13, 14] . Alternatively, they may be induced from the deformation of the canonical commutation relation through the use of the noncommutative fields method [15, 16] .\n\nRecently, the renormalizability of the Yang Mills (YM) theory with a four-dimensional non-Abelian Lorentz-breaking Chern-Simons (CS) term was studied in [17] . The induction of such Lorentz-breaking CS term starting from a pure YM was investigated within the noncommutative fields method in [16] . In the present work we show how the same CS term can be induced through radiative corrections starting from a YM theory coupled with fermions in the presence of an interaction of the fermions with a constant external field at zero and at finite temperture.\n\nWe start with the following model which represents a non-Abelian generalization of the spinor electrodynamics with the Lorentz-breaking coupling\n\nL f = ψ i (i∂ / -m -γ 5 b /)δ ij -gγ µ A a µ (Ω a ) ij ψ j . (1)\n\nHere b ρ is a constant four-vector. The A µ = A a µ Ω a is a Yang-Mills field coupled to spinors ψ which carry the isotopic indices, ψ = (ψ i ), with i taking values from 1 to N with N being the dimension of the chosen representation of the Lie algebra. The Ω a = (Ω a ) ij are the Lie group generators in this representation satisfying the relations:\n\n[Ω a , Ω b ] = if abc Ω c and Tr[Ω a Ω b ] = δ ab .\n\nThe one-loop effective action of the gauge field\n\nA a µ is S Y M +S f [b, A]\n\nwhere S Y M is the Yang-Mills action and S f [b, A] can be expressed in the form of the following functional trace:\n\nS f [b, A] = -i Tr ln(i∂ / -m -γ 5 b / -gγ µ A a µ Ω a ). (2)\n\nThis functional trace can be rewritten as\n\nS f [b, A] = S f [b] + S ′ f [b, A], with the first term being S f [b] = -i Tr ln(i∂ / -m -γ 5 b /)\n\n. The nontrivial dynamics is concentrated in the second term\n\nS ′ f [b, A],\n\nwhich is given by the power series:\n\nS ′ f [b, A] = i Tr ∞ n=1 1 n 1 i∂ / -m -γ 5 b / gγ µ A a µ Ω a n . (3)\n\nTo make explicit the non-Abelian Chern-Simons term we should expand this expression up to the third order in the gauge field:\n\nS ′ f [b, A] = S (2) f [b, A] + S (3) f [b, A] + . . . (4)\n\nwhere\n\nS (2) f [b, A] = ig 2 2 Tr[ 1 i∂ / -m -γ 5 b / γ µ A a µ Ω a 1 i∂ / -m -γ 5 b / γ ν A b ν Ω b ], (5)\n\nS (3) f [b, A] = ig 3 2 Tr[ 1 i∂ / -m -γ 5 b / γ µ A a µ Ω a 1 i∂ / -m -γ 5 b / γ λ A b λ Ω b 1 i∂ / -m -γ 5 b / γ ν A c ν Ω c ]. ( 6\n\n)\n\nand the elipsis stands for higher order terms in the gauge field.\n\nUsing the above expressions, it is easy now to verify that the one loop effective action expanded up to first order in b µ may be written as\n\nS ′ f [b, A] = d 4 x k ρ ǫ ρµνλ (∂ λ A a µ A a ν - 2 3 igA a µ A b ν A c λ f abc ) (7)\n\nwhere k ρ is\n\nk ρ = 2ig 2 d 4 p (2π) 4 b ρ (p 2 + 3m 2 ) -4p ρ (b • p) (p 2 -m 2 ) 3 . (8)\n\nThis result exactly reproduces the structure of the non-Abelian Lorentz-breaking Chern-Simons term described in [17] . One can observe that the expressions (7, 8) , after reduction to the Abelian case, coincide with the known Abelian results [14, 18] . Apparently, there is a relation between the induced Lorentz-breaking Chern-Simons term and Adler-Bell-Jackiw anomaly as both situation are observed for the well known triangle graph. This issue has been discursed in Ref. [6] . Also, the interesting discussion of the problem of ambiguities in the Lorentz-breaking theories is presented in [7] . By power counting, the momentum integral in expression (8) involves terms with logarithmic divergence so that different regularization prescriptions will produce diverse outcomes. Lorentz preserving regularizations, more precisely any regularization in which we can make: p µ p ν → gµν D p 2 , will produce finite results. By adopting the method of dimensional regularization [19] , the above integral is promoted to D dimensions and a straightforward calculation yields\n\nk ρ = 2ig 2 b ρ d D p (2π) D 1 (p 2 -m 2 ) 3 [(1 - 4 D )p 2 + 3m 2 ] = 4g 2 (4 -D) Γ((4 -D)/2) Γ(3)(4π) D/2 b ρ = g 2 4π 2 b ρ , (9)\n\nwhich coincides with the result found in [18] for the Abelian situation. If, instead of dimensional regularization, the integral in Eq. ( 8 ) is kept in four dimensions the regularization enforced the replacement\n\nk ρ = 6ig 2 m 2 b ρ d 4 p (2π) 4 1 (p 2 -m 2 ) 3 = 3g 2 16π 2 b ρ , (10)\n\nwhich now agrees with the Abelian result obtained in [20] .\n\nTo develop calculations in the finite temperature case, let us now assume that the system is in the state of thermal equilibrium at a temperature T = 1/β. In this case, we can use the Matsubara formalism for fermions, which consists in taking p 0 ≡ iω n = (n + 1/2) 2πi β and replacing the integration over the zeroth component of the momentum by a discrete sum (1/2π) dp 0 → i β n . Thus, the Eq. ( 7 ) can be written as\n\nS ′ f [b, A] = d 4 x k ρ (β)ǫ ρµνλ (∂ λ A a µ A a ν - 2 3 igA a µ A b ν A c λ f abc ). ( 11\n\n)\n\nHereafter all expressions are in the Euclidean space (all greek indices run from 1 to 4). The vector\n\nk ρ (β) is given by k ρ (β) = 2g 2 β ∞ n=-∞ d 3 p (2π) 3 b ρ (3m 2 -p 2 ) + 4p ρ (b • p) (p 2 + m 2 ) 3 . (12)\n\nBy extending the p integration to d dimensions it follows that the time-like component of k ρ (β) is\n\nk 4 (β) = 2g 2 β b 4 ∞ n=-∞ d d p (2π) d 3M 2 n -p 2 ( p 2 + M 2 n ) 3 , (13)\n\nwhere M 2 n = ω 2 n + m 2 . Using the prescription of dimensional regularization [19] , we have\n\nk 4 (β) = - g 2 2β b 4 (4π) d/2 [dΓ(2 -d/2) -6Γ(3 -d/2)] ∞ n=-∞ 1 (M 2 n ) 2-d/2 = g 2 b 4 m 3 π m 2 √ π d (a 2 ) λ-1/2 (3 -d) Γ(λ) ∞ n=-∞ 1 [(n + 1/2) 2 + a 2 ] λ , ( 14\n\n)\n\nwhere a = mβ/2π and λ = 2 -d/2. At this point the following identity [21] :\n\nn 1 [(n + b) 2 + a 2 ] λ = √ πΓ(λ -1/2) Γ(λ)(a 2 ) λ-1/2 + 4 sin(πλ) ∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 , (15)\n\nvalid for 1/2 < λ < 1 can be used to get\n\nk 4 (β) = g 2 b 4 m 3 π m 2 √ π d 2 √ π+ +4 (3 -d)(a 2 ) 3-d 2 Γ(λ) sin(πλ) ∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 . ( 16\n\n)\n\nk 4 (β) = g 2 4π 2 b 4 , (17)\n\ni.e., the same result ( 9 ) without any dependence on the temperature, which agrees with the one obtained in [22] for the Abelian situation. If instead of (12) we use (10) as the starting point for the computation of finite temperature effects we get\n\nk 4 (β) = b 4 ( 3 32π 2 + 3 16 F (a)), (18)\n\nwhere\n\nF (a) = ∞ |a| dz(z 2 -a 2 ) 1/2 tanh(πz) cosh 2 (πz) , (19)\n\nhas the following asymptotics: 0.0035 0.004 0.0045 0.005 0.0055 0.006 0 0.2 0.4 0.6 0.8 1 1.2 beta FIG. 1: The function F (a) is different from zero everywhere. At zero temperature (β → ∞), the function tends to a nonzero value 1 2π 2 .\n\nF (a → ∞) → 0 (T → 0) and F (a → 0) → 1/2π 2 (T → ∞), see\n\nLet us now consider the space part, k i (β), of the vector k ρ (β). In this case, the expression (12) can be rewritten as:\n\nk i (β) = 2g 2 β ∞ n=-∞ d 3 p (2π) 3 b i (3m 2 -p 2 ) + 4p i (b • p) (p 2 + m 2 ) 3 , (20)\n\nThen, considering this expression formally in d space dimensions, we can replace p i p j by p 2 d δ ij , hence we get\n\nk i (β) = 2g 2 β b i ∞ n=-∞ d d p (2π) d 4m 2 -( d-4 d ) p 2 -M 2 n ( p 2 + M 2 n ) 3 , (21)\n\nwhich now furnishes\n\nk i (β) = 4m 2 g 2 β b i Γ(3 -d 2 ) (4π) d/2 ∞ n=-∞ 1 (m 2 + ω 2 n ) 3-d 2 = = g 2 b i 2mπ 3 m 2 d/2 (a 2 ) λ-1/2 Γ(λ) ∞ n=-∞ 1 [(n + 1 2 ) 2 + a 2 ] λ , (22)\n\nwhere we have introduced λ = 3 -d 2 . We cannot apply the relation (15) for d = 3, because the integral in that expression does not converge. Thus, let us perform the analytic continuation of that relation; we obtain [13]\n\n∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 = 1 2a 2 3 -2λ 1 -λ ∞ |a| dz (z 2 -a 2 ) λ-1 × (23) × Re 1 exp 2π(z + ib) -1 - - 1 4a 2 1 (2 -λ)(1 -λ) ∞ |a| dz (z 2 -a 2 ) λ-2 d 2 dz 2 Re 1 exp 2π(z + ib) -1\n\n.\n\nThus for d = 3 the Eq. ( 22 ) takes the form\n\nk i (β) = b i ( 1 4π 2 + 1 2 F (a)), (24)\n\nwhere F (a) was defined in (19) . Thus, we see that at high temperature the Chern-Simons coefficient is twice its value at zero temperature, i.e., k i (β → 0) = 1 2π 2 . On the other hand, at zero temperature, one recovers the result k i (β → ∞) = 1 4π 2 . We have generated the non-Abelian Lorentz-breaking Chern-Simons term via the Lorentzbreaking coupling of the Yang-Mills field with the spinor field at zero and at finite temperature.\n\nThe essential property of the result is that within the framework of dimensional regularization this term turns out to be finite. We note that the derivative expansion approach naturally allows to preserve the gauge invariance for the quantum corrections. It is natural to expect that at least some of other Lorentz-breaking terms which existence was predicted in [4] also can be generated via appropriate couplings of the gauge or gravity fields with some matter fields.\n\nWe have also obtained the coefficient k ρ for the non-Abelian Lorentz-breaking Chern-Simons term at the finite temperature. We found that the results for this term turn out to be dependent on the regularization scheme both at zero and at finite temperature (in a particular regularization scheme the time-like component was found to be temperature independent). Considering the dependence on the regularization scheme, one should note that the momentum integral determining the value of the vector k ρ is formally superficially divergent, thus dependence of its finite part on the renormalization procedure is very natural. However, in the regularization schemes suggested in the paper the divergent part identically disappears as a consequence of the rotational invariance of the relevant integrands."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is generated as a one-loop perturbative correction via an appropriate Lorentz-breaking coupling of the non-Abelian gauge field to a spinor field. This term is shown to be regularization dependent but nevertheless it can be found unambiguously in different regularization schemes at zero and at finite temperature."},{"section_type":"OTHER","section_title":"Untitled Section","text":"During the last years, different aspects of the Lorentz symmetry breaking have been intensively studied [1] . One of the theoretical consequences of this effect is the birefringence of light in vacuum. After the formulation of the concept of noncommutativity of the space-time [2] , which implies in Lorentz symmetry breaking (see the discussion in [3] ), The interest in this subject has greatly increased . One of the implications of the Lorentz symmetry breaking is the possibility of introducing a lot of new couplings in the Standard Model [4] . These terms may arise from radiative corrections to some Lorentz-breaking field theories at zero [5, 6, 7, 8, 9, 10] and at finite temperature [11, 12, 13, 14] . Alternatively, they may be induced from the deformation of the canonical commutation relation through the use of the noncommutative fields method [15, 16] .\n\nRecently, the renormalizability of the Yang Mills (YM) theory with a four-dimensional non-Abelian Lorentz-breaking Chern-Simons (CS) term was studied in [17] . The induction of such Lorentz-breaking CS term starting from a pure YM was investigated within the noncommutative fields method in [16] . In the present work we show how the same CS term can be induced through radiative corrections starting from a YM theory coupled with fermions in the presence of an interaction of the fermions with a constant external field at zero and at finite temperture.\n\nWe start with the following model which represents a non-Abelian generalization of the spinor electrodynamics with the Lorentz-breaking coupling\n\nL f = ψ i (i∂ / -m -γ 5 b /)δ ij -gγ µ A a µ (Ω a ) ij ψ j . (1)\n\nHere b ρ is a constant four-vector. The A µ = A a µ Ω a is a Yang-Mills field coupled to spinors ψ which carry the isotopic indices, ψ = (ψ i ), with i taking values from 1 to N with N being the dimension of the chosen representation of the Lie algebra. The Ω a = (Ω a ) ij are the Lie group generators in this representation satisfying the relations:\n\n[Ω a , Ω b ] = if abc Ω c and Tr[Ω a Ω b ] = δ ab .\n\nThe one-loop effective action of the gauge field\n\nA a µ is S Y M +S f [b, A]\n\nwhere S Y M is the Yang-Mills action and S f [b, A] can be expressed in the form of the following functional trace:\n\nS f [b, A] = -i Tr ln(i∂ / -m -γ 5 b / -gγ µ A a µ Ω a ). (2)\n\nThis functional trace can be rewritten as\n\nS f [b, A] = S f [b] + S ′ f [b, A], with the first term being S f [b] = -i Tr ln(i∂ / -m -γ 5 b /)\n\n. The nontrivial dynamics is concentrated in the second term\n\nS ′ f [b, A],\n\nwhich is given by the power series:\n\nS ′ f [b, A] = i Tr ∞ n=1 1 n 1 i∂ / -m -γ 5 b / gγ µ A a µ Ω a n . (3)\n\nTo make explicit the non-Abelian Chern-Simons term we should expand this expression up to the third order in the gauge field:\n\nS ′ f [b, A] = S (2) f [b, A] + S (3) f [b, A] + . . . (4)\n\nwhere\n\nS (2) f [b, A] = ig 2 2 Tr[ 1 i∂ / -m -γ 5 b / γ µ A a µ Ω a 1 i∂ / -m -γ 5 b / γ ν A b ν Ω b ], (5)\n\nS (3) f [b, A] = ig 3 2 Tr[ 1 i∂ / -m -γ 5 b / γ µ A a µ Ω a 1 i∂ / -m -γ 5 b / γ λ A b λ Ω b 1 i∂ / -m -γ 5 b / γ ν A c ν Ω c ]. ( 6\n\n)\n\nand the elipsis stands for higher order terms in the gauge field.\n\nUsing the above expressions, it is easy now to verify that the one loop effective action expanded up to first order in b µ may be written as\n\nS ′ f [b, A] = d 4 x k ρ ǫ ρµνλ (∂ λ A a µ A a ν - 2 3 igA a µ A b ν A c λ f abc ) (7)\n\nwhere k ρ is\n\nk ρ = 2ig 2 d 4 p (2π) 4 b ρ (p 2 + 3m 2 ) -4p ρ (b • p) (p 2 -m 2 ) 3 . (8)\n\nThis result exactly reproduces the structure of the non-Abelian Lorentz-breaking Chern-Simons term described in [17] . One can observe that the expressions (7, 8) , after reduction to the Abelian case, coincide with the known Abelian results [14, 18] . Apparently, there is a relation between the induced Lorentz-breaking Chern-Simons term and Adler-Bell-Jackiw anomaly as both situation are observed for the well known triangle graph. This issue has been discursed in Ref. [6] . Also, the interesting discussion of the problem of ambiguities in the Lorentz-breaking theories is presented in [7] . By power counting, the momentum integral in expression (8) involves terms with logarithmic divergence so that different regularization prescriptions will produce diverse outcomes. Lorentz preserving regularizations, more precisely any regularization in which we can make: p µ p ν → gµν D p 2 , will produce finite results. By adopting the method of dimensional regularization [19] , the above integral is promoted to D dimensions and a straightforward calculation yields\n\nk ρ = 2ig 2 b ρ d D p (2π) D 1 (p 2 -m 2 ) 3 [(1 - 4 D )p 2 + 3m 2 ] = 4g 2 (4 -D) Γ((4 -D)/2) Γ(3)(4π) D/2 b ρ = g 2 4π 2 b ρ , (9)\n\nwhich coincides with the result found in [18] for the Abelian situation. If, instead of dimensional regularization, the integral in Eq. ( 8 ) is kept in four dimensions the regularization enforced the replacement\n\nk ρ = 6ig 2 m 2 b ρ d 4 p (2π) 4 1 (p 2 -m 2 ) 3 = 3g 2 16π 2 b ρ , (10)\n\nwhich now agrees with the Abelian result obtained in [20] .\n\nTo develop calculations in the finite temperature case, let us now assume that the system is in the state of thermal equilibrium at a temperature T = 1/β. In this case, we can use the Matsubara formalism for fermions, which consists in taking p 0 ≡ iω n = (n + 1/2) 2πi β and replacing the integration over the zeroth component of the momentum by a discrete sum (1/2π) dp 0 → i β n . Thus, the Eq. ( 7 ) can be written as\n\nS ′ f [b, A] = d 4 x k ρ (β)ǫ ρµνλ (∂ λ A a µ A a ν - 2 3 igA a µ A b ν A c λ f abc ). ( 11\n\n)\n\nHereafter all expressions are in the Euclidean space (all greek indices run from 1 to 4). The vector\n\nk ρ (β) is given by k ρ (β) = 2g 2 β ∞ n=-∞ d 3 p (2π) 3 b ρ (3m 2 -p 2 ) + 4p ρ (b • p) (p 2 + m 2 ) 3 . (12)\n\nBy extending the p integration to d dimensions it follows that the time-like component of k ρ (β) is\n\nk 4 (β) = 2g 2 β b 4 ∞ n=-∞ d d p (2π) d 3M 2 n -p 2 ( p 2 + M 2 n ) 3 , (13)\n\nwhere M 2 n = ω 2 n + m 2 . Using the prescription of dimensional regularization [19] , we have\n\nk 4 (β) = - g 2 2β b 4 (4π) d/2 [dΓ(2 -d/2) -6Γ(3 -d/2)] ∞ n=-∞ 1 (M 2 n ) 2-d/2 = g 2 b 4 m 3 π m 2 √ π d (a 2 ) λ-1/2 (3 -d) Γ(λ) ∞ n=-∞ 1 [(n + 1/2) 2 + a 2 ] λ , ( 14\n\n)\n\nwhere a = mβ/2π and λ = 2 -d/2. At this point the following identity [21] :\n\nn 1 [(n + b) 2 + a 2 ] λ = √ πΓ(λ -1/2) Γ(λ)(a 2 ) λ-1/2 + 4 sin(πλ) ∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 , (15)\n\nvalid for 1/2 < λ < 1 can be used to get\n\nk 4 (β) = g 2 b 4 m 3 π m 2 √ π d 2 √ π+ +4 (3 -d)(a 2 ) 3-d 2 Γ(λ) sin(πλ) ∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 . ( 16\n\n)\n\nk 4 (β) = g 2 4π 2 b 4 , (17)\n\ni.e., the same result ( 9 ) without any dependence on the temperature, which agrees with the one obtained in [22] for the Abelian situation. If instead of (12) we use (10) as the starting point for the computation of finite temperature effects we get\n\nk 4 (β) = b 4 ( 3 32π 2 + 3 16 F (a)), (18)\n\nwhere\n\nF (a) = ∞ |a| dz(z 2 -a 2 ) 1/2 tanh(πz) cosh 2 (πz) , (19)\n\nhas the following asymptotics: 0.0035 0.004 0.0045 0.005 0.0055 0.006 0 0.2 0.4 0.6 0.8 1 1.2 beta FIG. 1: The function F (a) is different from zero everywhere. At zero temperature (β → ∞), the function tends to a nonzero value 1 2π 2 .\n\nF (a → ∞) → 0 (T → 0) and F (a → 0) → 1/2π 2 (T → ∞), see\n\nLet us now consider the space part, k i (β), of the vector k ρ (β). In this case, the expression (12) can be rewritten as:\n\nk i (β) = 2g 2 β ∞ n=-∞ d 3 p (2π) 3 b i (3m 2 -p 2 ) + 4p i (b • p) (p 2 + m 2 ) 3 , (20)\n\nThen, considering this expression formally in d space dimensions, we can replace p i p j by p 2 d δ ij , hence we get\n\nk i (β) = 2g 2 β b i ∞ n=-∞ d d p (2π) d 4m 2 -( d-4 d ) p 2 -M 2 n ( p 2 + M 2 n ) 3 , (21)\n\nwhich now furnishes\n\nk i (β) = 4m 2 g 2 β b i Γ(3 -d 2 ) (4π) d/2 ∞ n=-∞ 1 (m 2 + ω 2 n ) 3-d 2 = = g 2 b i 2mπ 3 m 2 d/2 (a 2 ) λ-1/2 Γ(λ) ∞ n=-∞ 1 [(n + 1 2 ) 2 + a 2 ] λ , (22)\n\nwhere we have introduced λ = 3 -d 2 . We cannot apply the relation (15) for d = 3, because the integral in that expression does not converge. Thus, let us perform the analytic continuation of that relation; we obtain [13]\n\n∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 = 1 2a 2 3 -2λ 1 -λ ∞ |a| dz (z 2 -a 2 ) λ-1 × (23) × Re 1 exp 2π(z + ib) -1 - - 1 4a 2 1 (2 -λ)(1 -λ) ∞ |a| dz (z 2 -a 2 ) λ-2 d 2 dz 2 Re 1 exp 2π(z + ib) -1\n\n.\n\nThus for d = 3 the Eq. ( 22 ) takes the form\n\nk i (β) = b i ( 1 4π 2 + 1 2 F (a)), (24)\n\nwhere F (a) was defined in (19) . Thus, we see that at high temperature the Chern-Simons coefficient is twice its value at zero temperature, i.e., k i (β → 0) = 1 2π 2 . On the other hand, at zero temperature, one recovers the result k i (β → ∞) = 1 4π 2 . We have generated the non-Abelian Lorentz-breaking Chern-Simons term via the Lorentzbreaking coupling of the Yang-Mills field with the spinor field at zero and at finite temperature.\n\nThe essential property of the result is that within the framework of dimensional regularization this term turns out to be finite. We note that the derivative expansion approach naturally allows to preserve the gauge invariance for the quantum corrections. It is natural to expect that at least some of other Lorentz-breaking terms which existence was predicted in [4] also can be generated via appropriate couplings of the gauge or gravity fields with some matter fields.\n\nWe have also obtained the coefficient k ρ for the non-Abelian Lorentz-breaking Chern-Simons term at the finite temperature. We found that the results for this term turn out to be dependent on the regularization scheme both at zero and at finite temperature (in a particular regularization scheme the time-like component was found to be temperature independent). Considering the dependence on the regularization scheme, one should note that the momentum integral determining the value of the vector k ρ is formally superficially divergent, thus dependence of its finite part on the renormalization procedure is very natural. However, in the regularization schemes suggested in the paper the divergent part identically disappears as a consequence of the rotational invariance of the relevant integrands."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is generated as a one-loop perturbative correction via an appropriate Lorentz-breaking coupling of the non-Abelian gauge field to a spinor field. This term is shown to be regularization dependent but nevertheless it can be found unambiguously in different regularization schemes at zero and at finite temperature.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"During the last years, different aspects of the Lorentz symmetry breaking have been intensively studied [1] . One of the theoretical consequences of this effect is the birefringence of light in vacuum. After the formulation of the concept of noncommutativity of the space-time [2] , which implies in Lorentz symmetry breaking (see the discussion in [3] ), The interest in this subject has greatly increased . One of the implications of the Lorentz symmetry breaking is the possibility of introducing a lot of new couplings in the Standard Model [4] . These terms may arise from radiative corrections to some Lorentz-breaking field theories at zero [5, 6, 7, 8, 9, 10] and at finite temperature [11, 12, 13, 14] . Alternatively, they may be induced from the deformation of the canonical commutation relation through the use of the noncommutative fields method [15, 16] .\\n\\nRecently, the renormalizability of the Yang Mills (YM) theory with a four-dimensional non-Abelian Lorentz-breaking Chern-Simons (CS) term was studied in [17] . The induction of such Lorentz-breaking CS term starting from a pure YM was investigated within the noncommutative fields method in [16] . In the present work we show how the same CS term can be induced through radiative corrections starting from a YM theory coupled with fermions in the presence of an interaction of the fermions with a constant external field at zero and at finite temperture.\\n\\nWe start with the following model which represents a non-Abelian generalization of the spinor electrodynamics with the Lorentz-breaking coupling\\n\\nL f = ψ i (i∂ / -m -γ 5 b /)δ ij -gγ µ A a µ (Ω a ) ij ψ j . (1)\\n\\nHere b ρ is a constant four-vector. The A µ = A a µ Ω a is a Yang-Mills field coupled to spinors ψ which carry the isotopic indices, ψ = (ψ i ), with i taking values from 1 to N with N being the dimension of the chosen representation of the Lie algebra. The Ω a = (Ω a ) ij are the Lie group generators in this representation satisfying the relations:\\n\\n[Ω a , Ω b ] = if abc Ω c and Tr[Ω a Ω b ] = δ ab .\\n\\nThe one-loop effective action of the gauge field\\n\\nA a µ is S Y M +S f [b, A]\\n\\nwhere S Y M is the Yang-Mills action and S f [b, A] can be expressed in the form of the following functional trace:\\n\\nS f [b, A] = -i Tr ln(i∂ / -m -γ 5 b / -gγ µ A a µ Ω a ). (2)\\n\\nThis functional trace can be rewritten as\\n\\nS f [b, A] = S f [b] + S ′ f [b, A], with the first term being S f [b] = -i Tr ln(i∂ / -m -γ 5 b /)\\n\\n. The nontrivial dynamics is concentrated in the second term\\n\\nS ′ f [b, A],\\n\\nwhich is given by the power series:\\n\\nS ′ f [b, A] = i Tr ∞ n=1 1 n 1 i∂ / -m -γ 5 b / gγ µ A a µ Ω a n . (3)\\n\\nTo make explicit the non-Abelian Chern-Simons term we should expand this expression up to the third order in the gauge field:\\n\\nS ′ f [b, A] = S (2) f [b, A] + S (3) f [b, A] + . . . (4)\\n\\nwhere\\n\\nS (2) f [b, A] = ig 2 2 Tr[ 1 i∂ / -m -γ 5 b / γ µ A a µ Ω a 1 i∂ / -m -γ 5 b / γ ν A b ν Ω b ], (5)\\n\\nS (3) f [b, A] = ig 3 2 Tr[ 1 i∂ / -m -γ 5 b / γ µ A a µ Ω a 1 i∂ / -m -γ 5 b / γ λ A b λ Ω b 1 i∂ / -m -γ 5 b / γ ν A c ν Ω c ]. ( 6\\n\\n)\\n\\nand the elipsis stands for higher order terms in the gauge field.\\n\\nUsing the above expressions, it is easy now to verify that the one loop effective action expanded up to first order in b µ may be written as\\n\\nS ′ f [b, A] = d 4 x k ρ ǫ ρµνλ (∂ λ A a µ A a ν - 2 3 igA a µ A b ν A c λ f abc ) (7)\\n\\nwhere k ρ is\\n\\nk ρ = 2ig 2 d 4 p (2π) 4 b ρ (p 2 + 3m 2 ) -4p ρ (b • p) (p 2 -m 2 ) 3 . (8)\\n\\nThis result exactly reproduces the structure of the non-Abelian Lorentz-breaking Chern-Simons term described in [17] . One can observe that the expressions (7, 8) , after reduction to the Abelian case, coincide with the known Abelian results [14, 18] . Apparently, there is a relation between the induced Lorentz-breaking Chern-Simons term and Adler-Bell-Jackiw anomaly as both situation are observed for the well known triangle graph. This issue has been discursed in Ref. [6] . Also, the interesting discussion of the problem of ambiguities in the Lorentz-breaking theories is presented in [7] . By power counting, the momentum integral in expression (8) involves terms with logarithmic divergence so that different regularization prescriptions will produce diverse outcomes. Lorentz preserving regularizations, more precisely any regularization in which we can make: p µ p ν → gµν D p 2 , will produce finite results. By adopting the method of dimensional regularization [19] , the above integral is promoted to D dimensions and a straightforward calculation yields\\n\\nk ρ = 2ig 2 b ρ d D p (2π) D 1 (p 2 -m 2 ) 3 [(1 - 4 D )p 2 + 3m 2 ] = 4g 2 (4 -D) Γ((4 -D)/2) Γ(3)(4π) D/2 b ρ = g 2 4π 2 b ρ , (9)\\n\\nwhich coincides with the result found in [18] for the Abelian situation. If, instead of dimensional regularization, the integral in Eq. ( 8 ) is kept in four dimensions the regularization enforced the replacement\\n\\nk ρ = 6ig 2 m 2 b ρ d 4 p (2π) 4 1 (p 2 -m 2 ) 3 = 3g 2 16π 2 b ρ , (10)\\n\\nwhich now agrees with the Abelian result obtained in [20] .\\n\\nTo develop calculations in the finite temperature case, let us now assume that the system is in the state of thermal equilibrium at a temperature T = 1/β. In this case, we can use the Matsubara formalism for fermions, which consists in taking p 0 ≡ iω n = (n + 1/2) 2πi β and replacing the integration over the zeroth component of the momentum by a discrete sum (1/2π) dp 0 → i β n . Thus, the Eq. ( 7 ) can be written as\\n\\nS ′ f [b, A] = d 4 x k ρ (β)ǫ ρµνλ (∂ λ A a µ A a ν - 2 3 igA a µ A b ν A c λ f abc ). ( 11\\n\\n)\\n\\nHereafter all expressions are in the Euclidean space (all greek indices run from 1 to 4). The vector\\n\\nk ρ (β) is given by k ρ (β) = 2g 2 β ∞ n=-∞ d 3 p (2π) 3 b ρ (3m 2 -p 2 ) + 4p ρ (b • p) (p 2 + m 2 ) 3 . (12)\\n\\nBy extending the p integration to d dimensions it follows that the time-like component of k ρ (β) is\\n\\nk 4 (β) = 2g 2 β b 4 ∞ n=-∞ d d p (2π) d 3M 2 n -p 2 ( p 2 + M 2 n ) 3 , (13)\\n\\nwhere M 2 n = ω 2 n + m 2 . Using the prescription of dimensional regularization [19] , we have\\n\\nk 4 (β) = - g 2 2β b 4 (4π) d/2 [dΓ(2 -d/2) -6Γ(3 -d/2)] ∞ n=-∞ 1 (M 2 n ) 2-d/2 = g 2 b 4 m 3 π m 2 √ π d (a 2 ) λ-1/2 (3 -d) Γ(λ) ∞ n=-∞ 1 [(n + 1/2) 2 + a 2 ] λ , ( 14\\n\\n)\\n\\nwhere a = mβ/2π and λ = 2 -d/2. At this point the following identity [21] :\\n\\nn 1 [(n + b) 2 + a 2 ] λ = √ πΓ(λ -1/2) Γ(λ)(a 2 ) λ-1/2 + 4 sin(πλ) ∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 , (15)\\n\\nvalid for 1/2 < λ < 1 can be used to get\\n\\nk 4 (β) = g 2 b 4 m 3 π m 2 √ π d 2 √ π+ +4 (3 -d)(a 2 ) 3-d 2 Γ(λ) sin(πλ) ∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 . ( 16\\n\\n)\\n\\nk 4 (β) = g 2 4π 2 b 4 , (17)\\n\\ni.e., the same result ( 9 ) without any dependence on the temperature, which agrees with the one obtained in [22] for the Abelian situation. If instead of (12) we use (10) as the starting point for the computation of finite temperature effects we get\\n\\nk 4 (β) = b 4 ( 3 32π 2 + 3 16 F (a)), (18)\\n\\nwhere\\n\\nF (a) = ∞ |a| dz(z 2 -a 2 ) 1/2 tanh(πz) cosh 2 (πz) , (19)\\n\\nhas the following asymptotics: 0.0035 0.004 0.0045 0.005 0.0055 0.006 0 0.2 0.4 0.6 0.8 1 1.2 beta FIG. 1: The function F (a) is different from zero everywhere. At zero temperature (β → ∞), the function tends to a nonzero value 1 2π 2 .\\n\\nF (a → ∞) → 0 (T → 0) and F (a → 0) → 1/2π 2 (T → ∞), see\\n\\nLet us now consider the space part, k i (β), of the vector k ρ (β). In this case, the expression (12) can be rewritten as:\\n\\nk i (β) = 2g 2 β ∞ n=-∞ d 3 p (2π) 3 b i (3m 2 -p 2 ) + 4p i (b • p) (p 2 + m 2 ) 3 , (20)\\n\\nThen, considering this expression formally in d space dimensions, we can replace p i p j by p 2 d δ ij , hence we get\\n\\nk i (β) = 2g 2 β b i ∞ n=-∞ d d p (2π) d 4m 2 -( d-4 d ) p 2 -M 2 n ( p 2 + M 2 n ) 3 , (21)\\n\\nwhich now furnishes\\n\\nk i (β) = 4m 2 g 2 β b i Γ(3 -d 2 ) (4π) d/2 ∞ n=-∞ 1 (m 2 + ω 2 n ) 3-d 2 = = g 2 b i 2mπ 3 m 2 d/2 (a 2 ) λ-1/2 Γ(λ) ∞ n=-∞ 1 [(n + 1 2 ) 2 + a 2 ] λ , (22)\\n\\nwhere we have introduced λ = 3 -d 2 . We cannot apply the relation (15) for d = 3, because the integral in that expression does not converge. Thus, let us perform the analytic continuation of that relation; we obtain [13]\\n\\n∞ |a| dz (z 2 -a 2 ) λ Re 1 exp 2π(z + ib) -1 = 1 2a 2 3 -2λ 1 -λ ∞ |a| dz (z 2 -a 2 ) λ-1 × (23) × Re 1 exp 2π(z + ib) -1 - - 1 4a 2 1 (2 -λ)(1 -λ) ∞ |a| dz (z 2 -a 2 ) λ-2 d 2 dz 2 Re 1 exp 2π(z + ib) -1\\n\\n.\\n\\nThus for d = 3 the Eq. ( 22 ) takes the form\\n\\nk i (β) = b i ( 1 4π 2 + 1 2 F (a)), (24)\\n\\nwhere F (a) was defined in (19) . Thus, we see that at high temperature the Chern-Simons coefficient is twice its value at zero temperature, i.e., k i (β → 0) = 1 2π 2 . On the other hand, at zero temperature, one recovers the result k i (β → ∞) = 1 4π 2 . We have generated the non-Abelian Lorentz-breaking Chern-Simons term via the Lorentzbreaking coupling of the Yang-Mills field with the spinor field at zero and at finite temperature.\\n\\nThe essential property of the result is that within the framework of dimensional regularization this term turns out to be finite. We note that the derivative expansion approach naturally allows to preserve the gauge invariance for the quantum corrections. It is natural to expect that at least some of other Lorentz-breaking terms which existence was predicted in [4] also can be generated via appropriate couplings of the gauge or gravity fields with some matter fields.\\n\\nWe have also obtained the coefficient k ρ for the non-Abelian Lorentz-breaking Chern-Simons term at the finite temperature. We found that the results for this term turn out to be dependent on the regularization scheme both at zero and at finite temperature (in a particular regularization scheme the time-like component was found to be temperature independent). Considering the dependence on the regularization scheme, one should note that the momentum integral determining the value of the vector k ρ is formally superficially divergent, thus dependence of its finite part on the renormalization procedure is very natural. However, in the regularization schemes suggested in the paper the divergent part identically disappears as a consequence of the rotational invariance of the relevant integrands.\"\n }\n]","isConversational":false}}},{"rowIdx":198,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1109"},"arxiv_id":{"kind":"string","value":"0704.1109"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1007/s10773-009-0204-x"},"content_hash":{"kind":"string","value":"0150dbf53a2658b7aa805f3b8407fa9fb78488a4b5c888bd44f7c9a11f6c9fcb"},"title":{"kind":"string","value":"Spinning Strings, Black Holes and Stable Closed Timelike Geodesics"},"abstract":{"kind":"string","value":"The existence and stability under linear perturbation of closed timelike\ncurves in the spacetime associated to Schwarzschild black hole pierced by a\nspinning string are studied. Due to the superposition of the black hole, we\nfind that the spinning string spacetime is deformed in such a way to allow the\nexistence of closed timelike geodesics."},"authors":{"kind":"list like","value":["Valeria M. Rosa and Patricio S. letelier"],"string":"[\n \"Valeria M. Rosa and Patricio S. letelier\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["gr-qc"],"string":"[\n \"gr-qc\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"gr-qc"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the superposition of the black hole, we find that the spinning string spacetime is deformed in such a way to allow the existence of closed timelike geodesics.\n\nThe existance of closed timelike curves (CTCs) in the Gödel universe and other apacetimes is a worrying fact since these curves show a clear violation of causality. In some cases these CTCs can be disregarded by energy considerations. Their existance requires an external force acting along the whole CTC, process that may consume a great amount of energy. The energy needed to travel along a CTC in Gödel's universe is computed in [1] . When the external force is null the energy needed to travel is also null. Therefore, in principle, the existence of closed timelike geodesics (CTGs) presents a bigger problem of breakdown of causality.\n\nThe classical problem of the existence of closed geodesics in Riemannian geometry was solved by Hadamard [2] in two dimensions and by Cartan [3] in an arbitrary number of dimensions. As a topological problem, the existence of CTGs was proved by Tipler [4] in a class of four-dimensional compact Lorentz manifolds with covering space containing a compact Cauchy surface. In a compact pseudo-Riemaniann manifold with Lorentzian signature (Lorentzian manifold) Galloway [5] found sufficient conditions to have CTGs, see also [6] .\n\nTo the best of our knowledge there are four solution to the Einstein equations generated by matter with positive mass density that contain CTGs: a) Soares [7] found a class of cosmological models, solutions of Einstein-Maxwell equations, with a subclass where the timelike paths of matter are closed. For these models the existence of CTGs is demonstrated and explicit examples are given. These CTGs are not linearly stable [8] . b) Steadman [9] described the behavior of CTGs in a vacuum exterior for the van Stockum solution that represents an infinite rotating dust cylinder. For this solution explicit examples of CTCs and CTGs are shown. There are stable CTGs in this spacetime [8] . c) Bonnor and Steadman [10] studied the existence of CTGs in a spacetime with two spinning particles each one with magnetic moment equal to angular moment and mass equal to charge (Perjeons), in particular, they present a explicit CTG. This particular CTG is not stable, but there exist many other that are stable [11] . d) There are linearly stable CTGs [8] in one of the Gödel-type metrics with not flat background studied by Gürses et al. [12][13] . For CTGs in a spacetime associated to a cloud of strings with negative mass density see [14] . These CTGs are not stable [8] .\n\nThe existence of CTCs in a spacetime whose source is a spinning string has been investigated by many authors (see for instance [15] - [19] ). The interpretation of these strings as torsion line defects can be found in [20] , [21] , see also [22] [23] . These torsion line defects appear when one tries to stabilize two rotating black holes kept apart by spin repulsion [24] . Also, the black hole thermodynamics associated to a static black hole pierced by a non rotating string was studied some time ago by Aryal et al. [25] .\n\nIn the present work we study the existence and stability of CTCs under linear perturbations in the spacetime associated to Schwarzschild black hole (BH) pierced by a spinning string. Even though this spacetime is more a mathematical curiosity than an example of a real spacetime we believe that the study of stability of CTCs and CTGs can shed some light into the existence of this rather pathological curves. In particular, we study sufficient conditions to have linearly stable CTGs. We find that these conditions are not very restrictive and can be easily satisfied. Furthermore, we compared them with the same conditions studied by Galloway [5] for a compact Lorentzian manifold.\n\nLet us consider the spacetime with metric,\n\nds 2 = (1- 2m r )(dt-αdϕ) 2 - dr 2 1 -2m r -r 2 (dθ 2 +β 2 sin 2 θdϕ 2 ), (1)\n\nwhere α = 4S and S is the string's spin angular momentum per unit of length, β = 1 -4λ and λ is the string's linear mass density that is equal to its tension (λ ≤ 1/4).\n\nIn the particular case, α = 0 and β = 1, the metric (1) reduces to the Schwarzschild solution. When m = 0, Eq. ( 1 ) represents a spinning string, with the further special-ization β = 1 (not deficit angle) we have a pure massless torsion line defect [20] [21] . Therefore the metric (1) can be considered as representing the spacetime associated to a Schwarzschild black hole pierced by a spinning string.\n\nLet us denote by γ a closed curve given in its parametric form by,\n\nt = t 0 , r = r 0 , ϕ ∈ [0, 2π], θ = π 2 , (2)\n\nwhere t 0 and r 0 are constants. When γ is parametrized with an arbitrary parameter σ, we have a timelike curve when dx µ dσ dxµ dσ > 0. This condition reduces to g ϕϕ > 0, i.e.,\n\n(1 -2m/r 0 )α 2 -r 2 0 β 2 > 0. (3)\n\nA generic CTC γ satisfies the system of equations given by\n\nẍµ + Γ µ αβ ẋα ẋβ = F µ (x), (4)\n\nwhere the overdot indicates derivation with respect to s, Γ µ αβ are the Christoffel symbols and F µ is a specific external force (a µ = F µ ). The nonzero component of the four-acceleration of γ is\n\na r = 1 r 3 0 (r 0 -2m)(α 2 m -r 3 0 β 2 ) φ2 . (5)\n\nOur goal is to study the behavior of closed timelike geodesics. Therefore taking α as one of the two solutions of\n\nα 2 m -r 3 0 β 2 = 0, (6)\n\nwe have a r = 0. Under this condition (3) is satisfied when r 0 > 3 m, that put the CTG outside the black hole. Let γ be the curve obtained from γ after a small perturbation ξ, i.e., xµ = x µ + ξ µ . From equations (4) one finds that the system of differential equations satisfied by the perturbation ξ is [26] ,\n\nd 2 ξ α ds 2 + 2Γ α βµ dξ β ds u µ + Γ α βµ,λ ξ λ u β u µ = F α ,λ ξ λ . ( 7\n\n)\n\nFor the above mentioned closed timelike geodesic the system (7)\n\nreduces to ξ0 + k 1 ξ1 = 0, (8) ξ1 + k 2 ξ0 + k 3 ξ 1 = 0, (9) ξ2 + k 4 ξ1 = 0, ( 10\n\n) ξ3 + k 5 ξ 3 = 0, (11)\n\nwhere\n\nk 1 = 2Γ 0 21 φ, k 2 = 2Γ 1 20 φ, k 3 = Γ 1 22,1 φ2 , k 4 = 2Γ 2 21 φ, k 5 = Γ 3 22,3 φ2 . (12)\n\nA curve γ parametrized by the proper time, s, is timelike when ẋµ ẋµ = 1. For the curve γ(s) we have that this last condition gives us,\n\nφ2 = m β 2 r 2 0 (r 0 -3m) . ( 13\n\n)\n\nThe solution of ( 9 )-( 11 ) is\n\nξ 0 = -k 1 (c 3 sin(ωs + c 4 )/ω + λs) + c 1 s + c 5 , ξ 1 = c 3 cos(ωs + c 4 ) + λ, ξ 2 = -k 4 (c 3 sin(ωs + c 4 )/ω + λs) + c 2 s + c 6 , ξ 3 = c 7 cos( √ k 5 s + c 8 ), (14)\n\nwhere c i , i = 1, . . . , 8 are integration constants,\n\nω = k 3 -k 1 k 2 = [β 2 (r 0 -6m) φ2 /r 0 ] 1/2 , (15)\n\nand λ = -k 2 c 1 /ω 2 . Thus when r 0 > 6 m, the constant ω is real and the solution (14) shows the typical behavior for stability, i.e., vibrational modes untangled with translational ones that can be eliminated by a suitable choice of the initial conditions. When the black hole is removed, we are left with the spacetime of the spinning string whose line element is,\n\nds 2 = (dt -αdϕ) 2 -dr 2 -r 2 (dθ 2 + β 2 sin 2 θdϕ 2 ). ( 16\n\n)\n\nThe closed curve, γ, is timelike when α 2 -r 2 0 β 2 > 0. The a r -component of the four-acceleration is given by a r = -β 2 r 0 φ2 . Thus for r < |α/β| we have closed timelike curves, which are not geodesics.\n\nFor the closed curve (2) the system (7) is written now as in ( 9 )- (11) replacing equation (10) by\n\nξ1 + k 2 ξ2 + k 3 ξ 1 = ∂ r (Γ 1 22 φ2 )ξ 1 , (17)\n\nwhere now k 2 = 2Γ 1 22 φ and φ2 = (α 2 -r 2 0 β 2 ) -1 . In this particular case the solution of (7) has the same form that (14) with ω 2 = 2β 2 φ2 (2 + β 2 r 2 0 φ2 ). Therefore, the CTCs are stable.\n\nIn summary, there exist linearly stable CTCs in the spacetime related to a spinning string and these curves are restricted to a small region of the spacetime. Closed timelike geodesics do not exist in this spacetime.\n\nFor the nonlinear superposition of a spinning string with a Schwarzschild black hole the new spacetime has linearly stable CTGs. The region of stability is the same of the usual circular geodesics in the Schwarzschild black hole alone. The presence of the spinning string does not affect the stability of the orbits. It seems that torsion lines defects superposed to matter (not strings, β = 1) is a main ingredient to have stable CTGs. Loosely speaking, we have that a torsion line defect alone makes possible the existence of CTCs. When the black hole is present the spinning string spacetime is deformed in such a way to allow the existence of a CTG. This fact is also confirmed in the case of the two Perjeons solutions studied\n\n0 5 10 15 20 25 30 35 0 50 100 150 200 m= 0.3; β= 0.9; α= 25; r m = 6.1401 r 0 s(r 0 ) 0 5 10 15 20 25 30 35 0 50 100 150 200 m= 1; β= 0.9; α= 25; r m = 9.172 r 0 s(r 0 ) 0 5 10 15 20 25 30 35 0 50 100 150 200 m= 4; β= 0.9; α= 25; r m = 14.5597 r 0 s(r 0 )\n\nFIG. 1: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the presence of the mass shift the maximum of s(r0) for the string that is located at r0 = 0 to a position outside the black hole horizon. The maximum, rm represent the radius of the CTGs, the first two are stable and the second is not.\n\nin [10] wherein the torsion line defect is a main ingredient to have CTCs and CTGs.\n\nIt is instructive to look the previous results in a more direct and graphic way. The length of CTC in (2) only depends on the value of r = r 0 . We find,\n\ns(r 0 ) = 2π g ϕϕ (r 0 ), = 2π[(1 -2m/r 0 )α 2 -r 2 0 β 2 ] 1/2 . ( 18\n\n)\n\nThis function has a local maximum for\n\nr m = (mα 2 /β 2 ) 1/3 . ( 19\n\n)\n\nNote that this equation is equivalent to (6) , the condition to have a geodesic. The role of the black hole mass, in the appearance of CTGs, is to produce a local maximum in the length function, s(r 0 ). This maximum gives us the position of the CTG that in our case is located outside of the source of the spacetime, beyond the black hole horizon.\n\nIn Fig. 1 we present, as a solid line the function s(r 0 ) for a spinning string, and as a dashed line the same function for the superposition of the black hole with the previously mentioned string for the same values of the pa-\n\n0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.4; α= 25; r m = 15.749 r 0 s(r 0 ) 0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.7; α= 25; r m = 10.845 r 0 s(r 0 ) 0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.9; α= 25; r m = 9.172 r 0 s(r 0 )\n\nFIG. 2: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the size of the deficit angle parameter β changes the region for CTCs and the value of rm.\n\nrameters α = 25 (spin parameter) and deficit angle parameter β = 0.9 and different values of the black hole mass (m = 0.3, 1, 4). We see how the presence of the mass shift the maximum for the string located at r 0 = 0 to a position r 0 > 3m. Also the points under the curves represent the pairs [r 0 , s(r 0 )] for CTCs in each case. We note that the region for CTCs for the black hole pierced by the string diminishes when the mass increases. The maximum of the dashed line represents the CTG. We see, that in the first two cases the CTGs are stable (r m > 6m) and in the last case the CTG is not stable (r m < 6m). In Fig. 2 we keep the value of the black hole mass constant, m = 1, as well as, the spin parameter, α = 25, and change the deficit angle parameter β = 0.4, 0.7, 0.9. We see that the larger the string density, λ = (1 -β)/4, the larger the region for CTCs.\n\nIn Fig. 3 we keep the value of the black hole mass constant, m = 1, as well as, the deficit angle parameter β = 0.9 and change the spin parameter α = 15, 20, 25. We see that the regions where the CTCs appear are larger for bigger spin parameter. This parameter is essential to have CTCs and CTGs in this case.\n\nAs we said before the existence of a CTGs does not put restrictions on the energy to travel along this curve. Furthermore, the force needed to move near a stable geodesic is small. Therefore, the energy required will be also small. In principle this small force can be provided by and engine, say a rocket. Hence there will be not a severe energy restriction to travel near to a geodesic. Furthermore, when moving along a stable CTG the control problem is a trivial one. Small trajectory corrections require small energy, also we do not have the danger to enter into a run away situation.\n\nA result from Galloway [5] states that in a compact Lorentzian manifold, each stable free t-homotopy class contains a longest closed timelike curve, and this curve is necessarily a closed timelike geodesic. The assumption that M be compact can be weakened, it is sufficient to assume that there exists an open set U in M with compact closure such that each curve γ ∈ C (the free t-homotopy class) is contained in U . In our case th Gödel universe and other apacetimes e region containing the CTCs in C is not compact. Therefore Galloway's conditions do not apply in this case, they too strong.\n\nWe want to point out that the stability of the circular orbits does not depend on the fact of the orbit be a CTG. We found the same region of stability of the usual circular geodesics. This result is not surprising since our pierced black hole is locally identical to a usual black hole. Moreover one can consider black holes surrounded by different axially symmetric distributions of matter [27] pierced by a spinning string. In this case, depending on the different parameters of the solution, we can also have CTGs and their stability will be the same as the usual circular orbits considered in [27] .\n\nFurthermore, we analyze if the CTGs studied in the present work satisfy the sufficient conditions of Galloway's theorem for the existence of CTGs. We found that ours CTGs do not satisfy these conditions. The possibility of an example that satisfy exactly the conditions of this theorem is under study. We want to mention that the solution of Einstein equations considered in this work is much simpler that the ones listed in the introduction.\n\nFinally, we notice that the spacetime associated to the black hole pierced by a spinning string is not a counter example to the Chronology Protection Conjecture [28] that essentially says that the laws of the physics do not allow the appearance of closed timelike curves. A valid dynamic to built this spacetime is not known.\n\nV.M.R. thanks Departamento de Matemática-UFV for giving the conditions to finish this work which was partially supported by PICDT-UFV/CAPES. P.S.L. thanks the partial financial support of FAPESP and CNPq."},"sections":{"kind":"list like","value":[{"section_type":"ABSTRACT","section_title":"Abstract","text":"The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the superposition of the black hole, we find that the spinning string spacetime is deformed in such a way to allow the existence of closed timelike geodesics."},{"section_type":"OTHER","section_title":"Untitled Section","text":"The existance of closed timelike curves (CTCs) in the Gödel universe and other apacetimes is a worrying fact since these curves show a clear violation of causality. In some cases these CTCs can be disregarded by energy considerations. Their existance requires an external force acting along the whole CTC, process that may consume a great amount of energy. The energy needed to travel along a CTC in Gödel's universe is computed in [1] . When the external force is null the energy needed to travel is also null. Therefore, in principle, the existence of closed timelike geodesics (CTGs) presents a bigger problem of breakdown of causality.\n\nThe classical problem of the existence of closed geodesics in Riemannian geometry was solved by Hadamard [2] in two dimensions and by Cartan [3] in an arbitrary number of dimensions. As a topological problem, the existence of CTGs was proved by Tipler [4] in a class of four-dimensional compact Lorentz manifolds with covering space containing a compact Cauchy surface. In a compact pseudo-Riemaniann manifold with Lorentzian signature (Lorentzian manifold) Galloway [5] found sufficient conditions to have CTGs, see also [6] .\n\nTo the best of our knowledge there are four solution to the Einstein equations generated by matter with positive mass density that contain CTGs: a) Soares [7] found a class of cosmological models, solutions of Einstein-Maxwell equations, with a subclass where the timelike paths of matter are closed. For these models the existence of CTGs is demonstrated and explicit examples are given. These CTGs are not linearly stable [8] . b) Steadman [9] described the behavior of CTGs in a vacuum exterior for the van Stockum solution that represents an infinite rotating dust cylinder. For this solution explicit examples of CTCs and CTGs are shown. There are stable CTGs in this spacetime [8] . c) Bonnor and Steadman [10] studied the existence of CTGs in a spacetime with two spinning particles each one with magnetic moment equal to angular moment and mass equal to charge (Perjeons), in particular, they present a explicit CTG. This particular CTG is not stable, but there exist many other that are stable [11] . d) There are linearly stable CTGs [8] in one of the Gödel-type metrics with not flat background studied by Gürses et al. [12][13] . For CTGs in a spacetime associated to a cloud of strings with negative mass density see [14] . These CTGs are not stable [8] .\n\nThe existence of CTCs in a spacetime whose source is a spinning string has been investigated by many authors (see for instance [15] - [19] ). The interpretation of these strings as torsion line defects can be found in [20] , [21] , see also [22] [23] . These torsion line defects appear when one tries to stabilize two rotating black holes kept apart by spin repulsion [24] . Also, the black hole thermodynamics associated to a static black hole pierced by a non rotating string was studied some time ago by Aryal et al. [25] .\n\nIn the present work we study the existence and stability of CTCs under linear perturbations in the spacetime associated to Schwarzschild black hole (BH) pierced by a spinning string. Even though this spacetime is more a mathematical curiosity than an example of a real spacetime we believe that the study of stability of CTCs and CTGs can shed some light into the existence of this rather pathological curves. In particular, we study sufficient conditions to have linearly stable CTGs. We find that these conditions are not very restrictive and can be easily satisfied. Furthermore, we compared them with the same conditions studied by Galloway [5] for a compact Lorentzian manifold.\n\nLet us consider the spacetime with metric,\n\nds 2 = (1- 2m r )(dt-αdϕ) 2 - dr 2 1 -2m r -r 2 (dθ 2 +β 2 sin 2 θdϕ 2 ), (1)\n\nwhere α = 4S and S is the string's spin angular momentum per unit of length, β = 1 -4λ and λ is the string's linear mass density that is equal to its tension (λ ≤ 1/4).\n\nIn the particular case, α = 0 and β = 1, the metric (1) reduces to the Schwarzschild solution. When m = 0, Eq. ( 1 ) represents a spinning string, with the further special-ization β = 1 (not deficit angle) we have a pure massless torsion line defect [20] [21] . Therefore the metric (1) can be considered as representing the spacetime associated to a Schwarzschild black hole pierced by a spinning string.\n\nLet us denote by γ a closed curve given in its parametric form by,\n\nt = t 0 , r = r 0 , ϕ ∈ [0, 2π], θ = π 2 , (2)\n\nwhere t 0 and r 0 are constants. When γ is parametrized with an arbitrary parameter σ, we have a timelike curve when dx µ dσ dxµ dσ > 0. This condition reduces to g ϕϕ > 0, i.e.,\n\n(1 -2m/r 0 )α 2 -r 2 0 β 2 > 0. (3)\n\nA generic CTC γ satisfies the system of equations given by\n\nẍµ + Γ µ αβ ẋα ẋβ = F µ (x), (4)\n\nwhere the overdot indicates derivation with respect to s, Γ µ αβ are the Christoffel symbols and F µ is a specific external force (a µ = F µ ). The nonzero component of the four-acceleration of γ is\n\na r = 1 r 3 0 (r 0 -2m)(α 2 m -r 3 0 β 2 ) φ2 . (5)\n\nOur goal is to study the behavior of closed timelike geodesics. Therefore taking α as one of the two solutions of\n\nα 2 m -r 3 0 β 2 = 0, (6)\n\nwe have a r = 0. Under this condition (3) is satisfied when r 0 > 3 m, that put the CTG outside the black hole. Let γ be the curve obtained from γ after a small perturbation ξ, i.e., xµ = x µ + ξ µ . From equations (4) one finds that the system of differential equations satisfied by the perturbation ξ is [26] ,\n\nd 2 ξ α ds 2 + 2Γ α βµ dξ β ds u µ + Γ α βµ,λ ξ λ u β u µ = F α ,λ ξ λ . ( 7\n\n)\n\nFor the above mentioned closed timelike geodesic the system (7)\n\nreduces to ξ0 + k 1 ξ1 = 0, (8) ξ1 + k 2 ξ0 + k 3 ξ 1 = 0, (9) ξ2 + k 4 ξ1 = 0, ( 10\n\n) ξ3 + k 5 ξ 3 = 0, (11)\n\nwhere\n\nk 1 = 2Γ 0 21 φ, k 2 = 2Γ 1 20 φ, k 3 = Γ 1 22,1 φ2 , k 4 = 2Γ 2 21 φ, k 5 = Γ 3 22,3 φ2 . (12)\n\nA curve γ parametrized by the proper time, s, is timelike when ẋµ ẋµ = 1. For the curve γ(s) we have that this last condition gives us,\n\nφ2 = m β 2 r 2 0 (r 0 -3m) . ( 13\n\n)\n\nThe solution of ( 9 )-( 11 ) is\n\nξ 0 = -k 1 (c 3 sin(ωs + c 4 )/ω + λs) + c 1 s + c 5 , ξ 1 = c 3 cos(ωs + c 4 ) + λ, ξ 2 = -k 4 (c 3 sin(ωs + c 4 )/ω + λs) + c 2 s + c 6 , ξ 3 = c 7 cos( √ k 5 s + c 8 ), (14)\n\nwhere c i , i = 1, . . . , 8 are integration constants,\n\nω = k 3 -k 1 k 2 = [β 2 (r 0 -6m) φ2 /r 0 ] 1/2 , (15)\n\nand λ = -k 2 c 1 /ω 2 . Thus when r 0 > 6 m, the constant ω is real and the solution (14) shows the typical behavior for stability, i.e., vibrational modes untangled with translational ones that can be eliminated by a suitable choice of the initial conditions. When the black hole is removed, we are left with the spacetime of the spinning string whose line element is,\n\nds 2 = (dt -αdϕ) 2 -dr 2 -r 2 (dθ 2 + β 2 sin 2 θdϕ 2 ). ( 16\n\n)\n\nThe closed curve, γ, is timelike when α 2 -r 2 0 β 2 > 0. The a r -component of the four-acceleration is given by a r = -β 2 r 0 φ2 . Thus for r < |α/β| we have closed timelike curves, which are not geodesics.\n\nFor the closed curve (2) the system (7) is written now as in ( 9 )- (11) replacing equation (10) by\n\nξ1 + k 2 ξ2 + k 3 ξ 1 = ∂ r (Γ 1 22 φ2 )ξ 1 , (17)\n\nwhere now k 2 = 2Γ 1 22 φ and φ2 = (α 2 -r 2 0 β 2 ) -1 . In this particular case the solution of (7) has the same form that (14) with ω 2 = 2β 2 φ2 (2 + β 2 r 2 0 φ2 ). Therefore, the CTCs are stable.\n\nIn summary, there exist linearly stable CTCs in the spacetime related to a spinning string and these curves are restricted to a small region of the spacetime. Closed timelike geodesics do not exist in this spacetime.\n\nFor the nonlinear superposition of a spinning string with a Schwarzschild black hole the new spacetime has linearly stable CTGs. The region of stability is the same of the usual circular geodesics in the Schwarzschild black hole alone. The presence of the spinning string does not affect the stability of the orbits. It seems that torsion lines defects superposed to matter (not strings, β = 1) is a main ingredient to have stable CTGs. Loosely speaking, we have that a torsion line defect alone makes possible the existence of CTCs. When the black hole is present the spinning string spacetime is deformed in such a way to allow the existence of a CTG. This fact is also confirmed in the case of the two Perjeons solutions studied\n\n0 5 10 15 20 25 30 35 0 50 100 150 200 m= 0.3; β= 0.9; α= 25; r m = 6.1401 r 0 s(r 0 ) 0 5 10 15 20 25 30 35 0 50 100 150 200 m= 1; β= 0.9; α= 25; r m = 9.172 r 0 s(r 0 ) 0 5 10 15 20 25 30 35 0 50 100 150 200 m= 4; β= 0.9; α= 25; r m = 14.5597 r 0 s(r 0 )\n\nFIG. 1: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the presence of the mass shift the maximum of s(r0) for the string that is located at r0 = 0 to a position outside the black hole horizon. The maximum, rm represent the radius of the CTGs, the first two are stable and the second is not.\n\nin [10] wherein the torsion line defect is a main ingredient to have CTCs and CTGs.\n\nIt is instructive to look the previous results in a more direct and graphic way. The length of CTC in (2) only depends on the value of r = r 0 . We find,\n\ns(r 0 ) = 2π g ϕϕ (r 0 ), = 2π[(1 -2m/r 0 )α 2 -r 2 0 β 2 ] 1/2 . ( 18\n\n)\n\nThis function has a local maximum for\n\nr m = (mα 2 /β 2 ) 1/3 . ( 19\n\n)\n\nNote that this equation is equivalent to (6) , the condition to have a geodesic. The role of the black hole mass, in the appearance of CTGs, is to produce a local maximum in the length function, s(r 0 ). This maximum gives us the position of the CTG that in our case is located outside of the source of the spacetime, beyond the black hole horizon.\n\nIn Fig. 1 we present, as a solid line the function s(r 0 ) for a spinning string, and as a dashed line the same function for the superposition of the black hole with the previously mentioned string for the same values of the pa-\n\n0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.4; α= 25; r m = 15.749 r 0 s(r 0 ) 0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.7; α= 25; r m = 10.845 r 0 s(r 0 ) 0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.9; α= 25; r m = 9.172 r 0 s(r 0 )\n\nFIG. 2: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the size of the deficit angle parameter β changes the region for CTCs and the value of rm.\n\nrameters α = 25 (spin parameter) and deficit angle parameter β = 0.9 and different values of the black hole mass (m = 0.3, 1, 4). We see how the presence of the mass shift the maximum for the string located at r 0 = 0 to a position r 0 > 3m. Also the points under the curves represent the pairs [r 0 , s(r 0 )] for CTCs in each case. We note that the region for CTCs for the black hole pierced by the string diminishes when the mass increases. The maximum of the dashed line represents the CTG. We see, that in the first two cases the CTGs are stable (r m > 6m) and in the last case the CTG is not stable (r m < 6m). In Fig. 2 we keep the value of the black hole mass constant, m = 1, as well as, the spin parameter, α = 25, and change the deficit angle parameter β = 0.4, 0.7, 0.9. We see that the larger the string density, λ = (1 -β)/4, the larger the region for CTCs.\n\nIn Fig. 3 we keep the value of the black hole mass constant, m = 1, as well as, the deficit angle parameter β = 0.9 and change the spin parameter α = 15, 20, 25. We see that the regions where the CTCs appear are larger for bigger spin parameter. This parameter is essential to have CTCs and CTGs in this case.\n\nAs we said before the existence of a CTGs does not put restrictions on the energy to travel along this curve. Furthermore, the force needed to move near a stable geodesic is small. Therefore, the energy required will be also small. In principle this small force can be provided by and engine, say a rocket. Hence there will be not a severe energy restriction to travel near to a geodesic. Furthermore, when moving along a stable CTG the control problem is a trivial one. Small trajectory corrections require small energy, also we do not have the danger to enter into a run away situation.\n\nA result from Galloway [5] states that in a compact Lorentzian manifold, each stable free t-homotopy class contains a longest closed timelike curve, and this curve is necessarily a closed timelike geodesic. The assumption that M be compact can be weakened, it is sufficient to assume that there exists an open set U in M with compact closure such that each curve γ ∈ C (the free t-homotopy class) is contained in U . In our case th Gödel universe and other apacetimes e region containing the CTCs in C is not compact. Therefore Galloway's conditions do not apply in this case, they too strong.\n\nWe want to point out that the stability of the circular orbits does not depend on the fact of the orbit be a CTG. We found the same region of stability of the usual circular geodesics. This result is not surprising since our pierced black hole is locally identical to a usual black hole. Moreover one can consider black holes surrounded by different axially symmetric distributions of matter [27] pierced by a spinning string. In this case, depending on the different parameters of the solution, we can also have CTGs and their stability will be the same as the usual circular orbits considered in [27] .\n\nFurthermore, we analyze if the CTGs studied in the present work satisfy the sufficient conditions of Galloway's theorem for the existence of CTGs. We found that ours CTGs do not satisfy these conditions. The possibility of an example that satisfy exactly the conditions of this theorem is under study. We want to mention that the solution of Einstein equations considered in this work is much simpler that the ones listed in the introduction.\n\nFinally, we notice that the spacetime associated to the black hole pierced by a spinning string is not a counter example to the Chronology Protection Conjecture [28] that essentially says that the laws of the physics do not allow the appearance of closed timelike curves. A valid dynamic to built this spacetime is not known.\n\nV.M.R. thanks Departamento de Matemática-UFV for giving the conditions to finish this work which was partially supported by PICDT-UFV/CAPES. P.S.L. thanks the partial financial support of FAPESP and CNPq."}],"string":"[\n {\n \"section_type\": \"ABSTRACT\",\n \"section_title\": \"Abstract\",\n \"text\": \"The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the superposition of the black hole, we find that the spinning string spacetime is deformed in such a way to allow the existence of closed timelike geodesics.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Untitled Section\",\n \"text\": \"The existance of closed timelike curves (CTCs) in the Gödel universe and other apacetimes is a worrying fact since these curves show a clear violation of causality. In some cases these CTCs can be disregarded by energy considerations. Their existance requires an external force acting along the whole CTC, process that may consume a great amount of energy. The energy needed to travel along a CTC in Gödel's universe is computed in [1] . When the external force is null the energy needed to travel is also null. Therefore, in principle, the existence of closed timelike geodesics (CTGs) presents a bigger problem of breakdown of causality.\\n\\nThe classical problem of the existence of closed geodesics in Riemannian geometry was solved by Hadamard [2] in two dimensions and by Cartan [3] in an arbitrary number of dimensions. As a topological problem, the existence of CTGs was proved by Tipler [4] in a class of four-dimensional compact Lorentz manifolds with covering space containing a compact Cauchy surface. In a compact pseudo-Riemaniann manifold with Lorentzian signature (Lorentzian manifold) Galloway [5] found sufficient conditions to have CTGs, see also [6] .\\n\\nTo the best of our knowledge there are four solution to the Einstein equations generated by matter with positive mass density that contain CTGs: a) Soares [7] found a class of cosmological models, solutions of Einstein-Maxwell equations, with a subclass where the timelike paths of matter are closed. For these models the existence of CTGs is demonstrated and explicit examples are given. These CTGs are not linearly stable [8] . b) Steadman [9] described the behavior of CTGs in a vacuum exterior for the van Stockum solution that represents an infinite rotating dust cylinder. For this solution explicit examples of CTCs and CTGs are shown. There are stable CTGs in this spacetime [8] . c) Bonnor and Steadman [10] studied the existence of CTGs in a spacetime with two spinning particles each one with magnetic moment equal to angular moment and mass equal to charge (Perjeons), in particular, they present a explicit CTG. This particular CTG is not stable, but there exist many other that are stable [11] . d) There are linearly stable CTGs [8] in one of the Gödel-type metrics with not flat background studied by Gürses et al. [12][13] . For CTGs in a spacetime associated to a cloud of strings with negative mass density see [14] . These CTGs are not stable [8] .\\n\\nThe existence of CTCs in a spacetime whose source is a spinning string has been investigated by many authors (see for instance [15] - [19] ). The interpretation of these strings as torsion line defects can be found in [20] , [21] , see also [22] [23] . These torsion line defects appear when one tries to stabilize two rotating black holes kept apart by spin repulsion [24] . Also, the black hole thermodynamics associated to a static black hole pierced by a non rotating string was studied some time ago by Aryal et al. [25] .\\n\\nIn the present work we study the existence and stability of CTCs under linear perturbations in the spacetime associated to Schwarzschild black hole (BH) pierced by a spinning string. Even though this spacetime is more a mathematical curiosity than an example of a real spacetime we believe that the study of stability of CTCs and CTGs can shed some light into the existence of this rather pathological curves. In particular, we study sufficient conditions to have linearly stable CTGs. We find that these conditions are not very restrictive and can be easily satisfied. Furthermore, we compared them with the same conditions studied by Galloway [5] for a compact Lorentzian manifold.\\n\\nLet us consider the spacetime with metric,\\n\\nds 2 = (1- 2m r )(dt-αdϕ) 2 - dr 2 1 -2m r -r 2 (dθ 2 +β 2 sin 2 θdϕ 2 ), (1)\\n\\nwhere α = 4S and S is the string's spin angular momentum per unit of length, β = 1 -4λ and λ is the string's linear mass density that is equal to its tension (λ ≤ 1/4).\\n\\nIn the particular case, α = 0 and β = 1, the metric (1) reduces to the Schwarzschild solution. When m = 0, Eq. ( 1 ) represents a spinning string, with the further special-ization β = 1 (not deficit angle) we have a pure massless torsion line defect [20] [21] . Therefore the metric (1) can be considered as representing the spacetime associated to a Schwarzschild black hole pierced by a spinning string.\\n\\nLet us denote by γ a closed curve given in its parametric form by,\\n\\nt = t 0 , r = r 0 , ϕ ∈ [0, 2π], θ = π 2 , (2)\\n\\nwhere t 0 and r 0 are constants. When γ is parametrized with an arbitrary parameter σ, we have a timelike curve when dx µ dσ dxµ dσ > 0. This condition reduces to g ϕϕ > 0, i.e.,\\n\\n(1 -2m/r 0 )α 2 -r 2 0 β 2 > 0. (3)\\n\\nA generic CTC γ satisfies the system of equations given by\\n\\nẍµ + Γ µ αβ ẋα ẋβ = F µ (x), (4)\\n\\nwhere the overdot indicates derivation with respect to s, Γ µ αβ are the Christoffel symbols and F µ is a specific external force (a µ = F µ ). The nonzero component of the four-acceleration of γ is\\n\\na r = 1 r 3 0 (r 0 -2m)(α 2 m -r 3 0 β 2 ) φ2 . (5)\\n\\nOur goal is to study the behavior of closed timelike geodesics. Therefore taking α as one of the two solutions of\\n\\nα 2 m -r 3 0 β 2 = 0, (6)\\n\\nwe have a r = 0. Under this condition (3) is satisfied when r 0 > 3 m, that put the CTG outside the black hole. Let γ be the curve obtained from γ after a small perturbation ξ, i.e., xµ = x µ + ξ µ . From equations (4) one finds that the system of differential equations satisfied by the perturbation ξ is [26] ,\\n\\nd 2 ξ α ds 2 + 2Γ α βµ dξ β ds u µ + Γ α βµ,λ ξ λ u β u µ = F α ,λ ξ λ . ( 7\\n\\n)\\n\\nFor the above mentioned closed timelike geodesic the system (7)\\n\\nreduces to ξ0 + k 1 ξ1 = 0, (8) ξ1 + k 2 ξ0 + k 3 ξ 1 = 0, (9) ξ2 + k 4 ξ1 = 0, ( 10\\n\\n) ξ3 + k 5 ξ 3 = 0, (11)\\n\\nwhere\\n\\nk 1 = 2Γ 0 21 φ, k 2 = 2Γ 1 20 φ, k 3 = Γ 1 22,1 φ2 , k 4 = 2Γ 2 21 φ, k 5 = Γ 3 22,3 φ2 . (12)\\n\\nA curve γ parametrized by the proper time, s, is timelike when ẋµ ẋµ = 1. For the curve γ(s) we have that this last condition gives us,\\n\\nφ2 = m β 2 r 2 0 (r 0 -3m) . ( 13\\n\\n)\\n\\nThe solution of ( 9 )-( 11 ) is\\n\\nξ 0 = -k 1 (c 3 sin(ωs + c 4 )/ω + λs) + c 1 s + c 5 , ξ 1 = c 3 cos(ωs + c 4 ) + λ, ξ 2 = -k 4 (c 3 sin(ωs + c 4 )/ω + λs) + c 2 s + c 6 , ξ 3 = c 7 cos( √ k 5 s + c 8 ), (14)\\n\\nwhere c i , i = 1, . . . , 8 are integration constants,\\n\\nω = k 3 -k 1 k 2 = [β 2 (r 0 -6m) φ2 /r 0 ] 1/2 , (15)\\n\\nand λ = -k 2 c 1 /ω 2 . Thus when r 0 > 6 m, the constant ω is real and the solution (14) shows the typical behavior for stability, i.e., vibrational modes untangled with translational ones that can be eliminated by a suitable choice of the initial conditions. When the black hole is removed, we are left with the spacetime of the spinning string whose line element is,\\n\\nds 2 = (dt -αdϕ) 2 -dr 2 -r 2 (dθ 2 + β 2 sin 2 θdϕ 2 ). ( 16\\n\\n)\\n\\nThe closed curve, γ, is timelike when α 2 -r 2 0 β 2 > 0. The a r -component of the four-acceleration is given by a r = -β 2 r 0 φ2 . Thus for r < |α/β| we have closed timelike curves, which are not geodesics.\\n\\nFor the closed curve (2) the system (7) is written now as in ( 9 )- (11) replacing equation (10) by\\n\\nξ1 + k 2 ξ2 + k 3 ξ 1 = ∂ r (Γ 1 22 φ2 )ξ 1 , (17)\\n\\nwhere now k 2 = 2Γ 1 22 φ and φ2 = (α 2 -r 2 0 β 2 ) -1 . In this particular case the solution of (7) has the same form that (14) with ω 2 = 2β 2 φ2 (2 + β 2 r 2 0 φ2 ). Therefore, the CTCs are stable.\\n\\nIn summary, there exist linearly stable CTCs in the spacetime related to a spinning string and these curves are restricted to a small region of the spacetime. Closed timelike geodesics do not exist in this spacetime.\\n\\nFor the nonlinear superposition of a spinning string with a Schwarzschild black hole the new spacetime has linearly stable CTGs. The region of stability is the same of the usual circular geodesics in the Schwarzschild black hole alone. The presence of the spinning string does not affect the stability of the orbits. It seems that torsion lines defects superposed to matter (not strings, β = 1) is a main ingredient to have stable CTGs. Loosely speaking, we have that a torsion line defect alone makes possible the existence of CTCs. When the black hole is present the spinning string spacetime is deformed in such a way to allow the existence of a CTG. This fact is also confirmed in the case of the two Perjeons solutions studied\\n\\n0 5 10 15 20 25 30 35 0 50 100 150 200 m= 0.3; β= 0.9; α= 25; r m = 6.1401 r 0 s(r 0 ) 0 5 10 15 20 25 30 35 0 50 100 150 200 m= 1; β= 0.9; α= 25; r m = 9.172 r 0 s(r 0 ) 0 5 10 15 20 25 30 35 0 50 100 150 200 m= 4; β= 0.9; α= 25; r m = 14.5597 r 0 s(r 0 )\\n\\nFIG. 1: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the presence of the mass shift the maximum of s(r0) for the string that is located at r0 = 0 to a position outside the black hole horizon. The maximum, rm represent the radius of the CTGs, the first two are stable and the second is not.\\n\\nin [10] wherein the torsion line defect is a main ingredient to have CTCs and CTGs.\\n\\nIt is instructive to look the previous results in a more direct and graphic way. The length of CTC in (2) only depends on the value of r = r 0 . We find,\\n\\ns(r 0 ) = 2π g ϕϕ (r 0 ), = 2π[(1 -2m/r 0 )α 2 -r 2 0 β 2 ] 1/2 . ( 18\\n\\n)\\n\\nThis function has a local maximum for\\n\\nr m = (mα 2 /β 2 ) 1/3 . ( 19\\n\\n)\\n\\nNote that this equation is equivalent to (6) , the condition to have a geodesic. The role of the black hole mass, in the appearance of CTGs, is to produce a local maximum in the length function, s(r 0 ). This maximum gives us the position of the CTG that in our case is located outside of the source of the spacetime, beyond the black hole horizon.\\n\\nIn Fig. 1 we present, as a solid line the function s(r 0 ) for a spinning string, and as a dashed line the same function for the superposition of the black hole with the previously mentioned string for the same values of the pa-\\n\\n0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.4; α= 25; r m = 15.749 r 0 s(r 0 ) 0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.7; α= 25; r m = 10.845 r 0 s(r 0 ) 0 10 20 30 40 50 60 70 0 50 100 150 200 m= 1; β= 0.9; α= 25; r m = 9.172 r 0 s(r 0 )\\n\\nFIG. 2: The function s(r0) for a spinning string (solid line) and for a black hole pierced by the string (dashed line). We see how the size of the deficit angle parameter β changes the region for CTCs and the value of rm.\\n\\nrameters α = 25 (spin parameter) and deficit angle parameter β = 0.9 and different values of the black hole mass (m = 0.3, 1, 4). We see how the presence of the mass shift the maximum for the string located at r 0 = 0 to a position r 0 > 3m. Also the points under the curves represent the pairs [r 0 , s(r 0 )] for CTCs in each case. We note that the region for CTCs for the black hole pierced by the string diminishes when the mass increases. The maximum of the dashed line represents the CTG. We see, that in the first two cases the CTGs are stable (r m > 6m) and in the last case the CTG is not stable (r m < 6m). In Fig. 2 we keep the value of the black hole mass constant, m = 1, as well as, the spin parameter, α = 25, and change the deficit angle parameter β = 0.4, 0.7, 0.9. We see that the larger the string density, λ = (1 -β)/4, the larger the region for CTCs.\\n\\nIn Fig. 3 we keep the value of the black hole mass constant, m = 1, as well as, the deficit angle parameter β = 0.9 and change the spin parameter α = 15, 20, 25. We see that the regions where the CTCs appear are larger for bigger spin parameter. This parameter is essential to have CTCs and CTGs in this case.\\n\\nAs we said before the existence of a CTGs does not put restrictions on the energy to travel along this curve. Furthermore, the force needed to move near a stable geodesic is small. Therefore, the energy required will be also small. In principle this small force can be provided by and engine, say a rocket. Hence there will be not a severe energy restriction to travel near to a geodesic. Furthermore, when moving along a stable CTG the control problem is a trivial one. Small trajectory corrections require small energy, also we do not have the danger to enter into a run away situation.\\n\\nA result from Galloway [5] states that in a compact Lorentzian manifold, each stable free t-homotopy class contains a longest closed timelike curve, and this curve is necessarily a closed timelike geodesic. The assumption that M be compact can be weakened, it is sufficient to assume that there exists an open set U in M with compact closure such that each curve γ ∈ C (the free t-homotopy class) is contained in U . In our case th Gödel universe and other apacetimes e region containing the CTCs in C is not compact. Therefore Galloway's conditions do not apply in this case, they too strong.\\n\\nWe want to point out that the stability of the circular orbits does not depend on the fact of the orbit be a CTG. We found the same region of stability of the usual circular geodesics. This result is not surprising since our pierced black hole is locally identical to a usual black hole. Moreover one can consider black holes surrounded by different axially symmetric distributions of matter [27] pierced by a spinning string. In this case, depending on the different parameters of the solution, we can also have CTGs and their stability will be the same as the usual circular orbits considered in [27] .\\n\\nFurthermore, we analyze if the CTGs studied in the present work satisfy the sufficient conditions of Galloway's theorem for the existence of CTGs. We found that ours CTGs do not satisfy these conditions. The possibility of an example that satisfy exactly the conditions of this theorem is under study. We want to mention that the solution of Einstein equations considered in this work is much simpler that the ones listed in the introduction.\\n\\nFinally, we notice that the spacetime associated to the black hole pierced by a spinning string is not a counter example to the Chronology Protection Conjecture [28] that essentially says that the laws of the physics do not allow the appearance of closed timelike curves. A valid dynamic to built this spacetime is not known.\\n\\nV.M.R. thanks Departamento de Matemática-UFV for giving the conditions to finish this work which was partially supported by PICDT-UFV/CAPES. P.S.L. thanks the partial financial support of FAPESP and CNPq.\"\n }\n]","isConversational":false}}},{"rowIdx":199,"cells":{"paper_id":{"kind":"string","value":"arxiv:0704.1111"},"arxiv_id":{"kind":"string","value":"0704.1111"},"arxiv_version":{"kind":"number","value":1,"string":"1"},"doi":{"kind":"string","value":"10.1103/PhysRevD.76.054506"},"content_hash":{"kind":"string","value":"c1e7189384c09b0d8453508ecbcfe6308ebf06fe3577cf0b8d3327ef3e292ae1"},"title":{"kind":"string","value":"QCD in One Dimension at Nonzero Chemical Potential"},"abstract":{"kind":"string","value":"Using an integration formula recently derived by Conrey, Farmer and\nZirnbauer, we calculate the expectation value of the phase factor of the\nfermion determinant for the staggered lattice QCD action in one dimension. We\nshow that the chemical potential can be absorbed into the quark masses; the\ntheory is in the same chiral symmetry class as QCD in three dimensions at zero\nchemical potential. In the limit of a large number of colors and fixed number\nof lattice points, chiral symmetry is broken spontaneously, and our results are\nin agreement with expressions based on a chiral Lagrangian. In this limit, the\neigenvalues of the Dirac operator are correlated according to random matrix\ntheory for QCD in three dimensions. The discontinuity of the chiral condensate\nis due to an alternative to the Banks-Casher formula recently discovered for\nQCD in four dimensions at nonzero chemical potential. The effect of temperature\non the average phase factor is discussed in a schematic random matrix model."},"authors":{"kind":"list like","value":["L. Ravagli and J.J.M. Verbaarschot"],"string":"[\n \"L. Ravagli and J.J.M. Verbaarschot\"\n]","isConversational":false},"arxiv_categories":{"kind":"list like","value":["hep-th","hep-lat"],"string":"[\n \"hep-th\",\n \"hep-lat\"\n]","isConversational":false},"primary_category":{"kind":"string","value":"hep-th"},"journal":{"kind":"string","value":""},"keywords":{"kind":"list like","value":[],"string":"[]","isConversational":false},"submission_date":{"kind":"string","value":"2007-04-09"},"processed_date":{"kind":"string","value":"2026-02-26"},"full_text":{"kind":"string","value":"In spite of significant recent progress, QCD at nonzero chemical potential remains a notoriously hard problem [1, 2, 3] . In particular, first principle nonperturbative results at low temperature are absent because the phase of the fermion determinant invalidates probabilistic methods to evaluate the partition function. This problem is known as the sign problem. The sign problem is particularly severe if the phase of the fermion determinant results in a different free energy, i.e.\n\ndet N f D |det N f D| = e V (Fpq-FN f ) with F N f > F pq . ( 1\n\n)\n\nIf the phase quenched free energy, F pq differs from the free energy of QCD with N f flavors, F N f , the number of required gauge field configurations grows exponentially with the volume. The ratio defined in equation (1) can also be interpreted as the average phase factor of the fermion determinant [4, 5]\n\ne iN f θ pq = e iN f θ |det N f D| |det N f D| . ( 2\n\n)\n\nAlternatively, one can define the average phase factor with respect to the full QCD partition function [4, 5]\n\ne 2iθ N f = e 2iθ det N f D det N f D , ( 3\n\n)\n\nwhich has also been used in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12] as a measure for the severity of the sign problem. When the chemical potential is sufficiently small, the average phase factor can be studied by means of chiral perturbation theory [4, 5] . Recently, the average phase factor was analyzed in the microscopic domain of QCD [13, 14] , where only the constant fields in the chiral Lagrangian contribute to the mass and chemical potential dependence of the partition function. It was found that the sign problem is not serious for µ < m π /2. For µ > m π /2 the chiral condensate of the phase quenched theory rotates into a pion condensate resulting in a free energy that is different from the full theory and a severe sign problem.\n\nIn this paper we study the sign problem for Euclidean QCD in one dimension. In one dimension, the only effect of the gauge field is in the boundary conditions, and lattice QCD simplifies to a matrix integral. A general formula for exactly this type of integrals was recently derived by Conrey, Farmer and Zirnbauer [15] . Using what we will call the CFZ formula, exact analytical results for the one dimensional QCD partition function and the average phase factor will be obtained.\n\nThe study of QCD in one dimension at finite chemical potential has had a long history. One reason to study this model is that it is an effective model for the large N c limit of strong coupling QCD [16, 17, 18, 19, 20, 21] . Among 2 others, it has been successfully used to explain [22, 23, 24] puzzling lattice results for quenched QCD at nonzero chemical potential. More recently, one dimensional QCD was used to study the complex zeros of the partition function [1, 25, 26] .\n\nThe sign problem of QCD at nonzero chemical potential arises because of the nonhermiticity of the Dirac operator. QCD in one dimension is not truly nonhermitean: Instead of being scattered in a two-dimensional domain of the complex plane [27] , the eigenvalues of the Dirac operator are localized on an ellipse in the complex plane [6, 22] . Another manifestation of the mild nonhermiticity is that the chemical potential can be absorbed into the quark masses. For each flavor in one dimension with quark mass m, we can associate flavors with mass m + µ and -m + µ. In the large N c limit, this results in spontaneous chiral symmetry breaking according to\n\nU (2N f ) → U (N f ) × U (N f ).\n\nTherefore, QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. The microscopic limit of QCD in one dimension is equivalent to the microscopic limit of QCD in three dimensions [28] . Another peculiarity of QCD in one dimension is that, already in the free theory, the eigenvalues near zero are spaced inversely proportional to the volume. Therefore, in the thermodynamic limit (i.e. at zero temperature) the chiral condensate is discontinuous across the imaginary axis. However, this type of symmetry breaking is not a collective phenomenon, and there are no associated Goldstone bosons. It is reminiscent to the alternative to Goldstone's theorem proposed by McKane and Stone [29] . Spontaneous symmetry breaking with Goldstone bosons takes place for a fixed number of lattice points in the limit N c → ∞.\n\nA chemical potential excites color singlet excitations with baryon charge q k and mass M k when q k µ > M k . This is also the case for one-dimensional QCD. For gauge group U (N c ), the only color singlet excitations are mesons that are uncharged with respect to the chemical potential, whereas for SU (N c ) gauge group we have both neutral mesonic and charged baryonic color singlet excitations. The complex conjugate of the fermion determinant can be interpreted in terms of conjugate quarks [23, 30] which have a baryon charge that is opposite to that of regular quarks. Therefore, both for U (N c ) and SU (N c ), the phase quenched partition function, where the fermion determinant has been replaced by its absolute value, has charged mesonic excitations made out of quarks and conjugate anti-quarks. This results in a phase transition at µ = µ c = m π /2. The SU (N c ) theory with N f flavors has only charged baryonic excitations and will have a phase transition at µ = µ c = m B /N c . In one dimension it turns out that the pion and the baryon have the same mass per quark number so that the critical chemical potential of the phase quenched U (N c ) theory and the SU (N c ) theory is the same.\n\nFor the U (N c ) partition function we expect the sign problem to be severe when µ > µ c because the phase quenched partition function has a phase transition at µ = µ c whereas the normal theory remains in the same phase. For SU (N c ), both the normal partition function and the phase quenched partition function do have a phase transition to a phase of free quarks at µ = µ c and have the same free energy not only for µ < µ c but also for µ > µ c . Therefore, in one dimension there is no severe sign problem for SU (N c ) QCD.\n\nBecause one dimensional QCD with gauge group U (N c ) does not have a phase transition, the chiral condensate in the thermodynamic limit or large N c limit is discontinuous at m = 0, independent of the value of the chemical potential. Because the Dirac eigenvalues are located on an ellipse for µ = 0, this discontinuity cannot be related to the Dirac spectrum by means of the Banks-Casher formula. In [31] a different mechanism to explain the discontinuity was discovered . It was found that an oscillating contribution to the spectral density with an amplitude that diverges exponentially with the volume, is responsible for the discontinuity of the chiral condensate. We will show that a similar mechanism is at work for U (N c ) gauge theory in one dimension.\n\nLattice QCD in one dimension will be introduced in section II, where we also discuss its continuum limit, mean field results and the Conrey-Farmer-Zirnbauer formula. In section III we will evaluate the U (N c ) partition function and its phase quenched version. The average phase factor for U (N c ) is calculated in section IV. In section V we study one dimensional QCD with SU (N c ) as gauge group and evaluate the regular and phase quenched partition functions and the average phase factor. The effect of temperature will be illustrated with results from a schematic random matrix model in section VI. The connection with the microscopic domain of QCD in three dimensions is discussed in section VII, and the Dirac spectrum and its relation with chiral symmetry breaking is analyzed in section VIII. Concluding remarks are made in section IX. 3 II. QCD IN ONE DIMENSION A. Lattice QCD The staggered lattice QCD Dirac operator in one dimension is given by D = mI e µ U 12 /2 . . . e -µ U † n1 /2 -e -µ U † 12 /2 mI • • • 0 . . . . . . 0 • • • mI e µ U n-1 n /2 -e µ U n1 /2 • • • -e -µ U † n-1 n /2 mI \n\n. ( 4\n\n)\n\nWe have used anti-periodic boundary conditions and the gauge fields on the links are taken to be in U (N c ) or SU (N c ). The chemical potential µ is an imaginary vector field introduced according to the Hasenfratz-Karsch prescription [32] .\n\nIn general, the lattice QCD partition function for N f flavors with mass m is given by\n\nZ N f (µ c , µ) = l dU l det N f D e -SYM/g 2 , ( 5\n\n)\n\nwhere the product is over the links and S YM is the Yang-Mills action. In one dimension, the partition function simplifies substantially. First, there is no Yang-Mills action, and second, by unitary transformations the integral over the links can be reduced to a single integral. In the gauge where all gauge fields except U n 1 ≡ U are equal the unity, the fermion determinant reduces to [24] det D = 2 -nNc det[e nµc + e -nµc + e nµ U + e -nµ U † ], (\n\n) 6\n\nwith µ c given by\n\nµ c = sinh -1 m. ( 7\n\n)\n\nThis value will turn out to be the critical value of the chemical potential. From now on, the overall factor 2 -nNc will be absorbed into the normalization of the partition function. To avoid sign factors, n is taken to be even throughout this paper. The standard method to evaluate the QCD partition function in one dimension is to use the eigenvalues of U as integration variables [6, 16, 17, 18, 24] . For example, one finds this way that the result for N f = 1 with U (N c ) as gauge group is given by [24]\n\nZ N f =1 (µ c , µ) = U(Nc) dU det D = sinh((N c + 1)nµ c ) sinh(nµ c ) . ( 8\n\n)\n\nFor most partition functions that are considered in this paper it is not possible to obtain analytical results by means of this method. Instead we will use powerful integration formulas for unitary integrals that were recently derived in [15] . These integrals are based on the color-flavor transformation [33] which has also been applied to lattice QCD with baryons in the canonical ensemble [34, 35] .\n\nThe continuum limit of the staggered lattice action is given by\n\nD cont = m ∂ 0 + iA 0 + µ ∂ 0 + iA 0 + µ m , ( 9\n\n)\n\nwhere A 0 is a Hermitean N c × N c matrix and the off-diagonal blocks connect even and odd lattice sites. What is special in one dimension is that the off-diagonal blocks are identical. The eigenvalue equation (∂ 0 + iA 0 )ψ k = iu k ψ k is solved by ψ = P e -i R t\n\n0 A0dt χ 0 with ∂ 0 χ 0 = iEχ 0 , ( 10\n\n)\n\n4 with P e the path ordered exponent. Nontrivial eigenvalues are obtained by imposing boundary conditions on ψ. At nonzero chemical potential the eigenvalues of the Dirac operator are given by λ k = m ± (iu k + µ) with u k ∈ R. (11) This is very different from QCD with d ≥ 2 where the eigenvalues of the QCD Dirac operator at µ = 0 are scattered in the complex plane. Also the eigenvalues of the one dimensional staggered lattice Dirac at nonzero chemical potential are localized on a curve in the complex plane. Another consequence of the structure of the Dirac operator (9) is that the fermion determinant can be rewritten as\n\ndet D cont = det(∂ 0 + iA 0 + µ + m) det(∂ 0 + iA 0 + µ -m), ( 12\n\n)\n\nwhich can be interpreted as a two-flavor partition function with masses m + µ and -m + µ. This Dirac operator has the same structure as QCD in three dimensions: In the large N c limit, chiral symmetry is broken spontaneously according to U (2) → U (1) × U (1) with the squared Goldstone masses proportional to the difference of the positive and negative quark masses, i.e. m 2 π ∼ m -(-m) = 2m. At low energy, these Goldstone modes interact according to a chiral Lagrangian determined by the pattern of chiral symmetry breaking. In the large N c limit and m π β ≪ 1 (with β the length of the one-dimensional box) the QCD partition function in one dimension is therefore equivalent to the low-energy limit of QCD in three dimensions. In section VII this will be worked out explicitly for the microscopic limit of the partition function. This is the limit ρ(0) → ∞ with mπρ(0) = fixed and µπρ(0) = fixed, (\n\n) 13\n\nwhere ρ(0) is density of the eigenvalues, or their projections onto the imaginary axis, close to zero. Since ρ(0) = nN c /2π (see section VIII), this is the limit nN c → ∞ with nN c m and nN c µ fixed. When we use the term microscopic limit, we always mean the universal microscopic limit. This is the limit (13) associated with the formation of Goldstone bosons, i.e. the limit N c → ∞ at fixed n.\n\nIn the large N c limit, chiral symmetry is broken spontaneously even in one dimension, so that its low-energy limit is a theory of Goldstone bosons. In this section we give general arguments that determine the chemical potential dependence of the partition function in the microscopic domain where the Compton wavelength of the Goldstone bosons is much larger than β.\n\nAs was argued in [4, 5] , in the microscopic domain, the mean field limit of the partition function is given by\n\nZ = J k 1 m π (µ) e -V F , ( 14\n\n)\n\nwhere J is the value of the integration measure at the saddle point, F is the free energy, V is the space time volume, and m π are the masses of the Goldstone bosons. Let us apply this result to the average phase factor in one dimension. For µ < µ c , in the limit T → 0, only the vacuum state contributes to the partition function, so that the free energy is independent of µ. For quark mass m, the equivalent QCD 3 mass matrix of the Dirac operator D in (12) is given by diag(-m + µ, m + µ), whereas the hermitean conjugate Dirac operator D † , has diag(-m -µ, m -µ) as equivalent QCD 3 mass matrix. For N f flavors, the average phase factor is the partition function with N f + 1 fermionic quarks and one conjugate bosonic quark. We thus have 2(N f + 1) 2 Goldstone bosons made out of two fermionic quarks with squared mass m 2 π = 2mG (with G a constant), 4(N f + 1) fermionic Goldstone modes, half of them with squared mass m 2 π = 2(m -µ)G, and the other half with squared mass m 2 π = 2(m + µ)G. Finally, we have 2 Goldstone bosons composed out of two bosonic quarks with squared mass equal to m 2 π = 2mG. The fermionic partition function Z N f has 2N 2 f Goldstone bosons all with squared mass m 2 π = 2mG. Using (14) we thus find\n\ne 2iθ N f = Z N f +1|1 * (µ c , µ) Z N f (µ c , µ) = 1 - µ 2 m 2 N f +1 for µ < µ c . ( 15\n\n)\n\nFor phase quenched partition functions with N f flavors and N f conjugate flavors, the equivalent QCD 3 symmetry breaking pattern is\n\nU (4N f )/(U (2N f ) × U (2N f ))\n\n, so that the total number of Goldstone bosons is equal to 8N 2 f . Of these, 4N 2 f Goldstone bosons have squared mass m 2 π = 2mG, half made out of two quarks and the other half out of two 5 conjugate quarks. The other 4N 2 f Goldstone bosons are composed out of a quark and a conjugate anti-quark quark, half with squared mass m 2 π = 2(m -µ)G and the other half with squared mass m 2 π = 2(m + µ)G. All equivalent QCD 3 Goldstone bosons of the full QCD partition function with 2N f flavors have a squared mass equal to m 2 π = 2mG. For µ < µ c the free energy of the phase quenched partition function and the full QCD partition function is the same so that, using (14), the phase quenched average phase factor is given by\n\ne 2iθ N f +N * f = Z N f +1+(N f -1) * (µ c , µ) Z 2N f (µ c , µ) = 1 - µ 2 m 2 N 2 f for µ < µ c . ( 16\n\n)\n\nBelow we will show that the results derived in this section also follow from the zero temperature microscopic limit of the exact evaluation of the partition function.\n\nExactly the integrals that are required for the evaluation of the average phase factor of QCD at nonzero chemical potential in one dimension were studied in a recent paper by Conrey, Farmer and Zirnbauer [15] . They considered the partition function\n\nZ({ψ k , φ k }) = U(Nc) dU p j=1 det(1 -e ψj U ) det(1 -e φj U ) p+q l=p+1 det(1 -e -ψ l U † ) det(1 -e -φ l U † ) , ( 17\n\n)\n\nwith dU the Haar measure of U (N c ) and ψ k , φ k complex parameters with Re(φ j ) < 0 < Re(φ l ). Using the color flavor transformation [33] and Howe's theory of supersymmetric dual pairs, they derived the following formula\n\nZ({ψ k , φ k }) = π∈Sp+q/(Sp×Sq) p+q l+1 e Nc(π(ψ l )-ψ l ) p j=1 (1 -e φj -π(ψ l ) )(1 -e π(ψj)-φ l ) (1 -e π(ψj)-π(ψ l ) )(1 -e φj-φ l ) . ( 18\n\n)\n\nThe sum is over permutations in\n\nS p+q /(S p ×S q ) that interchange any of the ψ 1 , • • • , ψ p with any of the ψ p+1 , • • • , ψ p+q .\n\nPartition functions with an unequal number of bosonic and fermionic determinants can be obtained from special limits of (18) . In the case of only fermionic determinants, an equivalent expression was first obtained in [36, 37] . Expressions for degenerate parameters can be derived by carefully taking limits of the above formula.\n\nUnitary matrix integrals can also be calculated by using an eigenvalue representation of the unitary matrices. The integrals we are interested in are of the form\n\nZ = U(Nc) dU k F (e iθ k ), ( 19\n\n)\n\nwhere exp(iθ k ) are the eigenvalues of U . Using an eigenvalue representation of the unitary matrices, the orthogonal polynomial method can be used to express Z as [17]\n\nZ = det(B k-l ) k,l=0,•••,Nc-1 , ( 20\n\n)\n\nwhere\n\nB k = 1 2π π -π dθe ikθ F ({e iθ }). ( 21\n\n)\n\nThis is the method that was used in the literature on one-dimensional QCD prior to this paper [6, 16, 17, 18, 24] . In a few cases the determinant in (20) could be evaluated explicitly resulting in expressions that are similar to those derived directly from the CFZ-formula. We have used (20) to numerically check the results obtained by means of the CFZ-formula.\n\nIII. EXACT EVALUATION OF THE ONE-DIMENSIONAL U (Nc) PARTITION FUNCTION A. Partition Function for arbitrary N f\n\nIn this section we evaluate the one-dimensional U (N c ) QCD partition function for N f flavors.\n\n6 To apply the CFZ formula (18) we rewrite the determinant (6) as det D = e µcnNc det(1 + e n(µ-µc) U ) det(1 + e n(-µ-µc) U † ). (\n\n) 22\n\nFor arbitrary N f we then find the remarkably simple answer\n\nZ N f (µ c , µ) ≡ U(Nc) dU det N f D = σ∈S2N f /SN f ×SN f N f k=1 N f l=1 e Ncm σ(+ k) 1 -exp(m σ(-l) -m σ(+ k) ) , ( 23\n\n) with m -k = -nµ c k and m + k = nµ c k . ( 24\n\n)\n\nThe sum is over all permutations that interchange positive and negative masses. Notice that the µ-dependence has canceled from this expression. This also follows from an expansion of the determinant in powers of U and U † . Only terms with an equal number of factors U and factors U † are non-vanishing. Although the result for degenerate positive and negative masses can be obtained by carefully taking limits of (23), it is simpler to start from a different representation [36, 37] of partition function (23) given by\n\nZ N f (µ c , µ) = 1 1≤k<l≤2N f (e M l -e M k ) 1 e m-1 • • • e (N f -1)m-1 e (Nc+N f )m-1 • • • e (Nc+2N f -1)m-1 . . . . . . . . . . . . . . . 1 e m-N f • • • e (N f -1)m-N f e (Nc+N f )m-N f • • • e (Nc+2N f -1)m-N f 1 e m+ 1 • • • e (N f -1)m+ 1 e (Nc+N f )m+ 1 • • • e (Nc+2N f -1)m+ 1 . . . . . . . . . . . . . . . 1 e m+ N f • • • e (N f -1)m+ N f e (Nc+N f )m+ N f • • • e (Nc+2N f -1)m+ N f . ( 25\n\n)\n\nFor convenience we have introduced the mass matrix\n\nM k = (m 1 1 , • • • , m -N f , m + 1 , • • • , m + N f ).\n\nTo obtain an expression for degenerate masses, we Taylor expand the exponential functions exp(m + k ) and exp(m -k ) to order N f -1 about exp(-nµ c ) and exp(nµ c ), respectively, and write the resulting matrix as the product of two matrices, one containing the Taylor coefficients, and the other containing powers of the expansion parameters. The determinant of the second matrix can be written as a Vandermonde determinant which cancels against part of the prefactor in (25) . Our final expression for the partition function with degenerate masses is given by\n\nZ N f (µ c , µ) = 1 ( N f -1 k=0 k!) 2 1 (e nµc -e -nµc ) N 2 f × 1 • • • 0 1 • • • 0 e -nµc • • • e -nµc e nµc • • • e nµc e -2nµc • • • δ N f -1 - e -2nµc e 2nµc • • • δ N f -1 + e 2nµc . . . . . . . . . . . . e -n(N f -1)µc • • • δ N f -1 - e -n(N f -1)µc e n(N f -1)µc • • • δ N f -1 + e n(N f -1)µc e -n(Nc+N f )µc • • • δ N f -1 - e -n(Nc+N f )µc e n(Nc+N f )µc • • • δ N f -1 + e n(Nc+N f )µc . . . . . . . . . . . . e -n(Nc+2N f -1)µc • • • δ N f -1 - e -n(Nc+2N f -1)µc e n(Nc+2N f -1)µc • • • δ N f -1 + e n(Nc+2N f -1)µc , ( 26\n\n) with δ -= d d(-nµ c ) , δ + = d d(nµ c ) . ( 27\n\n)\n\nThe k'th column (with k ≤ N f ) is given by the δ\n\n(k-1) -\n\nderivative of the first column, and the N f + k'th column is given by the δ (k-1) + derivative of the N f 'th column. From this result one can easily derive explicit expressions for small values of N f . The partition function for N f = 2 reads\n\nZ N f =2 (µ c , µ) = (e n(Nc+2)µc -e -n(Nc+2)µc ) 2 (e nµc -e -nµc ) 4 - (N c + 2) 2 (e nµc -e -nµc ) 2 . ( 28\n\n)\n\n7 The microscopic limit of this partition function is given by\n\nZ micro N f =2 (µ c , µ) = (e Ncµc -e -Ncµc ) 2 16µ 4 c . ( 29\n\n)\n\nFor N f = 3 the partition function is given by:\n\nZ N f =3 (µ c , µ) = (e n(Nc+3)µc -e -n(Nc+3)µc ) 3 (e nµc -e -nµc ) 9 -(N c + 3) 3 (e n(Nc+1)µc -e -n(Nc+1)µc ) (e nµc -e -nµc ) 7 ( 30\n\n) +N c (N c + 2) 2 (e n(Nc+3)µc -e -n(Nc+3)µc ) (e nµc -e -nµc ) 7 - 1 4 (N c + 2) 2 (N c + 3) 2 (e n(Nc+3)µc -e -n(Nc+3)µc ) (e nµc -e -nµc ) 5 . ( 31\n\n)\n\nTo calculate the average phase factor according to the definition (2) for N f = 2, we also need the one-dimensional phase quenched QCD partition function for two flavors which will be evaluated in this subsection. It is defined by\n\nZ 1+1 * (µ c , µ) = U∈U(Nc) dU det D det D † ( 32\n\n) = e 2µnNc U∈U(Nc) dU det(1 -e n(µ-µc) U ) det(1 -e n(-µ-µc) U † ) det(1 -e n(µ-µc) U † ) det(1 -e n(-µ-µc) U ).\n\nThis partition function is easily evaluated using the integration formulae of [15] . We find\n\nZ 1+1 * (µ c , µ) = cosh(2n(N c + 2)µ c ) 8 sinh(n(µ c -µ)) sinh(n(µ + µ c )) sinh 2 (nµ c ) + cosh(2n(N c + 2)µ)) 8 sinh(n(µ -µ c )) sinh(n(µ + µ c )) sinh 2 (nµ) + 1 8 sinh 2 (nµ) sinh 2 (nµ c ) . ( 33\n\n)\n\nFor µ = 0 this result agrees with the N f = 2 partition function given in (28) . Its microscopic limit simplifies to\n\nZ micro 1+1 * (µ c , µ) = e 2nNcµc + e -2nNcµc 16n 4 (µ 2 c -µ 2 )µ 2 c + 2 16n 4 µ 2 µ 2 c - e 2nNcµ + e -2nNcµ 16n 4 (µ 2 c -µ 2 )µ 2 . ( 34\n\n)\n\nIn section VII we will show that this result is equal to the microscopic limit of the\n\nQCD 3 partition function with masses -(µ c + µ), -(µ c -µ), µ c -µ, µ c + µ.\n\nThe large N c limit of the phase quenched partition function is given by\n\nZ 1+1 * (µ c , µ) = e 2n(Nc+2)µc 16 sinh(n(µ c -µ)) sinh(n(µ + µ c ))(sinh(nµ c )) 2 for µ < µ c , Z 1+1 * (µ c , µ) = e 2n(Nc+2)µ 16 sinh(n(µ -µ c )) sinh(n(µ + µ c ))(sinh(nµ)) 2 for µ > µ c , Z 1+1 * (µ c , µ) = N c e 2n(Nc+2)µc 8 sinh 2 (nµ c ) sinh(2nµ c ) 1 + O(N -1 c ) for µ = µ c , ( 35\n\n)\n\nwhich will be used to calculate the phase quenched average phase factor in this limit.\n\nIV. AVERAGE PHASE FACTOR FOR U (Nc)\n\nIn this section we will evaluate the average phase factor, first from the ratio of the full QCD partition function and the phase quenched partition function and in the next subsection starting from the definition in Eq. (3). 8 A. Phase Quenched Average Phase Factor For N f = 2, the average phase factor with the absolute value of the fermion determinant as weight is given by\n\ne 2iθ pq = e 2iθ | det(D)| 2 | det(D)| 2 = Z N f =2 (µ c , µ) Z 1+1 * (µ c , µ) . ( 36\n\n)\n\nThe two-flavor partition function was evaluated in subsection III A, and the phase quenched partition function was calculated in subsection III B. The large N c limit of the phase quenched average phase factor is given by\n\ne 2iθ pq = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc)\n\nfor µ < µ c , sinh 2 (nµ) sinh(n(µ-µc)) sinh(n(µ+µc)) sinh 4 (nµc) e -2n(Nc+2)(µ-µc) for µ > µ c , 2 cosh(nµc) Nc sinh(nµc)\n\nfor µ = µ c . ( 37\n\n)\n\nThe large µ c N c , µN c limit of the microscopic phase quenched partition function (34) for µ < µ c is given by\n\nZ pq (µ c , µ) = e 2nNcµc 16n 4 (µ 2 c -µ 2 )µ 2 c , ( 38\n\n)\n\nwhereas the two-flavor partition function in this limit reads\n\nZ N f =2 (µ c , µ) = e 2nNcµc 16n 4 µ 4 c , ( 39\n\n)\n\nso that the average phase factor simplifies to\n\ne 2iθ pq = 1 - µ 2 µ 2 c for µ < µ c . ( 40\n\n)\n\nThis result is in agreement with the mean field result (16) . Indeed, the denominator of (38) is the product of the Goldstone masses µ c -µ, µ c + µ, µ c and µ c , whereas the denominator of (40) is the product of four Goldstone masses µ c . The free energy of both partition functions is equal to 2nN c µ c . For µ > µ c , the large N c limit of the two flavor partition function is still given by (39), but the large N c µ limit of the microscopic phase quenched partition function is now given by\n\nZ pq = e 2nNcµ 16n 4 (µ 2 -µ 2 c )µ 2 , ( 41\n\n)\n\nresulting in the average phase factor\n\ne 2iθ pq = Z N f =2 Z pq = µ 2 µ 2 c ( µ 2 µ 2 c -1)e -2nNc(µ-µc) for µ > µ c . ( 42\n\n)\n\nThis result can also be obtained from the small nµ, nµ c expansion of the large N c limit of the average phase factor (see Eq. ( 37 )) at fixed nN c µ c . We conclude that the sign problem becomes exponentially hard for µ > µ c . In Fig. 1 we show the average phase factor for different values of n and N c and a critical chemical potential of µ c = 0.1. In the thermodynamic limit at fixed N c , the average phase factor is one for µ < µ c and jumps to zero for µ > µ c . In the right figure we observe a rapid convergence to the mean field result discussed in section II C.\n\nThe average phase factor with the N f -flavor quark determinant as weight is defined by\n\ne 2iθ N f = 1 Z N f (µ c , µ) dU det D det D † det N f D = e nN f Ncµc Z N f (µ c , µ) dU det(1 -U e nµ-nµc ) det(1 -U † e -nµ-nµc ) det(1 -U e -nµ-nµc ) det(1 -U † e nµ-nµc ) det N f (1 -U e nµ-nµc )det N f (1 -U † e -nµ-nµc ). 9\n\nFIG. 1: The average phase factor for one-dimensional QCD with gauge group U (Nc) calculated as the ratio of the two flavor flavor partition function and the phase quenched partition function. In the left figure the number of lattice points varies as indicated for fixed Nc = 3, and in the right figure the number of colors varies as indicated for fixed n = 4.\n\nWe evaluate this integral exactly for N f = 0 and N f = 1 but only give its large N c limit for other values of N f . Using the CFZ-formula (18) [15] we obtain in the quenched case\n\ne 2iθ N f =0 = (1 -e -2n(µ+µc) )(1 -e 2n(µ-µc) ) (1 -e -2nµc ) 2 + e -2nNcµc (1 -e -2nµ )(1 -e 2nµ ) (1 -e 2nµc )(1 -e -2nµc ) for µ < µ c . ( 43\n\n)\n\nThe large N c limit of this result coincides with the large N c limit of the phase quenched average phase factor. In the microscopic limit where µ c nN c and µnN c remain fixed for N c → ∞ we obtain\n\ne 2iθ N f =0 = 1 - µ 2 µ 2 c - µ 2 µ 2 c e -2nNcµc . ( 44\n\n)\n\nIn the large nN c µ c limit the second term does not contribute resulting in\n\ne 2iθ N f =0 = 1 - µ 2 µ 2 c , ( 45\n\n)\n\nin agreement with the results obtained in [4] (see subsection II C). In the thermodynamic limit at fixed N c the average phase factor converges to one for µ < µ c . After rewriting the determinants in (43) as\n\ndet(1 -U † e -nµ-nµc ) det(1 -U † e nµ-nµc ) → e -2nNcµ det(1 -U e nµ+nµc ) det(1 -U e -nµ+nµc ) , ( 46\n\n)\n\nthe denominator of (43) can be expanded in powers of U for µ > µ c . For N f = 0 only the constant term in the integrand yields a non-vanishing result. We thus find\n\ne 2iθ N f =0 = e -2nNcµ for µ > µ c , ( 47\n\n)\n\nso that the average phase factor vanishes in the large nN c µ limit. In Fig. 2 we show the quenched average phase factor for various values of n and N c . Also in this case the average phase factor jumps from 1 to 0 at µ = µ c in the thermodynamic limit. The convergence to the mean field result for increasing N c (right figure) is very rapid. This can be understood from the expansion of the microscopic result given by 1\n\n-(µ 2 /µ 2 c )(1 -1 3 (nµ) 2 + • • •.\n\n10 FIG. 2: The quenched average phase factor for one-dimensional QCD with gauge group U (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\n\nFinally we calculate the average phase factor for an arbitrary number of flavors in the large N c limit. The CFZ formula can again be applied after inserting the factor\n\ndet N f (1 -U e -α )det N f (1 -U † e -β )\n\nin the denominator and taking the limit α → ∞ and β → ∞ at the end of the calculation.\n\nFor µ < µ c , the leading order large N c result is given by the identity permutation in the sum over permutations in the CFZ formula. This results in\n\ne 2iθ N f = (1 -e -2n(µ+µc) )(1 -e 2n(µ-µc) ) (1 -e -2nµc ) 2 N f +1 for µ < µ c . ( 48\n\n)\n\nThe microscopic limit of this result is given by\n\ne 2iθ N f = 1 - µ 2 µ 2 c N f +1 for µ < µ c , ( 49\n\n)\n\nin agreement with the discussion given in section II C. For µ > µ c the CFZ formula can again be applied after rewriting the partition function according to (46) and the insertion (48) . Because of degenerate prefactors, all critical chemical potentials that occur in the combination µ + µ c have to be taken different. After carefully taking limits we find\n\nZ N f +1|1 * (µ c , µ) = N N f c N f ! e -2µnNc e µcnNcN f (1 -e -2µcn ) N 2 f (1 -e -2n(µ+µc) ) N f (1 -e -2nµ ) N f (1 -e -2nµc ) N f . ( 50\n\n)\n\nThe average phase factor given by\n\ne 2iθ N f = Z N f +1|1 * Z N f ∼ e -2nNcµ for µ > µ c ( 51\n\n)\n\nvanishes in the thermodynamic limit. The exact result becomes rather cumbersome even for small values of N f . As an illustration we give the result for Z 2|1 * (µ c , µ) in Appendix A . 11 V. AVERAGE PHASE FACTOR FOR SU (Nc)\n\nTo calculate integrals over SU (N c ) we use the identity SU(Nc)\n\ndU • • • = ∞ p=-∞ U(Nc) dU det p (U ) • • • . ( 52\n\n)\n\nFor the partition functions discussed below, with a few exceptions the sum truncates to a small number of terms.\n\nA. SU (Nc) QCD Partition Function\n\nLet us first calculate the 1d SU (N c ) QCD partition function for N f = 1. The partition function is defined by\n\nZ SU(Nc) N f (µ c , µ) = SU(Nc) dU det N f D = ∞ p=-∞ U(Nc) dU det p U e nNcµc det N f (1 + U e nµ-nµc )det N f (1 + U † e -nµ-nµc ). ( 53\n\n)\n\nFor p ≥ N f the integrand can be rewritten in terms of determinant of matrices U only\n\nZ N f ,p (µ c , µ) = U(Nc) dU det p U det N f D = U(Nc) dU e nNcN f µc det p-N f (U )det N f (1 + U e nµ-nµc )det N f (U + e -nµ-nµc ) . ( 54\n\n)\n\nFor p > N f it follows immediately that the integral vanishes. For p = N f only the constant term inside the determinants contributes to integral resulting in\n\nZ N f ,p=N f (µ c , µ) = e -nN f Ncµ , Z N f ,p>N f (µ c , µ) = 0. ( 55\n\n)\n\nFor p < 0 we combine det p U = det -p U † with the factor det N f (1 -U e nµ-nµc ). We then find\n\nZ N f ,p=-N f (µ c , µ) = e nN f Ncµ , Z N f ,p<-N f (µ c , µ) = 0. ( 56\n\n)\n\nUsing that the hermitean conjugate of a unitary matrix is also unitary we obtain\n\nZ N f ,-p (µ c , µ) = Z N f ,p (µ c , -µ). ( 57\n\n)\n\nFor |p| < N f the integral can be calculated by means of the CFZ formula by choosing all µ c different and introducing the limit\n\ndet U = lim α→∞ det(U -e -α ). ( 58\n\n)\n\nThe limit of degenerate µ c and α → ∞ is taken at the end of the calculation. We will only give exact results for N f = 1 and N f = 2. Using (55,56) and the result for U (N c ) we obtain for N f = 1 Z SU(Nc)\n\nN f =1 (µ c , µ) = e -nN f N f µ + e nN f N f µ + sinh(n(N c + 1)µ c ) sinh(nµ c ) , ( 59\n\n)\n\nin agreement with earlier work by Bilic and Demeterfi [24] . For N f = 2, the only integrals that have not yet been calculated are those for p = ±1 which are related by (57). For the p = 1 contribution we find\n\nZ N f =2,p=1 (µ c , µ) = e nNc(µc-µ) N c e -2n(Nc+1)µc + e 2nµc (e nµc -e -nµc ) 2 - e -2nNcµc (e -3nµc -3e -nµc ) -e 3nµc + 3e nµc (e nµc -e -nµc ) 3 . ( 60\n\n)\n\n12 For µ < µ c the large nN c µ limit of the N f = 2 partition function is dominated by the p = 0 term, whereas for µ > µ c the large nN c µ c limit is given by the p = -2 term. For µ = µ c the large N c limit is dominated by the p = -1 with 1/N c corrections from the p = 0 and p = -2 terms. We thus find as leading large N c result Z SU(Nc)\n\nN f =2 (µ c , µ) = e 2n(Nc+2)µc 16 sinh 4 (nµ c ) for µ < µ c , Z SU(Nc) N f =2 (µ c , µ) = e 2nNcµ for µ > µ c , Z SU(Nc) N f =2 (µ c , µ) = N c e 2n(Nc+1)µc 4 sinh 2 (nµ c ) 1 + O(N -1 c ) for µ = µ c . ( 61\n\n)\n\nBoth for µ < µ c and µ > µ c the terms that are canceled by the integration over the unitary group are subleading in the thermodynamic limit. In other words there is no serious sign problem. By inspection one can easily show that the dominance of the p = 0 terms for µ < µ c and the p = -N f term for µ > µ c is a feature of the large N c limit that is valid for any number of flavors. Therefore, for µ < µ c , the chiral condensate is the same as for the U (N c ) theory. For µ > µ c though, the dominant p = -N f term is mass-independent. This results in a vanishing chiral condensate so that the large N c limit of the U (N c ) theory and the SU (N c ) theory is different.\n\nB. The Phase Quenched Partition Function for SU (Nc)\n\nA second ingredient for the average phase factor is the phase quenched SU (N c ) partition function. Using the same arguments as for the full QCD partition function, one easily obtains\n\nZ 1+1 * ,|p|>2 (µ c , µ) = 0, Z 1+1 * ,|p|=2 (µ c , µ) = 1, Z 1+1 * ,-p (µ c , µ) = Z 1+1 * ,p (µ c , -µ). ( 62\n\n)\n\nThe partition function for p = 0 is the U (N c ) partition function which was already given in (33) . What remains to be calculated is the partition function for p = +1. Using the CFZ formula one easily arrives at\n\nZ 1+1 * ,p=1 (µ c , µ) = 1 4 sinh(nµ) sinh(nµ c ) sinh(n(N c + 2)(µ + µ c )) sinh(n(µ + µ c )) - sinh(n(N c + 2)(µ -µ c )) sinh(n(µ -µ c ))) . ( 63\n\n)\n\nBoth for µ < µ c and µ > µ c , the leading large N c result of the phase quenched partition function resides in the p = 0 term given in Eq. (35).\n\nC. Average Phase Factor for SU (Nc)\n\nThe average phase factor can be either obtained from the ratio of the full QCD partition function and the phase quenched partition function or can be calculated with the full QCD partition function as weight (see Eqs. (2) and (3)). In the first case we find for large N c ,\n\ne 2iθ SU(Nc) pq = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc) for µ < µ c , 16 sinh(n(µ -µ c )) sinh(n(µ c + µ)) sinh 2 (nµ) for µ > µ c , 1 -e -4nµc for µ = µ c . ( 64\n\n)\n\nThe microscopic limit of the average phase factor is obtained by expanding this result for small µ c , µ. We again find the usual mean field result\n\ne 2iθ SU(Nc) pq = 1 - µ 2 µ 2 c θ(µ c -µ). ( 65\n\n)\n\nIn the thermodynamic limit, the average phase factor converges to one in both for µ < µ c and µ > µ c . 13 FIG. 3: The phase quenched average phase factor for one-dimensional QCD with gauge group SU (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\n\nExact results for the average phase factor are displayed in Fig. 3 . At finite N c , we observe a rapid convergence to the asymptotic value of 1. At fixed n, the approach to the large N c limit is slow. Also clearly visible is that for µ ≈ µ c the corrections to the microscopic limit are large. As was already discussed before, this is due to the p = -1 contribution.\n\nIn the quenched case the average phase factor for SU (N c ) defined according to Eq. ( 3 ) can be written as\n\ne 2iθ SU(Nc) N f =0 = ∞ p=-∞ U∈U(Nc) dU det p (U ) det(1 + U e nµ-nµc ) det(1 + U † e -nµ-nµc ) det(1 + U e -nµ-nµc ) det(1 + U † e nµ-nµc ) . ( 66\n\n)\n\nThe integrals can again be calculated by means of the CFZ formula. For N c ≥ 3 one can easily show that the limit introduced in (58) gives vanishing results for |p| > 2. However, for N c = 1 and N c = 2 additional terms may contribute to this limit. For N c = 2, where the average phase factor is equal to one, contributions for |p| > 2 vanish for µ < µ c , but terms with p < -2 are nonzero for µ > µ c . The formulae for |p| ≤ 2 given below are valid for N c = 2. For N c = 1 the integrals in (66) only vanish for p ≥ 2 and µ > µ c . Also the general answer for p = -2 is not correct in this case. For p = -1 we obtain Ncn(µc-µ) (1-e -2n(µc +µ) )(1-e -2nµ ) 1-e -2nµc for µ < µ c ,\n\ne 2iθ N f =0 = e -\n\ne nNc(µc -µ) (1-e -2nµ ) 1-e -2nµc [(1 -e -2nµc-2nµ ) -e -2nNcµc (e -2nµc -e -2nµ )] for µ > µ c . ( 67\n\n)\n\nFor p = 1 we find e 2iθ N f =0 = e -Ncn(µc+µ) (1-e -2n(µc -µ) )(1-e 2nµ ) 1-e -2nµc for µ < µ c ,\n\n0 for µ > µ c . ( 68\n\n)\n\nFinally, for p = -2 the result is\n\ne 2iθ N f =0 = 0 for µ < µ c , (1 -e -2nµ )(1 -e -2n(µ+µc) )(1 -e 2n(µc-µ) )(1 -e -2nµ ) for µ > µ c . ( 69\n\n)\n\nFor p = 2 the average phase factor vanishes. Notice that the relation\n\ne 2iθ N f =0,-p (µ c , µ) = e 2iθ N f =0,p (µ c , -µ) ( 70\n\n) 14 FIG. 4: Average phase factor for one-dimensional QCD with gauge group SU (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\n\nis also valid in this case. It can be used to derive results for negative values of µ. In the large N c limit, the average phase factor is dominated by the p = 0 contribution for µ < µ c and by the p = -2 contribution for µ > µ c . For µ = µ c , the p = 0, p = -1 and p = -2 contributions are of equal order in N c , but only the p = -1 term is nonvanishing. As large N c limit of the average phase factor we thus find e 2iθ SU(Nc) N f =0 = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc) for µ < µ c , 16 sinh(n(µ -µ c )) sinh(n(µ c + µ)) sinh 2 (nµ) for µ > µ c ,\n\n1 -e -4nµc for µ = µ c , ( 71\n\n)\n\nwhich is the same expression as obtained in (64) for the phase quenched average phase factor. The microscopic limit of the average phase factor is again given by the small µ-µ c expansion of this result which is equal to the usual mean field result\n\ne 2iθ SU(Nc) N f =0 = 1 - µ 2 µ 2 c θ(µ c -µ). ( 72\n\n)\n\nIn Fig. 4 we show the exact result for the SU (N c ) average phase factor. We observe a rapid approach to the thermodynamic limit at fixed N c just like in Fig. 3 . In the large N c limit the result for the quenched average phase calculated according to (66) is also given by (64). As already was argued in the introduction, we conclude that QCD in one dimension with SU (N c ) as gauge group does not have a serious sign problem, not only for µ < µ c , but also for µ > µ c . Indeed one-dimensional QCD at nonzero chemical potential could be simulated reliably by the Glasgow method [26] .\n\nThe results of the previous two sections show that the sign problem is severe for QCD in one dimension with gauge group U (N c ) and µ > µ c . Although this theory does not have a critical temperature, it is clear from Fig. 1 that the average phase factor for µ > µ c increases significantly for higher temperatures (i.e. for lower values of n). However, this is a geometric effect due to the Boltzmann factor. Recent lattice simulations [10, 11, 12, 26, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] suggest that the sign problem becomes much milder for T around T c . Because of the absence of a critical temperature, this temperature dependence of the average phase factor cannot be investigated in 15 FIG. 5: Scatter plot of the eigenvalues of the random matrix Dirac operator for µ =0.3 and T = 0, 0.7, 0.9, 1.0, 1.1 . The quark mass at m = 0.1 is indicated by the black dot.\n\none dimensional QCD. Instead we study this problem in a schematic random matrix model at nonzero temperature and chemical potential [30, 48, 49] . This model has a severe sign problem as well as a critical temperature and its eigenvalues are scattered in a finite domain of the complex plane, just like in QCD. The quenched average phase factor in this model is given by the matrix integral\n\ne 2iθ (m, µ, T ) = dCe -N TrCC † det(D + m) det(D † + m) , ( 73\n\n)\n\nwith Dirac operator given by\n\nD = 0 iC + µ + it iC † + µ + it 0 , ( 74\n\n)\n\nand t is the traceless diagonal matrix t = diag(-T, • • • , -T, T, • • • , T ). The integral is over the real and imaginary parts of the matrix elements of the complex N × N matrices C. The phase quenched average phase factor is defined by\n\ne 2iθ pq = dCdet 2 (D + m)e -N TrCC † dCdet(D + m) det(D † + m)e -N TrCC † ( 75\n\n)\n\nIf at all possible, it certainly takes a significant effort to evaluate these integrals analytically. Therefore we have studied the quenched (73) and phase quenched (75) average phase factor numerically. In Fig. 5 , we give scatter plots of the eigenvalues of the random matrix Dirac operator. For each of the temperatures T = 0, 0.7, 0.9, 1, 1.1, the figures show results of 40 matrices with N = 100 and µ = 0.3. The black dot represents the quark mass for which the average phase factor is calculated. In Fig. 6 we show the µ-dependence of the phase quenched (left) and quenched (right) average phase factor obtained by averaging over 10,000 to 80,000 matrices C with N = 100. The results for the phase quenched average phase factor for T = 0 and T = 0.7 and µ > µ c are not displayed because they do not converge because of the severity of the sign problem. The average phase factor depends only weakly on the size of 16 FIG. 6: Phase quenched (left) and quenched (right) average phase factor versus chemical potential. Results have been obtained from the chiral random matrix model (74).\n\nthe matrices. The conclusion of Figs. 5 and 6 is that the sign problem is not serious when the quark mass is outside the domain of the eigenvalues. This confirms the conjecture that was made in [4, 5, 50] . It also agrees with the observation in [30] that the free energy of the full theory and the phase quenched theory are the same for T = 0 if the quark mass is outside the domain of the eigenvalues (this is the case for µ < µ c ). The µ-dependence is due to the curvatures at the saddle point.\n\nAccording to the general arguments given in section II B, the microscopic limit of the one dimensional QCD partition function at nonzero chemical potential is equal to the microscopic limit of the QCD 3 partition function at zero chemical potential, but with shifted quark masses. In this section we show this explicitly for some of the results derived before. We first discuss the microscopic limit of the QCD 3 partition function which can be derived from either a chiral Lagrangian or a random matrix model with the same global symmetries [28] . For a discussion of QCD 3 in terms of chiral Lagrangians we refer to [51] .\n\nFrom the continuum Dirac operator (9) it follows that the random matrix model that describes the fluctuations of the low-lying Dirac eigenvalues is given by\n\nZ = dHP (H) det(iH + µ + m) det(iH -m + µ), ( 76\n\n)\n\nwhere the probability distribution can conveniently be taken to be the Gaussian distribution\n\nP (H) = e -2N Σ 2 TrH 2 . ( 77\n\n)\n\nSuch random matrix partition functions have been studied elaborately in the literature to analyze the microscopic limit of QCD in three dimensions [28, 52, 53, 54, 55] . The microscopic limit of the partition function (76)\n\nwith quark masses given by M = diag(-m 1 , • • • , -m N f , m 1 , • • • , m N f ) is equal to [53, 54] Z QCD3 = 1 ∆(M ) det A(m) A(-m) A(-m) A(m) , ( 78\n\n)\n\n17 where the matrix elements of the N f × N f matrix A(m) are given by\n\nA(m) kl = m l k e -m k , ( 79\n\n)\n\nand ∆ is the Vandermonde determinant ∆(M ) =\n\n2N f k>l (M kk -M ll ). ( 80\n\n)\n\nAlthough general expressions for QCD 3 partition functions with an arbitrary number of bosonic and fermionic determinants are also known [56] , for our purposes we only need the partition function\n\nZ N f +2|2 = N f k=1 det(D + m k ) det(D -m k ) det(D + z 1 ) det(D -z 2 ) det(D + z1 ) det(D -z2 ) . ( 81\n\n)\n\nThis partition function was evaluated in [57] by means of the supersymmetric method and is given by\n\nZ N f +2|2 = (z 1 -z1 )(z 2 -z2 ) ∆ 2 N f (m) exp N c (z 1 + z2 ) N c (z 1 + z2 ) N f k=1 (m k -z1 )(m k -z2 ) (m k -z 1 )(m k -z 2 ) det sinh Nc(m k +m l ) Nc(m k +m l ) sinh Nc(m k +z1) Nc(m k +z1) sinh Nc(z2+m l ) Nc(z2+m l )\n\nsinh Nc(z1+z2) Nc(z1+z2) + e -Nc(z1-z1)+Nc(z2-z2)\n\n∆ 2 N f (m) det sinh N c (m k + m l ) N c (m k + m l ) . ( 82\n\n)\n\nThe last term is a so-called Efetov-Wegner term, and the Vandermonde determinant is over positive masses only\n\n∆(m) = N f k>l (m k -m l ). ( 83\n\n)\n\nWe will now show that the microscopic limit of the partition function (25) is equal to the microscopic limit of the QCD 3 partition function. To this end we multiply row k of the first N f rows of the determinant by exp(-N c m -k /2) and row k of the second N f rows of the determinant in (25) by exp(-N c m + k /2). In the microscopic limit we keep N c m ±,k fixed so that we can expand the masses that do not occur in this combination. By subtracting successive columns starting with the first one we obtain\n\nZ N f = 1 1≤k<l≤2N f (M l -M k ) ( 84\n\n) × e -Ncm-1 /2 m -1 e -Ncm-1/2 • • • m N f -1 -1 e -Ncm-1/2 m N f -1 e +Ncm-1/2 • • • m 2N f -1 -1\n\ne Ncm-1 /2 . . . . . . . . . . . . . . .\n\ne -Ncm-N f /2 m -N f e -Ncm-N f /2 • • • m N f -1 -N f e -Ncm-N f /2 m N f -N f e +Ncm-N f /2 • • • m 2N f -1 -N f e Ncm-N f /2 e -Ncm+ 1/2 m + 1 e -Ncm+ 1/2 • • • m N f -1 + 1 e -Ncm+ 1 /2 m N f + 1 e +Ncm+ 1 /2 • • • m 2N f -1 + 1\n\ne Ncm+ 1/2 . . . . . . . . . . . . . . .\n\ne -Ncm+ N f /2 m + N f e -Ncm+ N f /2 • • • m N f -1 + N f e -Ncm+ N f /2 m N f + N f e +Ncm+ N f /2 • • • m 2N f -1 + N f e Ncm+ N f /2 ,\n\nwhere the masses M K are defined below Eq. ( 25 ). By multiplying the columns\n\nl = N f + 1, • • • , 2N f by (-1) l-1 and\n\nintroducing microscopic masses, we obtain exactly the expression for the QCD 3 partition function given in (78) . For masses -(µ c + µ), -(µ c -µ), µ c -µ, µ c + µ corresponding to the microscopic limit of the two-flavor phase quenched partition function given in (34) the determinant in (84) is given by\n\n4nµ 2 (e 2nNcµc + e -2nNcµc ) -4nµ 2 c (e 2nNcµ + e -2nNcµ ) + 8n(µ 2 c -µ 2 ). ( 85\n\n)\n\nand the prefactor is equal to\n\n1≤k<l≤2N f (M l -M k ) = 64n 6 (µ 2 c -µ 2 )µ 2 c µ 2 . ( 86\n\n) 18\n\nFIG. 7: The number variance for n = 4 and Nc = 100.\n\nTheir ratio coincides with the microscopic phase quenched partition function given in (34) .\n\nAs second example, we consider the quenched average phase factor for U (N c ) given in (43) . For N f = 0 the partition function (82) simplifies to Nc(z1-z1-z2+z2) .\n\nZ 2|2 = (z 1 -z1 )(z 2 -z2 ) e Nc(z1+z2) N c (z 1 + z2 ) sinh N c (z 1 + z 2 ) N c (z 1 + z 2 ) + e -\n\n( 87\n\n)\n\nThe masses in this partition function corresponding to (43) are given by\n\nz 1 = nµ c + nµ, z 2 = nµ c -nµ, z2 = nµ -nµ c , z1 = -nµ -nµ c . ( 88\n\n)\n\nSubstituting them into (87) we obtain\n\nZ 2|2 = 1 - µ 2 µ 2 c + µ 2 µ 2 c e -4Ncnµ ( 89\n\n)\n\nwhich is exactly the microscopic limit of the quenched average phase factor (43) given in Eq. (44).\n\nWe consider the Dirac operator (4) in a gauge where all gauge fields except U n1 ≡ U and U 1,n = U † are equal to unity. If the eigenvalues of U are equal to exp(iθ k ) the eigenvalues of the Dirac operator are given by\n\nλ k,l = 1 2 (e 2πi(k+1/2)+iθ l n +µ -e -2πi(k+1/2)-iθ l n -µ ), k = 1, • • • , n, l = 1, • • • , N c . ( 90\n\n)\n\nContrary to QCD at µ = 0 in more than one dimension, where the eigenvalues are scattered in the complex plane [27, 30, 58, 59, 60, 61] , the eigenvalues are located on an ellipse in the complex plane with real and imaginary parts related by\n\nRe(λ k,l ) e µ -e -µ 2 + Im(λ k,l ) e µ + e -µ 2 = 1. ( 91\n\n) 19\n\nA. Universal Fluctuations In section II B we have shown that the continuum limit of the staggered Dirac operator in one dimension is in the same universality class as QCD in three dimensions. The continuum limit of the staggered three dimensional Dirac operator, though, is in the chiral symmetry universality class of QCD in four dimensions. For example, in [62] this was found for the distribution of the small Dirac eigenvalues. The reason for the chiral structure is that the staggered Dirac operator only couples even and odd lattice sites. This is also the case in one dimension, but the off-diagonal blocks, containing the gauge fields, are the same in the continuum limit (or occur in the combination U n + U † n ), resulting in a two flavor theory with opposite masses (see (12)). The secular equation is given by\n\ndet[(∂ 0 + iA 0 + µ + λ)(∂ 0 + iA 0 + µ -λ)] = 0. ( 92\n\n)\n\nThis corresponds to the superposition of the spectrum of\n\n∂ 0 + A 0 + µ and -∂ 0 -iA 0 -µ.\n\nIn the domain of the Dirac spectrum where ∂ 0 can be neglected, the eigenvalues on each of the lines ±µ are given by the Hermitean random matrix ensemble A 0 . For µ = 0 we have the superposition of the Hermitean random matrix ensembles of A 0 and -A 0 . From the eigenvalues (90) it is clear that the staggered lattice Dirac eigenvalues show a similar superposition. One ensemble is localized on the left half of the ellipse and the other ensemble on its right half. Superpostions of more ensembles arise for correlations on scales larger than N c level spacings. Eigenvalues of U (N c ) matrices are correlated according to the circular unitary ensemble. Therefore, the eigenvalues λ k,l are correlated according the Gaussian Unitary Ensemble (GUE) on a scale where the variations in the average spectral density can be neglected, i.e. on a scale much less than N c level spacings. By correcting for the variation of the average level spacing, the scale on which universal random matrix correlations are found can be extended, but this scale should always remain well below N c level spacings. Then we are necessarily within one of the n successive copies of the eigenvalues along the ellipse. At a scale of a finite number of average level spacings, we expect convergence to universal random matrix correlations in the large N c limit.\n\nAs measure of the correlations between eigenvalues, we use the number variance Σ 2 (n) defined as the variance on the number of eigenvalues in an interval that contains n eigenvalues on average. We calculate the number variance for intervals starting from zero. Because of the superposition of an enemble and its negative, this is equal to the number variance of a single ensemble for an interval that is symmetric about zero which is given by the GUE result. In Fig. 7 we show the number variance obtained from an average over 10 5 gauge field configurations for N c = 100 and µ = 0. The number variance is calculated for an interval starting at zero. The discrepancy between the analytical result and the GUE (solid curve) is only barely visible.\n\nThe increase of the domain of validity of chiral random matrix theory with increasing N c was also observed for Dirac operator of QCD in four dimensions [63] . This can be explained as follows: The number of eigenvalues that is described by random matrix theory is equal to ≈ F 2 π √ V (with V the space-time volume), but F 2 π scales with N c in the large N c limit.\n\nWe first discuss the case of µ = 0. Then the average spacing of the eigenvalues scales as 1/(nN c ). This means that the chiral condensate develops a discontinuity at m = 0 when nN c → ∞. Two cases of interest where a nonzero chiral condensate can be obtained are the limit n → ∞ for fixed N c , i.e. at zero temperature, or at fixed n in the limit N c → ∞. In both cases the chiral condensate is discontinuous for nN c → ∞ when the quark mass crosses the imaginary axis where the eigenvalues are located. However, Goldstone bosons and universal behavior is only found in the large N c limit. At fixed N c , chiral symmetry breaking is not associated with universal behavior.\n\nFor µ = 0 the Dirac eigenvalues are located on the ellipse (91). When the quark mass is inside the ellipse of eigenvalues the chiral condensate is zero for nN c → ∞ for the SU (N c ) partition function and the phase quenched U (N c ) partition function [24] . For gauge group U (N c ) with N f ≥ 1 flavors, though, the partition function is µindependent so that the chiral condensate is also nonzero if the mass is inside the eigenvalue ellipse. The Banks-Casher formula fails in this case which is known as the \"Silver Blaze Problem\" [64] . The resolution is the same as for QCD in four dimensions [31] : the re-weighted eigenvalue distribution defined as\n\nρ full (z) = 1 Z N f U(Nc) dU det N f D k δ 2 (z -λ k ) ( 93\n\n)\n\nshows oscillations with an amplitude that diverges exponentially with n and a period that is proportional to 1/n. Below we will illustrate this for U (1).\n\n20 For N c = 1 one easily derives that the spectral density is given by ρ full (z) = 4 e nµc + e -nµc -e n(iα+µ) -e -n(iα+µ) (e nµc + e -nµc )(e 2r + e -2r + e 2iα + e -2iα ) δ(r -µ), (\n\n) 94\n\nwhere z is parameterized as\n\nz = 1 2 (e r+iα -e -r-iα ), r > 0, α ∈ [0, 2π], ( 95\n\n)\n\nand we have used that Z N f =1 = 2 cosh(nµ c ) (see eq. ( 8 )). For µ > µ c this spectral density has oscillations with an amplitude that diverges exponentially with n and a period of order 1/n which are the essential properties of the spectral density of the Dirac operator of QCD in four dimensions [31, 65, 66] . It can be decomposed as\n\nρ full (z) = ρ q (z) + ρ osc (z). ( 96\n\n)\n\nThe average quenched spectral density is defined by\n\nρ q (z) = 1 2π dθ 1 n n k=1 δ 2 (z -λ k ) ( 97\n\n)\n\nwith the eigenvalues λ k given by Eq. ( 90 ). After changing variables according to (95) and integrating over θ we find\n\nρ q (z) = 1 2π 4 (e 2r + e -2r + e 2iα + e -2iα ) δ(r -µ). ( 98\n\n)\n\nThe oscillatory part of the spectral density, ρ osc (z) is equal to the difference ρ full (z) -ρ q (z). The chiral condensate is given by\n\nΣ full (m) = d 2 z ρ full (z) z + m , ( 99\n\n)\n\nand can also be decomposed as\n\nΣ full (m) = Σ q (m) + Σ osc (m). ( 100\n\n)\n\nThe Jacobian for the transformation of d 2 z to drdα is given by\n\nd 2 z = 1 4 (e 2r + e -2r + e 2iα + e -2iα )drdα, ( 101\n\n)\n\nso that the chiral condensate after integration over r can be simplified to\n\nΣ full (m) = 1 π 2π 0 dα e nµc + e -nµc -e n(iα+µ) -e -n(iα+µ) (e nµc + e -nµc )(e µ+iα -e -µ-iα -2m) = tanh(nµ c ) cosh µ c . ( 102\n\n)\n\nThis result remains finite for n → ∞. We remind the reader that the mass is parameterized as m = sinh µ c . This expression represents the resolvent at the quark mass which was evaluated in [49] and is in agreement with earlier work [6, 24] . Decomposing the spectral density and the chiral condensate according to (96) we obtain\n\nΣ osc (m) = θ(sinh(µ) -m) Σ full (m), Σ q (m) = θ(m -sinh(µ)) Σ full (m), ( 103\n\n)\n\nso that when the quark mass is inside the ellipse of eigenvalues, the entire chiral condensate is due to the oscillatory part of the spectral density. The discontinuity in the chiral condensate is reminiscent to a Stokes phenomenon. The alternative to the Banks-Casher relation proposed in [31] is also at work for QCD in one dimension. This solves the \"Silver Blaze Problem\" [64]. 21 IX. CONCLUSIONS We have studied QCD in one dimension at nonzero chemical potential. Both the full theory, its quenched and phase quenched versions and gauge groups U (N c ) and SU (N c ) have been considered. In one dimension, the QCD or QCD-like partition functions can be reduced to a single matrix integral, which because of recent advances by Conrey, Farmer and Zirnbauer, could be evaluated analytically. In this paper we have analyzed the small mass behavior of the partition function, the nature of the sign problem, and the relation between the Dirac spectrum and chiral symmetry breaking.\n\nTo put our results in perspective, we emphasize that QCD in one dimension, is quite different from QCD in more dimensions. In particular, we wish to mention the following three points. First, instead of being scattered in the complex plane, the eigenvalues of the staggered Dirac operator are located on an ellipse in the complex plane. Second, phase transitions can only take place for zero temperature or for N c → ∞. At the critical chemical potential, a transition from the vacuum state to a state of free quarks takes place. Third, color singlets are made out of noninteracting quarks so that the critical chemical potential is given by the quark mass. In particular, the critical chemical potential for the meson state and the baryon state is the same.\n\nBecause the Dirac eigenvalues are located on a curve, the large N c limit of staggered lattice QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. We have shown this both by an explicit evaluation of the partition function, and by analyzing the fluctuations of the Dirac eigenvalues for large N c . We have used this equivalence to explain the behavior of the partition functions and the average phase factor in the microscopic limit.\n\nContrary to QCD in four dimensions, the sign problem for QCD in one dimension is not severe in the case of gauge group SU (N c ). One reason is that the critical chemical potential for full QCD and phase quenched QCD is the same, so that the parameter domain m π /2 < µ < m N /3, where the sign problem becomes severe in four dimensions, is absent in one dimension. A second reason is that for µ > µ c both the full theory and the phase quenched theory become a theory of free quarks with the same free energy in the thermodynamic limit.\n\nFor gauge group U (N c ), on the other hand, the sign problem is severe when µ > m π /2 both in one dimension and in four dimensions. The reason is that the U (N c ) theory does not have charged excitations, whereas the phase quenched theory has charged mesons resulting in a phase transition at µ = m π /2. We have evaluated the average phase factor both by averaging with respect to the phase quenched partition function and the full partition function, and similar conclusions have been reached.\n\nThe condititon µ > m π /2 for having a severe sign problem can be rephrased as the quark mass being inside the ellipse of eigenvalues. In more dimensions this condition is that the quark mass is inside the support of the Dirac spectrum. It also applies to nonzero temperature as we have demonstrated explicitly in the framework of a chiral random matrix model. Also for U (N c ) QCD in one dimension, the chiral condensate is discontinuous across the imaginary axis in spite of the fact that there are no Dirac eigenvalues. This can only mean that the phase of the fermion determinant is responsible for the discontinuity. This is what happens in four dimensions where the discontinuity is due to a contribution to the spectral density that oscillates with a period of the inverse volume and amplitude that diverges exponentially with the volume. Exactly the same mechanism, where the discontinuity in the chiral condensate arises due to a Stokes like phenomenon, is at work in one dimension. This suggests that this is a universal mechanism for theories with a sign problem.\n\nWe end by repeating that QCD in one dimension with SU (N c ) as gauge group has no serious sign problem. Our hope is that part of this conclusion translates to four dimensions ameliorating the sign problem for µ > M N /3.\n\nWe wish to thank K. Splittorff for numerous comments and criticism and P.H. Damgaard, H. Neuberger and R.Pisarski are thanked for valuable discussions. K. Splittorff is also thanked for a careful reading of the manuscript. This work was supported by U.S. DOE Grant No. DE-FG-88ER40388 (JV), the Angelo Della Riccia Foundation (LR), the Villum Kann Rassmussen Foundation (JV) and the Danish National Bank (JV).\n\nAPPENDIX A: AVERAGE U (Nc) PHASE FACTOR FOR N f = 1\n\nIn this appendix we give explicit results for the average phase factor for N f = 1 defined by\n\ne 2iθ N f =1 = Z 2|1 * (µ c , µ) Z N f =1 (µ c , µ) . ( A1\n\n)\n\nThe partition function Z N f =1 (µ c , µ) was already given in Eq. ( 8 ). The numerator can be obtained from the CFZ formula. After taking the limit of degenerate critical chemical potentials at the end of the calculation, we obtain for 22 µ < µ c Z 2|1 * (µ c , µ) = e n(Nc+1)µc e -4nµ (1 -e 2n(µ+µc) ) 2 (1 -e 2n(µ-µc) ) 2 (e nµc -e -nµc ) 5 + e -3n(Nc+1)µc (e nµ -e -nµ ) 4 (e nµc -e -nµc ) 5 +e -n(Nc+1)µc [f 0 + N c f 1 + N 2 c f 2 ]. (A2) with f 0 = 2e 2n(µ+µc) + 2e 2n(µ-µc) -e 4nµ -e -4nµ + 4e 2nµ + 4e -2nµ -e 4n(µ-µc) -e 4n(µ+µc) + 2e 6nµc-2nµ + 2e 6nµc+2nµ (e nµc -e -nµc ) 5 + -10e 2nµc + 6e 4nµc -2 -5e 6nµc -e -2nµc (e nµc -e -nµc ) 5 ,\n\nf 1 = (1 -e -2nµ ) 2 (3e 2nµc + e -2nµc -2e 2nµ -2e -2nµ ) (e nµc -e -nµc ) 3 , f 2 = - (1 -e -2nµ ) 2 (1 -e n(µ+µc ))(1 -e n(µ-µc )) (e nµc -e -nµc ) 3 . ( A3\n\n)\n\nFor µ > µ c a similar calculation results in\n\nZ 2|1 * = N c e nNc(µc-2µ) (1 -e -2n(µ+µc) )(1 -e -2nµ ) (1 -e -2nµc ) 2 (1 + g 1 N c ) + N c e -nNc(2µ+µc) (1 -e 2n(µc-µ) )(1 -e -2nµ ) (1 -e 2nµc ) 2 (1 + g 2 N c ) ( A4\n\n) with g 1 = 1 + 1 e 2nµ -1 + 1 e 2n(µ+µc) -1 - 2 e 2nµc -1 , g 2 = 1 + 1 e 2n(µ-µc) -1 + 1 e 2nµ -1 - 2 e -2nµc -1 . ( A5\n\n)\n\nWe have checked that the results in this appendix agree with the large N c limit of the average phase factor given in section IV B.\n\n[1] M. P. Lombardo, arXiv:hep-lat/0612017. [2] C. Schmidt, PoS LAT2006, 021 (2006) [arXiv:hep-lat/0610116]. [3] K. Splittorff, arXiv:hep-lat/0610072. [4] K. Splittorff and J. J. M. Verbaarschot, [arXiv:hep-lat/0702011]. [5] K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 98, 031601 (2007) [arXiv:hep-lat/0609076]. [6] P. E. Gibbs, Preprint PRINT-86-0389-GLASGOW, 1986; P. E. Gibbs, Phys. Lett. B 182 (1986) 369. [7] D. Toussaint, Nucl. Phys. Proc. Suppl. 17, 248 (1990). [8] P. de Forcrand and V. Laliena, Phys. Rev. D 61, 034502 (2000) [arXiv:hep-lat/9907004]. [9] Y. Sasai, A. Nakamura and T. Takaishi, Nucl. Phys. Proc. Suppl. 129, 539 (2004) [arXiv:hep-lat/0310046]. [10] S. Ejiri, Phys. Rev. D 69, 094506 (2004) [arXiv:hep-lat/0401012]; S. Ejiri, Phys. Rev. D 73, 054502 (2006) [arXiv:hep-lat/0506023]. [11] C. Schmidt, Z. Fodor and S. D. Katz, arXiv:hep-lat/0512032. [12] C. R. Allton et al., Phys. Rev. D 66, 074507 (2002) [arXiv:hep-lat/0204010]; C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]; C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [13] E. V. Shuryak and J. J. M. Verbaarschot, Nucl. Phys. A 560, 306 (1993) [arXiv:hep-th/9212088]. [14] J. J. M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994) [arXiv:hep-th/9401059]. [15] J.B. Conrey, D.W. Farmer and M.R. Zirnbauer, [arXiv:math-ph/0511024], 2005. [16] E. M. Ilgenfritz and J. Kripfganz, Z. Phys. C 29, 79 (1985). [17] P. H. Damgaard, N. Kawamoto and K. Shigemoto, Phys. Rev. Lett. 53 (1984) 2211; Nucl. Phys. B 264, 1 (1986). [18] A. Gocksch and M. Ogilvie, Phys. Rev. D 31 (1985) 877. [19] P. H. Damgaard, D. Hochberg and N. Kawamoto, Phys. Lett. B 158 (1985) 239. [20] N. Bilic, K. Demeterfi and B. Petersson, Nucl. Phys. B 377, 651 (1992). [21] P. de Forcrand and S. Kim, arXiv:hep-lat/0608012.\n\n[22] P. E. Gibbs, Phys. Lett. B 172, 53 (1986). [23] A. Gocksch, Phys. Rev. D 37, 1014 (1988). [24] N. Bilic and K. Demeterfi, Phys. Lett. B 212, 83 (1988). [25] S. Gupta, Phys. Lett. B 588, 136 (2004) [arXiv:hep-lat/0307007]. [26] M. P. Lombardo, Nucl. Phys. Proc. Suppl. 83, 375 (2000) [arXiv:hep-lat/9908006]. [27] I. Barbour, N. E. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H. W. Wyld, Nucl. Phys. B 275, 296 (1986). [28] J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 73, 2288 (1994) [arXiv:hep-th/9405005]. [29] A. J. McKane and M. Stone, Annals Phys. 131, 36 (1981). [30] M. A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996) [arXiv:hep-lat/9604003]. [31] J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 94, 202001 (2005) [arXiv:hep-th/0501210]. [32] P. Hasenfratz and F. Karsch, Phys. Lett. B 125, 308 (1983). [33] M.R. Zirnbauer, [arXiv:chao-dyn/9609007]. [34] B. Schlittgen and T. Wettig, Nucl. Phys. B 632, 155 (2002) [arXiv:hep-lat/0111039]. [35] J. Budczies, S. Nonnenmacher, Y. Shnir and M. R. Zirnbauer, Nucl. Phys. B 635, 309 (2002) [arXiv:hep-lat/0112018]. [36] J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein and N.C. Snaith, [arXiv:math-ph/0208007]. [37] E.L. Basor and P.J. Forrester, Mathematische Nachrichten, 170, 5 (1194). [38] Z. Fodor and S. D. Katz, JHEP 0203, 014 (2002) [arXiv:hep-lat/0106002]. [39] Z. Fodor and S. D. Katz, JHEP 0404, 050 (2004) [arXiv:hep-lat/0402006]. [40] P. de Forcrand and O. Philipsen, Nucl. Phys. B 642, 290 (2002) [arXiv:hep-lat/0205016]. [41] P. de Forcrand and O. Philipsen, Nucl. Phys. B 673, 170 (2003) [arXiv:hep-lat/0307020]. [42] M. D'Elia and M. P. Lombardo, Phys. Rev. D 67, 014505 (2003) [arXiv:hep-lat/0209146]. [43] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]. [44] C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [45] R. V. Gavai and S. Gupta, Phys. Rev. D 68, 034506 (2003) [arXiv:hep-lat/0303013]. [46] R. V. Gavai and S. Gupta, Phys. Rev. D 71, 114014 (2005) [arXiv:hep-lat/0412035]. [47] V. Azcoiti, G. di Carlo and A. F. Grillo, Phys. Rev. Lett. 65, 2239 (1990); J. Ambjorn, K. N. Anagnostopoulos, J. Nishimura and J. J. M. Verbaarschot, JHEP 0210, 062 (2002) [arXiv:hep-lat/0208025]. [48] A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 53, 7223 (1996) [arXiv:hep-ph/9509324]. [49] M. A. Halasz, A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 56 (1997) 5140 [arXiv:hep-lat/9703006]. [50] K. Splittorff, arXiv:hep-lat/0505001. [51] G. V. Dunne and S. M. Nishigaki, Nucl. Phys. B 670, 307 (2003) [arXiv:hep-ph/0306220]. [52] G. Akemann and P. H. Damgaard, Nucl. Phys. B 576, 597 (2000) [arXiv:hep-th/9910190]. [53] P. H. Damgaard and S. M. Nishigaki, Phys. Rev. D 57, 5299 (1998) [arXiv:hep-th/9711096]. [54] G. Akemann, D. Dalmazi, P. H. Damgaard and J. J. M. Verbaarschot, Nucl. Phys. B 601, 77 (2001) [arXiv:hep-th/0011072]. [55] T. Andersson, P. H. Damgaard and K. Splittorff, Nucl. Phys. B 707, 509 (2005) [arXiv:hep-th/0410163]. [56] G. Akemann and Y. V. Fyodorov, Nucl. Phys. B 664, 457 (2003) [arXiv:hep-th/0304095]. [57] R. J. Szabo, Nucl. Phys. B 598, 309 (2001) [arXiv:hep-th/0009237]. [58] D. Toublan and J. J. M. Verbaarschot, Int. J. Mod. Phys. B 15, 1404 (2001) [arXiv:hep-th/0001110]. [59] K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 683, 467 (2004) [arXiv:hep-th/0310271]. [60] G. Akemann and T. Wettig, Phys. Rev. Lett. 92, 102002 (2004) [Ibid. 96, 029902 (2006) ] [arXiv:hep-lat/0308003]. [61] J. Bloch and T. Wettig, Phys. Rev. Lett. 97, 012003 (2006) [arXiv:hep-lat/0604020]. [62] P. H. Damgaard, U. M. Heller, A. Krasnitz and T. Madsen, Phys. Lett. B 440, 129 (1998) [arXiv:hep-lat/9803012]. [63] R. Narayanan and H. Neuberger, Nucl. Phys. B 696, 107 (2004) [arXiv:hep-lat/0405025]. [64] T. D. Cohen, Phys. Rev. Lett. 91, 222001 (2003) [arXiv:hep-ph/0307089]. [65] J. C. Osborn, Phys. Rev. Lett. 93, 222001 (2004) [arXiv:hep-th/0403131]. [66] G. Akemann, J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 712, 287 (2005) [arXiv:hep-th/0411030]."},"sections":{"kind":"list like","value":[{"section_type":"BACKGROUND","section_title":"I. INTRODUCTION","text":"In spite of significant recent progress, QCD at nonzero chemical potential remains a notoriously hard problem [1, 2, 3] . In particular, first principle nonperturbative results at low temperature are absent because the phase of the fermion determinant invalidates probabilistic methods to evaluate the partition function. This problem is known as the sign problem. The sign problem is particularly severe if the phase of the fermion determinant results in a different free energy, i.e.\n\ndet N f D |det N f D| = e V (Fpq-FN f ) with F N f > F pq . ( 1\n\n)\n\nIf the phase quenched free energy, F pq differs from the free energy of QCD with N f flavors, F N f , the number of required gauge field configurations grows exponentially with the volume. The ratio defined in equation (1) can also be interpreted as the average phase factor of the fermion determinant [4, 5]\n\ne iN f θ pq = e iN f θ |det N f D| |det N f D| . ( 2\n\n)\n\nAlternatively, one can define the average phase factor with respect to the full QCD partition function [4, 5]\n\ne 2iθ N f = e 2iθ det N f D det N f D , ( 3\n\n)\n\nwhich has also been used in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12] as a measure for the severity of the sign problem. When the chemical potential is sufficiently small, the average phase factor can be studied by means of chiral perturbation theory [4, 5] . Recently, the average phase factor was analyzed in the microscopic domain of QCD [13, 14] , where only the constant fields in the chiral Lagrangian contribute to the mass and chemical potential dependence of the partition function. It was found that the sign problem is not serious for µ < m π /2. For µ > m π /2 the chiral condensate of the phase quenched theory rotates into a pion condensate resulting in a free energy that is different from the full theory and a severe sign problem.\n\nIn this paper we study the sign problem for Euclidean QCD in one dimension. In one dimension, the only effect of the gauge field is in the boundary conditions, and lattice QCD simplifies to a matrix integral. A general formula for exactly this type of integrals was recently derived by Conrey, Farmer and Zirnbauer [15] . Using what we will call the CFZ formula, exact analytical results for the one dimensional QCD partition function and the average phase factor will be obtained.\n\nThe study of QCD in one dimension at finite chemical potential has had a long history. One reason to study this model is that it is an effective model for the large N c limit of strong coupling QCD [16, 17, 18, 19, 20, 21] . Among 2 others, it has been successfully used to explain [22, 23, 24] puzzling lattice results for quenched QCD at nonzero chemical potential. More recently, one dimensional QCD was used to study the complex zeros of the partition function [1, 25, 26] .\n\nThe sign problem of QCD at nonzero chemical potential arises because of the nonhermiticity of the Dirac operator. QCD in one dimension is not truly nonhermitean: Instead of being scattered in a two-dimensional domain of the complex plane [27] , the eigenvalues of the Dirac operator are localized on an ellipse in the complex plane [6, 22] . Another manifestation of the mild nonhermiticity is that the chemical potential can be absorbed into the quark masses. For each flavor in one dimension with quark mass m, we can associate flavors with mass m + µ and -m + µ. In the large N c limit, this results in spontaneous chiral symmetry breaking according to\n\nU (2N f ) → U (N f ) × U (N f ).\n\nTherefore, QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. The microscopic limit of QCD in one dimension is equivalent to the microscopic limit of QCD in three dimensions [28] . Another peculiarity of QCD in one dimension is that, already in the free theory, the eigenvalues near zero are spaced inversely proportional to the volume. Therefore, in the thermodynamic limit (i.e. at zero temperature) the chiral condensate is discontinuous across the imaginary axis. However, this type of symmetry breaking is not a collective phenomenon, and there are no associated Goldstone bosons. It is reminiscent to the alternative to Goldstone's theorem proposed by McKane and Stone [29] . Spontaneous symmetry breaking with Goldstone bosons takes place for a fixed number of lattice points in the limit N c → ∞.\n\nA chemical potential excites color singlet excitations with baryon charge q k and mass M k when q k µ > M k . This is also the case for one-dimensional QCD. For gauge group U (N c ), the only color singlet excitations are mesons that are uncharged with respect to the chemical potential, whereas for SU (N c ) gauge group we have both neutral mesonic and charged baryonic color singlet excitations. The complex conjugate of the fermion determinant can be interpreted in terms of conjugate quarks [23, 30] which have a baryon charge that is opposite to that of regular quarks. Therefore, both for U (N c ) and SU (N c ), the phase quenched partition function, where the fermion determinant has been replaced by its absolute value, has charged mesonic excitations made out of quarks and conjugate anti-quarks. This results in a phase transition at µ = µ c = m π /2. The SU (N c ) theory with N f flavors has only charged baryonic excitations and will have a phase transition at µ = µ c = m B /N c . In one dimension it turns out that the pion and the baryon have the same mass per quark number so that the critical chemical potential of the phase quenched U (N c ) theory and the SU (N c ) theory is the same.\n\nFor the U (N c ) partition function we expect the sign problem to be severe when µ > µ c because the phase quenched partition function has a phase transition at µ = µ c whereas the normal theory remains in the same phase. For SU (N c ), both the normal partition function and the phase quenched partition function do have a phase transition to a phase of free quarks at µ = µ c and have the same free energy not only for µ < µ c but also for µ > µ c . Therefore, in one dimension there is no severe sign problem for SU (N c ) QCD.\n\nBecause one dimensional QCD with gauge group U (N c ) does not have a phase transition, the chiral condensate in the thermodynamic limit or large N c limit is discontinuous at m = 0, independent of the value of the chemical potential. Because the Dirac eigenvalues are located on an ellipse for µ = 0, this discontinuity cannot be related to the Dirac spectrum by means of the Banks-Casher formula. In [31] a different mechanism to explain the discontinuity was discovered . It was found that an oscillating contribution to the spectral density with an amplitude that diverges exponentially with the volume, is responsible for the discontinuity of the chiral condensate. We will show that a similar mechanism is at work for U (N c ) gauge theory in one dimension.\n\nLattice QCD in one dimension will be introduced in section II, where we also discuss its continuum limit, mean field results and the Conrey-Farmer-Zirnbauer formula. In section III we will evaluate the U (N c ) partition function and its phase quenched version. The average phase factor for U (N c ) is calculated in section IV. In section V we study one dimensional QCD with SU (N c ) as gauge group and evaluate the regular and phase quenched partition functions and the average phase factor. The effect of temperature will be illustrated with results from a schematic random matrix model in section VI. The connection with the microscopic domain of QCD in three dimensions is discussed in section VII, and the Dirac spectrum and its relation with chiral symmetry breaking is analyzed in section VIII. Concluding remarks are made in section IX. 3 II. QCD IN ONE DIMENSION A. Lattice QCD The staggered lattice QCD Dirac operator in one dimension is given by D = mI e µ U 12 /2 . . . e -µ U † n1 /2 -e -µ U † 12 /2 mI • • • 0 . . . . . . 0 • • • mI e µ U n-1 n /2 -e µ U n1 /2 • • • -e -µ U † n-1 n /2 mI \n\n. ( 4\n\n)\n\nWe have used anti-periodic boundary conditions and the gauge fields on the links are taken to be in U (N c ) or SU (N c ). The chemical potential µ is an imaginary vector field introduced according to the Hasenfratz-Karsch prescription [32] .\n\nIn general, the lattice QCD partition function for N f flavors with mass m is given by\n\nZ N f (µ c , µ) = l dU l det N f D e -SYM/g 2 , ( 5\n\n)\n\nwhere the product is over the links and S YM is the Yang-Mills action. In one dimension, the partition function simplifies substantially. First, there is no Yang-Mills action, and second, by unitary transformations the integral over the links can be reduced to a single integral. In the gauge where all gauge fields except U n 1 ≡ U are equal the unity, the fermion determinant reduces to [24] det D = 2 -nNc det[e nµc + e -nµc + e nµ U + e -nµ U † ], (\n\n) 6\n\nwith µ c given by\n\nµ c = sinh -1 m. ( 7\n\n)\n\nThis value will turn out to be the critical value of the chemical potential. From now on, the overall factor 2 -nNc will be absorbed into the normalization of the partition function. To avoid sign factors, n is taken to be even throughout this paper. The standard method to evaluate the QCD partition function in one dimension is to use the eigenvalues of U as integration variables [6, 16, 17, 18, 24] . For example, one finds this way that the result for N f = 1 with U (N c ) as gauge group is given by [24]\n\nZ N f =1 (µ c , µ) = U(Nc) dU det D = sinh((N c + 1)nµ c ) sinh(nµ c ) . ( 8\n\n)\n\nFor most partition functions that are considered in this paper it is not possible to obtain analytical results by means of this method. Instead we will use powerful integration formulas for unitary integrals that were recently derived in [15] . These integrals are based on the color-flavor transformation [33] which has also been applied to lattice QCD with baryons in the canonical ensemble [34, 35] ."},{"section_type":"OTHER","section_title":"B. Continuum Theory","text":"The continuum limit of the staggered lattice action is given by\n\nD cont = m ∂ 0 + iA 0 + µ ∂ 0 + iA 0 + µ m , ( 9\n\n)\n\nwhere A 0 is a Hermitean N c × N c matrix and the off-diagonal blocks connect even and odd lattice sites. What is special in one dimension is that the off-diagonal blocks are identical. The eigenvalue equation (∂ 0 + iA 0 )ψ k = iu k ψ k is solved by ψ = P e -i R t\n\n0 A0dt χ 0 with ∂ 0 χ 0 = iEχ 0 , ( 10\n\n)\n\n4 with P e the path ordered exponent. Nontrivial eigenvalues are obtained by imposing boundary conditions on ψ. At nonzero chemical potential the eigenvalues of the Dirac operator are given by λ k = m ± (iu k + µ) with u k ∈ R. (11) This is very different from QCD with d ≥ 2 where the eigenvalues of the QCD Dirac operator at µ = 0 are scattered in the complex plane. Also the eigenvalues of the one dimensional staggered lattice Dirac at nonzero chemical potential are localized on a curve in the complex plane. Another consequence of the structure of the Dirac operator (9) is that the fermion determinant can be rewritten as\n\ndet D cont = det(∂ 0 + iA 0 + µ + m) det(∂ 0 + iA 0 + µ -m), ( 12\n\n)\n\nwhich can be interpreted as a two-flavor partition function with masses m + µ and -m + µ. This Dirac operator has the same structure as QCD in three dimensions: In the large N c limit, chiral symmetry is broken spontaneously according to U (2) → U (1) × U (1) with the squared Goldstone masses proportional to the difference of the positive and negative quark masses, i.e. m 2 π ∼ m -(-m) = 2m. At low energy, these Goldstone modes interact according to a chiral Lagrangian determined by the pattern of chiral symmetry breaking. In the large N c limit and m π β ≪ 1 (with β the length of the one-dimensional box) the QCD partition function in one dimension is therefore equivalent to the low-energy limit of QCD in three dimensions. In section VII this will be worked out explicitly for the microscopic limit of the partition function. This is the limit ρ(0) → ∞ with mπρ(0) = fixed and µπρ(0) = fixed, (\n\n) 13\n\nwhere ρ(0) is density of the eigenvalues, or their projections onto the imaginary axis, close to zero. Since ρ(0) = nN c /2π (see section VIII), this is the limit nN c → ∞ with nN c m and nN c µ fixed. When we use the term microscopic limit, we always mean the universal microscopic limit. This is the limit (13) associated with the formation of Goldstone bosons, i.e. the limit N c → ∞ at fixed n."},{"section_type":"OTHER","section_title":"C. Mean Field Limit for Large Nc","text":"In the large N c limit, chiral symmetry is broken spontaneously even in one dimension, so that its low-energy limit is a theory of Goldstone bosons. In this section we give general arguments that determine the chemical potential dependence of the partition function in the microscopic domain where the Compton wavelength of the Goldstone bosons is much larger than β.\n\nAs was argued in [4, 5] , in the microscopic domain, the mean field limit of the partition function is given by\n\nZ = J k 1 m π (µ) e -V F , ( 14\n\n)\n\nwhere J is the value of the integration measure at the saddle point, F is the free energy, V is the space time volume, and m π are the masses of the Goldstone bosons. Let us apply this result to the average phase factor in one dimension. For µ < µ c , in the limit T → 0, only the vacuum state contributes to the partition function, so that the free energy is independent of µ. For quark mass m, the equivalent QCD 3 mass matrix of the Dirac operator D in (12) is given by diag(-m + µ, m + µ), whereas the hermitean conjugate Dirac operator D † , has diag(-m -µ, m -µ) as equivalent QCD 3 mass matrix. For N f flavors, the average phase factor is the partition function with N f + 1 fermionic quarks and one conjugate bosonic quark. We thus have 2(N f + 1) 2 Goldstone bosons made out of two fermionic quarks with squared mass m 2 π = 2mG (with G a constant), 4(N f + 1) fermionic Goldstone modes, half of them with squared mass m 2 π = 2(m -µ)G, and the other half with squared mass m 2 π = 2(m + µ)G. Finally, we have 2 Goldstone bosons composed out of two bosonic quarks with squared mass equal to m 2 π = 2mG. The fermionic partition function Z N f has 2N 2 f Goldstone bosons all with squared mass m 2 π = 2mG. Using (14) we thus find\n\ne 2iθ N f = Z N f +1|1 * (µ c , µ) Z N f (µ c , µ) = 1 - µ 2 m 2 N f +1 for µ < µ c . ( 15\n\n)\n\nFor phase quenched partition functions with N f flavors and N f conjugate flavors, the equivalent QCD 3 symmetry breaking pattern is\n\nU (4N f )/(U (2N f ) × U (2N f ))\n\n, so that the total number of Goldstone bosons is equal to 8N 2 f . Of these, 4N 2 f Goldstone bosons have squared mass m 2 π = 2mG, half made out of two quarks and the other half out of two 5 conjugate quarks. The other 4N 2 f Goldstone bosons are composed out of a quark and a conjugate anti-quark quark, half with squared mass m 2 π = 2(m -µ)G and the other half with squared mass m 2 π = 2(m + µ)G. All equivalent QCD 3 Goldstone bosons of the full QCD partition function with 2N f flavors have a squared mass equal to m 2 π = 2mG. For µ < µ c the free energy of the phase quenched partition function and the full QCD partition function is the same so that, using (14), the phase quenched average phase factor is given by\n\ne 2iθ N f +N * f = Z N f +1+(N f -1) * (µ c , µ) Z 2N f (µ c , µ) = 1 - µ 2 m 2 N 2 f for µ < µ c . ( 16\n\n)\n\nBelow we will show that the results derived in this section also follow from the zero temperature microscopic limit of the exact evaluation of the partition function."},{"section_type":"OTHER","section_title":"D. The Conrey-Farmer-Zirnbauer formula","text":"Exactly the integrals that are required for the evaluation of the average phase factor of QCD at nonzero chemical potential in one dimension were studied in a recent paper by Conrey, Farmer and Zirnbauer [15] . They considered the partition function\n\nZ({ψ k , φ k }) = U(Nc) dU p j=1 det(1 -e ψj U ) det(1 -e φj U ) p+q l=p+1 det(1 -e -ψ l U † ) det(1 -e -φ l U † ) , ( 17\n\n)\n\nwith dU the Haar measure of U (N c ) and ψ k , φ k complex parameters with Re(φ j ) < 0 < Re(φ l ). Using the color flavor transformation [33] and Howe's theory of supersymmetric dual pairs, they derived the following formula\n\nZ({ψ k , φ k }) = π∈Sp+q/(Sp×Sq) p+q l+1 e Nc(π(ψ l )-ψ l ) p j=1 (1 -e φj -π(ψ l ) )(1 -e π(ψj)-φ l ) (1 -e π(ψj)-π(ψ l ) )(1 -e φj-φ l ) . ( 18\n\n)\n\nThe sum is over permutations in\n\nS p+q /(S p ×S q ) that interchange any of the ψ 1 , • • • , ψ p with any of the ψ p+1 , • • • , ψ p+q .\n\nPartition functions with an unequal number of bosonic and fermionic determinants can be obtained from special limits of (18) . In the case of only fermionic determinants, an equivalent expression was first obtained in [36, 37] . Expressions for degenerate parameters can be derived by carefully taking limits of the above formula.\n\nUnitary matrix integrals can also be calculated by using an eigenvalue representation of the unitary matrices. The integrals we are interested in are of the form\n\nZ = U(Nc) dU k F (e iθ k ), ( 19\n\n)\n\nwhere exp(iθ k ) are the eigenvalues of U . Using an eigenvalue representation of the unitary matrices, the orthogonal polynomial method can be used to express Z as [17]\n\nZ = det(B k-l ) k,l=0,•••,Nc-1 , ( 20\n\n)\n\nwhere\n\nB k = 1 2π π -π dθe ikθ F ({e iθ }). ( 21\n\n)\n\nThis is the method that was used in the literature on one-dimensional QCD prior to this paper [6, 16, 17, 18, 24] . In a few cases the determinant in (20) could be evaluated explicitly resulting in expressions that are similar to those derived directly from the CFZ-formula. We have used (20) to numerically check the results obtained by means of the CFZ-formula.\n\nIII. EXACT EVALUATION OF THE ONE-DIMENSIONAL U (Nc) PARTITION FUNCTION A. Partition Function for arbitrary N f\n\nIn this section we evaluate the one-dimensional U (N c ) QCD partition function for N f flavors.\n\n6 To apply the CFZ formula (18) we rewrite the determinant (6) as det D = e µcnNc det(1 + e n(µ-µc) U ) det(1 + e n(-µ-µc) U † ). (\n\n) 22\n\nFor arbitrary N f we then find the remarkably simple answer\n\nZ N f (µ c , µ) ≡ U(Nc) dU det N f D = σ∈S2N f /SN f ×SN f N f k=1 N f l=1 e Ncm σ(+ k) 1 -exp(m σ(-l) -m σ(+ k) ) , ( 23\n\n) with m -k = -nµ c k and m + k = nµ c k . ( 24\n\n)\n\nThe sum is over all permutations that interchange positive and negative masses. Notice that the µ-dependence has canceled from this expression. This also follows from an expansion of the determinant in powers of U and U † . Only terms with an equal number of factors U and factors U † are non-vanishing. Although the result for degenerate positive and negative masses can be obtained by carefully taking limits of (23), it is simpler to start from a different representation [36, 37] of partition function (23) given by\n\nZ N f (µ c , µ) = 1 1≤k<l≤2N f (e M l -e M k ) 1 e m-1 • • • e (N f -1)m-1 e (Nc+N f )m-1 • • • e (Nc+2N f -1)m-1 . . . . . . . . . . . . . . . 1 e m-N f • • • e (N f -1)m-N f e (Nc+N f )m-N f • • • e (Nc+2N f -1)m-N f 1 e m+ 1 • • • e (N f -1)m+ 1 e (Nc+N f )m+ 1 • • • e (Nc+2N f -1)m+ 1 . . . . . . . . . . . . . . . 1 e m+ N f • • • e (N f -1)m+ N f e (Nc+N f )m+ N f • • • e (Nc+2N f -1)m+ N f . ( 25\n\n)\n\nFor convenience we have introduced the mass matrix\n\nM k = (m 1 1 , • • • , m -N f , m + 1 , • • • , m + N f ).\n\nTo obtain an expression for degenerate masses, we Taylor expand the exponential functions exp(m + k ) and exp(m -k ) to order N f -1 about exp(-nµ c ) and exp(nµ c ), respectively, and write the resulting matrix as the product of two matrices, one containing the Taylor coefficients, and the other containing powers of the expansion parameters. The determinant of the second matrix can be written as a Vandermonde determinant which cancels against part of the prefactor in (25) . Our final expression for the partition function with degenerate masses is given by\n\nZ N f (µ c , µ) = 1 ( N f -1 k=0 k!) 2 1 (e nµc -e -nµc ) N 2 f × 1 • • • 0 1 • • • 0 e -nµc • • • e -nµc e nµc • • • e nµc e -2nµc • • • δ N f -1 - e -2nµc e 2nµc • • • δ N f -1 + e 2nµc . . . . . . . . . . . . e -n(N f -1)µc • • • δ N f -1 - e -n(N f -1)µc e n(N f -1)µc • • • δ N f -1 + e n(N f -1)µc e -n(Nc+N f )µc • • • δ N f -1 - e -n(Nc+N f )µc e n(Nc+N f )µc • • • δ N f -1 + e n(Nc+N f )µc . . . . . . . . . . . . e -n(Nc+2N f -1)µc • • • δ N f -1 - e -n(Nc+2N f -1)µc e n(Nc+2N f -1)µc • • • δ N f -1 + e n(Nc+2N f -1)µc , ( 26\n\n) with δ -= d d(-nµ c ) , δ + = d d(nµ c ) . ( 27\n\n)\n\nThe k'th column (with k ≤ N f ) is given by the δ\n\n(k-1) -\n\nderivative of the first column, and the N f + k'th column is given by the δ (k-1) + derivative of the N f 'th column. From this result one can easily derive explicit expressions for small values of N f . The partition function for N f = 2 reads\n\nZ N f =2 (µ c , µ) = (e n(Nc+2)µc -e -n(Nc+2)µc ) 2 (e nµc -e -nµc ) 4 - (N c + 2) 2 (e nµc -e -nµc ) 2 . ( 28\n\n)\n\n7 The microscopic limit of this partition function is given by\n\nZ micro N f =2 (µ c , µ) = (e Ncµc -e -Ncµc ) 2 16µ 4 c . ( 29\n\n)\n\nFor N f = 3 the partition function is given by:\n\nZ N f =3 (µ c , µ) = (e n(Nc+3)µc -e -n(Nc+3)µc ) 3 (e nµc -e -nµc ) 9 -(N c + 3) 3 (e n(Nc+1)µc -e -n(Nc+1)µc ) (e nµc -e -nµc ) 7 ( 30\n\n) +N c (N c + 2) 2 (e n(Nc+3)µc -e -n(Nc+3)µc ) (e nµc -e -nµc ) 7 - 1 4 (N c + 2) 2 (N c + 3) 2 (e n(Nc+3)µc -e -n(Nc+3)µc ) (e nµc -e -nµc ) 5 . ( 31\n\n)"},{"section_type":"OTHER","section_title":"B. The Phase Quenched Partition Function","text":"To calculate the average phase factor according to the definition (2) for N f = 2, we also need the one-dimensional phase quenched QCD partition function for two flavors which will be evaluated in this subsection. It is defined by\n\nZ 1+1 * (µ c , µ) = U∈U(Nc) dU det D det D † ( 32\n\n) = e 2µnNc U∈U(Nc) dU det(1 -e n(µ-µc) U ) det(1 -e n(-µ-µc) U † ) det(1 -e n(µ-µc) U † ) det(1 -e n(-µ-µc) U ).\n\nThis partition function is easily evaluated using the integration formulae of [15] . We find\n\nZ 1+1 * (µ c , µ) = cosh(2n(N c + 2)µ c ) 8 sinh(n(µ c -µ)) sinh(n(µ + µ c )) sinh 2 (nµ c ) + cosh(2n(N c + 2)µ)) 8 sinh(n(µ -µ c )) sinh(n(µ + µ c )) sinh 2 (nµ) + 1 8 sinh 2 (nµ) sinh 2 (nµ c ) . ( 33\n\n)\n\nFor µ = 0 this result agrees with the N f = 2 partition function given in (28) . Its microscopic limit simplifies to\n\nZ micro 1+1 * (µ c , µ) = e 2nNcµc + e -2nNcµc 16n 4 (µ 2 c -µ 2 )µ 2 c + 2 16n 4 µ 2 µ 2 c - e 2nNcµ + e -2nNcµ 16n 4 (µ 2 c -µ 2 )µ 2 . ( 34\n\n)\n\nIn section VII we will show that this result is equal to the microscopic limit of the\n\nQCD 3 partition function with masses -(µ c + µ), -(µ c -µ), µ c -µ, µ c + µ.\n\nThe large N c limit of the phase quenched partition function is given by\n\nZ 1+1 * (µ c , µ) = e 2n(Nc+2)µc 16 sinh(n(µ c -µ)) sinh(n(µ + µ c ))(sinh(nµ c )) 2 for µ < µ c , Z 1+1 * (µ c , µ) = e 2n(Nc+2)µ 16 sinh(n(µ -µ c )) sinh(n(µ + µ c ))(sinh(nµ)) 2 for µ > µ c , Z 1+1 * (µ c , µ) = N c e 2n(Nc+2)µc 8 sinh 2 (nµ c ) sinh(2nµ c ) 1 + O(N -1 c ) for µ = µ c , ( 35\n\n)\n\nwhich will be used to calculate the phase quenched average phase factor in this limit.\n\nIV. AVERAGE PHASE FACTOR FOR U (Nc)\n\nIn this section we will evaluate the average phase factor, first from the ratio of the full QCD partition function and the phase quenched partition function and in the next subsection starting from the definition in Eq. (3). 8 A. Phase Quenched Average Phase Factor For N f = 2, the average phase factor with the absolute value of the fermion determinant as weight is given by\n\ne 2iθ pq = e 2iθ | det(D)| 2 | det(D)| 2 = Z N f =2 (µ c , µ) Z 1+1 * (µ c , µ) . ( 36\n\n)\n\nThe two-flavor partition function was evaluated in subsection III A, and the phase quenched partition function was calculated in subsection III B. The large N c limit of the phase quenched average phase factor is given by\n\ne 2iθ pq = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc)\n\nfor µ < µ c , sinh 2 (nµ) sinh(n(µ-µc)) sinh(n(µ+µc)) sinh 4 (nµc) e -2n(Nc+2)(µ-µc) for µ > µ c , 2 cosh(nµc) Nc sinh(nµc)\n\nfor µ = µ c . ( 37\n\n)\n\nThe large µ c N c , µN c limit of the microscopic phase quenched partition function (34) for µ < µ c is given by\n\nZ pq (µ c , µ) = e 2nNcµc 16n 4 (µ 2 c -µ 2 )µ 2 c , ( 38\n\n)\n\nwhereas the two-flavor partition function in this limit reads\n\nZ N f =2 (µ c , µ) = e 2nNcµc 16n 4 µ 4 c , ( 39\n\n)\n\nso that the average phase factor simplifies to\n\ne 2iθ pq = 1 - µ 2 µ 2 c for µ < µ c . ( 40\n\n)\n\nThis result is in agreement with the mean field result (16) . Indeed, the denominator of (38) is the product of the Goldstone masses µ c -µ, µ c + µ, µ c and µ c , whereas the denominator of (40) is the product of four Goldstone masses µ c . The free energy of both partition functions is equal to 2nN c µ c . For µ > µ c , the large N c limit of the two flavor partition function is still given by (39), but the large N c µ limit of the microscopic phase quenched partition function is now given by\n\nZ pq = e 2nNcµ 16n 4 (µ 2 -µ 2 c )µ 2 , ( 41\n\n)\n\nresulting in the average phase factor\n\ne 2iθ pq = Z N f =2 Z pq = µ 2 µ 2 c ( µ 2 µ 2 c -1)e -2nNc(µ-µc) for µ > µ c . ( 42\n\n)\n\nThis result can also be obtained from the small nµ, nµ c expansion of the large N c limit of the average phase factor (see Eq. ( 37 )) at fixed nN c µ c . We conclude that the sign problem becomes exponentially hard for µ > µ c . In Fig. 1 we show the average phase factor for different values of n and N c and a critical chemical potential of µ c = 0.1. In the thermodynamic limit at fixed N c , the average phase factor is one for µ < µ c and jumps to zero for µ > µ c . In the right figure we observe a rapid convergence to the mean field result discussed in section II C."},{"section_type":"OTHER","section_title":"B. Average phase factor for full QCD","text":"The average phase factor with the N f -flavor quark determinant as weight is defined by\n\ne 2iθ N f = 1 Z N f (µ c , µ) dU det D det D † det N f D = e nN f Ncµc Z N f (µ c , µ) dU det(1 -U e nµ-nµc ) det(1 -U † e -nµ-nµc ) det(1 -U e -nµ-nµc ) det(1 -U † e nµ-nµc ) det N f (1 -U e nµ-nµc )det N f (1 -U † e -nµ-nµc ). 9\n\nFIG. 1: The average phase factor for one-dimensional QCD with gauge group U (Nc) calculated as the ratio of the two flavor flavor partition function and the phase quenched partition function. In the left figure the number of lattice points varies as indicated for fixed Nc = 3, and in the right figure the number of colors varies as indicated for fixed n = 4.\n\nWe evaluate this integral exactly for N f = 0 and N f = 1 but only give its large N c limit for other values of N f . Using the CFZ-formula (18) [15] we obtain in the quenched case\n\ne 2iθ N f =0 = (1 -e -2n(µ+µc) )(1 -e 2n(µ-µc) ) (1 -e -2nµc ) 2 + e -2nNcµc (1 -e -2nµ )(1 -e 2nµ ) (1 -e 2nµc )(1 -e -2nµc ) for µ < µ c . ( 43\n\n)\n\nThe large N c limit of this result coincides with the large N c limit of the phase quenched average phase factor. In the microscopic limit where µ c nN c and µnN c remain fixed for N c → ∞ we obtain\n\ne 2iθ N f =0 = 1 - µ 2 µ 2 c - µ 2 µ 2 c e -2nNcµc . ( 44\n\n)\n\nIn the large nN c µ c limit the second term does not contribute resulting in\n\ne 2iθ N f =0 = 1 - µ 2 µ 2 c , ( 45\n\n)\n\nin agreement with the results obtained in [4] (see subsection II C). In the thermodynamic limit at fixed N c the average phase factor converges to one for µ < µ c . After rewriting the determinants in (43) as\n\ndet(1 -U † e -nµ-nµc ) det(1 -U † e nµ-nµc ) → e -2nNcµ det(1 -U e nµ+nµc ) det(1 -U e -nµ+nµc ) , ( 46\n\n)\n\nthe denominator of (43) can be expanded in powers of U for µ > µ c . For N f = 0 only the constant term in the integrand yields a non-vanishing result. We thus find\n\ne 2iθ N f =0 = e -2nNcµ for µ > µ c , ( 47\n\n)\n\nso that the average phase factor vanishes in the large nN c µ limit. In Fig. 2 we show the quenched average phase factor for various values of n and N c . Also in this case the average phase factor jumps from 1 to 0 at µ = µ c in the thermodynamic limit. The convergence to the mean field result for increasing N c (right figure) is very rapid. This can be understood from the expansion of the microscopic result given by 1\n\n-(µ 2 /µ 2 c )(1 -1 3 (nµ) 2 + • • •.\n\n10 FIG. 2: The quenched average phase factor for one-dimensional QCD with gauge group U (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\n\nFinally we calculate the average phase factor for an arbitrary number of flavors in the large N c limit. The CFZ formula can again be applied after inserting the factor\n\ndet N f (1 -U e -α )det N f (1 -U † e -β )\n\nin the denominator and taking the limit α → ∞ and β → ∞ at the end of the calculation.\n\nFor µ < µ c , the leading order large N c result is given by the identity permutation in the sum over permutations in the CFZ formula. This results in\n\ne 2iθ N f = (1 -e -2n(µ+µc) )(1 -e 2n(µ-µc) ) (1 -e -2nµc ) 2 N f +1 for µ < µ c . ( 48\n\n)\n\nThe microscopic limit of this result is given by\n\ne 2iθ N f = 1 - µ 2 µ 2 c N f +1 for µ < µ c , ( 49\n\n)\n\nin agreement with the discussion given in section II C. For µ > µ c the CFZ formula can again be applied after rewriting the partition function according to (46) and the insertion (48) . Because of degenerate prefactors, all critical chemical potentials that occur in the combination µ + µ c have to be taken different. After carefully taking limits we find\n\nZ N f +1|1 * (µ c , µ) = N N f c N f ! e -2µnNc e µcnNcN f (1 -e -2µcn ) N 2 f (1 -e -2n(µ+µc) ) N f (1 -e -2nµ ) N f (1 -e -2nµc ) N f . ( 50\n\n)\n\nThe average phase factor given by\n\ne 2iθ N f = Z N f +1|1 * Z N f ∼ e -2nNcµ for µ > µ c ( 51\n\n)\n\nvanishes in the thermodynamic limit. The exact result becomes rather cumbersome even for small values of N f . As an illustration we give the result for Z 2|1 * (µ c , µ) in Appendix A . 11 V. AVERAGE PHASE FACTOR FOR SU (Nc)\n\nTo calculate integrals over SU (N c ) we use the identity SU(Nc)\n\ndU • • • = ∞ p=-∞ U(Nc) dU det p (U ) • • • . ( 52\n\n)\n\nFor the partition functions discussed below, with a few exceptions the sum truncates to a small number of terms.\n\nA. SU (Nc) QCD Partition Function\n\nLet us first calculate the 1d SU (N c ) QCD partition function for N f = 1. The partition function is defined by\n\nZ SU(Nc) N f (µ c , µ) = SU(Nc) dU det N f D = ∞ p=-∞ U(Nc) dU det p U e nNcµc det N f (1 + U e nµ-nµc )det N f (1 + U † e -nµ-nµc ). ( 53\n\n)\n\nFor p ≥ N f the integrand can be rewritten in terms of determinant of matrices U only\n\nZ N f ,p (µ c , µ) = U(Nc) dU det p U det N f D = U(Nc) dU e nNcN f µc det p-N f (U )det N f (1 + U e nµ-nµc )det N f (U + e -nµ-nµc ) . ( 54\n\n)\n\nFor p > N f it follows immediately that the integral vanishes. For p = N f only the constant term inside the determinants contributes to integral resulting in\n\nZ N f ,p=N f (µ c , µ) = e -nN f Ncµ , Z N f ,p>N f (µ c , µ) = 0. ( 55\n\n)\n\nFor p < 0 we combine det p U = det -p U † with the factor det N f (1 -U e nµ-nµc ). We then find\n\nZ N f ,p=-N f (µ c , µ) = e nN f Ncµ , Z N f ,p<-N f (µ c , µ) = 0. ( 56\n\n)\n\nUsing that the hermitean conjugate of a unitary matrix is also unitary we obtain\n\nZ N f ,-p (µ c , µ) = Z N f ,p (µ c , -µ). ( 57\n\n)\n\nFor |p| < N f the integral can be calculated by means of the CFZ formula by choosing all µ c different and introducing the limit\n\ndet U = lim α→∞ det(U -e -α ). ( 58\n\n)\n\nThe limit of degenerate µ c and α → ∞ is taken at the end of the calculation. We will only give exact results for N f = 1 and N f = 2. Using (55,56) and the result for U (N c ) we obtain for N f = 1 Z SU(Nc)\n\nN f =1 (µ c , µ) = e -nN f N f µ + e nN f N f µ + sinh(n(N c + 1)µ c ) sinh(nµ c ) , ( 59\n\n)\n\nin agreement with earlier work by Bilic and Demeterfi [24] . For N f = 2, the only integrals that have not yet been calculated are those for p = ±1 which are related by (57). For the p = 1 contribution we find\n\nZ N f =2,p=1 (µ c , µ) = e nNc(µc-µ) N c e -2n(Nc+1)µc + e 2nµc (e nµc -e -nµc ) 2 - e -2nNcµc (e -3nµc -3e -nµc ) -e 3nµc + 3e nµc (e nµc -e -nµc ) 3 . ( 60\n\n)\n\n12 For µ < µ c the large nN c µ limit of the N f = 2 partition function is dominated by the p = 0 term, whereas for µ > µ c the large nN c µ c limit is given by the p = -2 term. For µ = µ c the large N c limit is dominated by the p = -1 with 1/N c corrections from the p = 0 and p = -2 terms. We thus find as leading large N c result Z SU(Nc)\n\nN f =2 (µ c , µ) = e 2n(Nc+2)µc 16 sinh 4 (nµ c ) for µ < µ c , Z SU(Nc) N f =2 (µ c , µ) = e 2nNcµ for µ > µ c , Z SU(Nc) N f =2 (µ c , µ) = N c e 2n(Nc+1)µc 4 sinh 2 (nµ c ) 1 + O(N -1 c ) for µ = µ c . ( 61\n\n)\n\nBoth for µ < µ c and µ > µ c the terms that are canceled by the integration over the unitary group are subleading in the thermodynamic limit. In other words there is no serious sign problem. By inspection one can easily show that the dominance of the p = 0 terms for µ < µ c and the p = -N f term for µ > µ c is a feature of the large N c limit that is valid for any number of flavors. Therefore, for µ < µ c , the chiral condensate is the same as for the U (N c ) theory. For µ > µ c though, the dominant p = -N f term is mass-independent. This results in a vanishing chiral condensate so that the large N c limit of the U (N c ) theory and the SU (N c ) theory is different.\n\nB. The Phase Quenched Partition Function for SU (Nc)\n\nA second ingredient for the average phase factor is the phase quenched SU (N c ) partition function. Using the same arguments as for the full QCD partition function, one easily obtains\n\nZ 1+1 * ,|p|>2 (µ c , µ) = 0, Z 1+1 * ,|p|=2 (µ c , µ) = 1, Z 1+1 * ,-p (µ c , µ) = Z 1+1 * ,p (µ c , -µ). ( 62\n\n)\n\nThe partition function for p = 0 is the U (N c ) partition function which was already given in (33) . What remains to be calculated is the partition function for p = +1. Using the CFZ formula one easily arrives at\n\nZ 1+1 * ,p=1 (µ c , µ) = 1 4 sinh(nµ) sinh(nµ c ) sinh(n(N c + 2)(µ + µ c )) sinh(n(µ + µ c )) - sinh(n(N c + 2)(µ -µ c )) sinh(n(µ -µ c ))) . ( 63\n\n)\n\nBoth for µ < µ c and µ > µ c , the leading large N c result of the phase quenched partition function resides in the p = 0 term given in Eq. (35).\n\nC. Average Phase Factor for SU (Nc)\n\nThe average phase factor can be either obtained from the ratio of the full QCD partition function and the phase quenched partition function or can be calculated with the full QCD partition function as weight (see Eqs. (2) and (3)). In the first case we find for large N c ,\n\ne 2iθ SU(Nc) pq = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc) for µ < µ c , 16 sinh(n(µ -µ c )) sinh(n(µ c + µ)) sinh 2 (nµ) for µ > µ c , 1 -e -4nµc for µ = µ c . ( 64\n\n)\n\nThe microscopic limit of the average phase factor is obtained by expanding this result for small µ c , µ. We again find the usual mean field result\n\ne 2iθ SU(Nc) pq = 1 - µ 2 µ 2 c θ(µ c -µ). ( 65\n\n)\n\nIn the thermodynamic limit, the average phase factor converges to one in both for µ < µ c and µ > µ c . 13 FIG. 3: The phase quenched average phase factor for one-dimensional QCD with gauge group SU (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\n\nExact results for the average phase factor are displayed in Fig. 3 . At finite N c , we observe a rapid convergence to the asymptotic value of 1. At fixed n, the approach to the large N c limit is slow. Also clearly visible is that for µ ≈ µ c the corrections to the microscopic limit are large. As was already discussed before, this is due to the p = -1 contribution.\n\nIn the quenched case the average phase factor for SU (N c ) defined according to Eq. ( 3 ) can be written as\n\ne 2iθ SU(Nc) N f =0 = ∞ p=-∞ U∈U(Nc) dU det p (U ) det(1 + U e nµ-nµc ) det(1 + U † e -nµ-nµc ) det(1 + U e -nµ-nµc ) det(1 + U † e nµ-nµc ) . ( 66\n\n)\n\nThe integrals can again be calculated by means of the CFZ formula. For N c ≥ 3 one can easily show that the limit introduced in (58) gives vanishing results for |p| > 2. However, for N c = 1 and N c = 2 additional terms may contribute to this limit. For N c = 2, where the average phase factor is equal to one, contributions for |p| > 2 vanish for µ < µ c , but terms with p < -2 are nonzero for µ > µ c . The formulae for |p| ≤ 2 given below are valid for N c = 2. For N c = 1 the integrals in (66) only vanish for p ≥ 2 and µ > µ c . Also the general answer for p = -2 is not correct in this case. For p = -1 we obtain Ncn(µc-µ) (1-e -2n(µc +µ) )(1-e -2nµ ) 1-e -2nµc for µ < µ c ,\n\ne 2iθ N f =0 = e -\n\ne nNc(µc -µ) (1-e -2nµ ) 1-e -2nµc [(1 -e -2nµc-2nµ ) -e -2nNcµc (e -2nµc -e -2nµ )] for µ > µ c . ( 67\n\n)\n\nFor p = 1 we find e 2iθ N f =0 = e -Ncn(µc+µ) (1-e -2n(µc -µ) )(1-e 2nµ ) 1-e -2nµc for µ < µ c ,\n\n0 for µ > µ c . ( 68\n\n)\n\nFinally, for p = -2 the result is\n\ne 2iθ N f =0 = 0 for µ < µ c , (1 -e -2nµ )(1 -e -2n(µ+µc) )(1 -e 2n(µc-µ) )(1 -e -2nµ ) for µ > µ c . ( 69\n\n)\n\nFor p = 2 the average phase factor vanishes. Notice that the relation\n\ne 2iθ N f =0,-p (µ c , µ) = e 2iθ N f =0,p (µ c , -µ) ( 70\n\n) 14 FIG. 4: Average phase factor for one-dimensional QCD with gauge group SU (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\n\nis also valid in this case. It can be used to derive results for negative values of µ. In the large N c limit, the average phase factor is dominated by the p = 0 contribution for µ < µ c and by the p = -2 contribution for µ > µ c . For µ = µ c , the p = 0, p = -1 and p = -2 contributions are of equal order in N c , but only the p = -1 term is nonvanishing. As large N c limit of the average phase factor we thus find e 2iθ SU(Nc) N f =0 = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc) for µ < µ c , 16 sinh(n(µ -µ c )) sinh(n(µ c + µ)) sinh 2 (nµ) for µ > µ c ,\n\n1 -e -4nµc for µ = µ c , ( 71\n\n)\n\nwhich is the same expression as obtained in (64) for the phase quenched average phase factor. The microscopic limit of the average phase factor is again given by the small µ-µ c expansion of this result which is equal to the usual mean field result\n\ne 2iθ SU(Nc) N f =0 = 1 - µ 2 µ 2 c θ(µ c -µ). ( 72\n\n)\n\nIn Fig. 4 we show the exact result for the SU (N c ) average phase factor. We observe a rapid approach to the thermodynamic limit at fixed N c just like in Fig. 3 . In the large N c limit the result for the quenched average phase calculated according to (66) is also given by (64). As already was argued in the introduction, we conclude that QCD in one dimension with SU (N c ) as gauge group does not have a serious sign problem, not only for µ < µ c , but also for µ > µ c . Indeed one-dimensional QCD at nonzero chemical potential could be simulated reliably by the Glasgow method [26] ."},{"section_type":"OTHER","section_title":"VI. THE SIGN PROBLEM AT NONZERO TEMPERATURE","text":"The results of the previous two sections show that the sign problem is severe for QCD in one dimension with gauge group U (N c ) and µ > µ c . Although this theory does not have a critical temperature, it is clear from Fig. 1 that the average phase factor for µ > µ c increases significantly for higher temperatures (i.e. for lower values of n). However, this is a geometric effect due to the Boltzmann factor. Recent lattice simulations [10, 11, 12, 26, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] suggest that the sign problem becomes much milder for T around T c . Because of the absence of a critical temperature, this temperature dependence of the average phase factor cannot be investigated in 15 FIG. 5: Scatter plot of the eigenvalues of the random matrix Dirac operator for µ =0.3 and T = 0, 0.7, 0.9, 1.0, 1.1 . The quark mass at m = 0.1 is indicated by the black dot.\n\none dimensional QCD. Instead we study this problem in a schematic random matrix model at nonzero temperature and chemical potential [30, 48, 49] . This model has a severe sign problem as well as a critical temperature and its eigenvalues are scattered in a finite domain of the complex plane, just like in QCD. The quenched average phase factor in this model is given by the matrix integral\n\ne 2iθ (m, µ, T ) = dCe -N TrCC † det(D + m) det(D † + m) , ( 73\n\n)\n\nwith Dirac operator given by\n\nD = 0 iC + µ + it iC † + µ + it 0 , ( 74\n\n)\n\nand t is the traceless diagonal matrix t = diag(-T, • • • , -T, T, • • • , T ). The integral is over the real and imaginary parts of the matrix elements of the complex N × N matrices C. The phase quenched average phase factor is defined by\n\ne 2iθ pq = dCdet 2 (D + m)e -N TrCC † dCdet(D + m) det(D † + m)e -N TrCC † ( 75\n\n)\n\nIf at all possible, it certainly takes a significant effort to evaluate these integrals analytically. Therefore we have studied the quenched (73) and phase quenched (75) average phase factor numerically. In Fig. 5 , we give scatter plots of the eigenvalues of the random matrix Dirac operator. For each of the temperatures T = 0, 0.7, 0.9, 1, 1.1, the figures show results of 40 matrices with N = 100 and µ = 0.3. The black dot represents the quark mass for which the average phase factor is calculated. In Fig. 6 we show the µ-dependence of the phase quenched (left) and quenched (right) average phase factor obtained by averaging over 10,000 to 80,000 matrices C with N = 100. The results for the phase quenched average phase factor for T = 0 and T = 0.7 and µ > µ c are not displayed because they do not converge because of the severity of the sign problem. The average phase factor depends only weakly on the size of 16 FIG. 6: Phase quenched (left) and quenched (right) average phase factor versus chemical potential. Results have been obtained from the chiral random matrix model (74).\n\nthe matrices. The conclusion of Figs. 5 and 6 is that the sign problem is not serious when the quark mass is outside the domain of the eigenvalues. This confirms the conjecture that was made in [4, 5, 50] . It also agrees with the observation in [30] that the free energy of the full theory and the phase quenched theory are the same for T = 0 if the quark mass is outside the domain of the eigenvalues (this is the case for µ < µ c ). The µ-dependence is due to the curvatures at the saddle point."},{"section_type":"OTHER","section_title":"VII. COMPARISON WITH THE QCD3 PARTITION FUNCTION","text":"According to the general arguments given in section II B, the microscopic limit of the one dimensional QCD partition function at nonzero chemical potential is equal to the microscopic limit of the QCD 3 partition function at zero chemical potential, but with shifted quark masses. In this section we show this explicitly for some of the results derived before. We first discuss the microscopic limit of the QCD 3 partition function which can be derived from either a chiral Lagrangian or a random matrix model with the same global symmetries [28] . For a discussion of QCD 3 in terms of chiral Lagrangians we refer to [51] ."},{"section_type":"OTHER","section_title":"A. Random Matrix Model","text":"From the continuum Dirac operator (9) it follows that the random matrix model that describes the fluctuations of the low-lying Dirac eigenvalues is given by\n\nZ = dHP (H) det(iH + µ + m) det(iH -m + µ), ( 76\n\n)\n\nwhere the probability distribution can conveniently be taken to be the Gaussian distribution\n\nP (H) = e -2N Σ 2 TrH 2 . ( 77\n\n)\n\nSuch random matrix partition functions have been studied elaborately in the literature to analyze the microscopic limit of QCD in three dimensions [28, 52, 53, 54, 55] . The microscopic limit of the partition function (76)\n\nwith quark masses given by M = diag(-m 1 , • • • , -m N f , m 1 , • • • , m N f ) is equal to [53, 54] Z QCD3 = 1 ∆(M ) det A(m) A(-m) A(-m) A(m) , ( 78\n\n)\n\n17 where the matrix elements of the N f × N f matrix A(m) are given by\n\nA(m) kl = m l k e -m k , ( 79\n\n)\n\nand ∆ is the Vandermonde determinant ∆(M ) =\n\n2N f k>l (M kk -M ll ). ( 80\n\n)\n\nAlthough general expressions for QCD 3 partition functions with an arbitrary number of bosonic and fermionic determinants are also known [56] , for our purposes we only need the partition function\n\nZ N f +2|2 = N f k=1 det(D + m k ) det(D -m k ) det(D + z 1 ) det(D -z 2 ) det(D + z1 ) det(D -z2 ) . ( 81\n\n)\n\nThis partition function was evaluated in [57] by means of the supersymmetric method and is given by\n\nZ N f +2|2 = (z 1 -z1 )(z 2 -z2 ) ∆ 2 N f (m) exp N c (z 1 + z2 ) N c (z 1 + z2 ) N f k=1 (m k -z1 )(m k -z2 ) (m k -z 1 )(m k -z 2 ) det sinh Nc(m k +m l ) Nc(m k +m l ) sinh Nc(m k +z1) Nc(m k +z1) sinh Nc(z2+m l ) Nc(z2+m l )\n\nsinh Nc(z1+z2) Nc(z1+z2) + e -Nc(z1-z1)+Nc(z2-z2)\n\n∆ 2 N f (m) det sinh N c (m k + m l ) N c (m k + m l ) . ( 82\n\n)\n\nThe last term is a so-called Efetov-Wegner term, and the Vandermonde determinant is over positive masses only\n\n∆(m) = N f k>l (m k -m l ). ( 83\n\n)"},{"section_type":"OTHER","section_title":"B. Microscopic Limit","text":"We will now show that the microscopic limit of the partition function (25) is equal to the microscopic limit of the QCD 3 partition function. To this end we multiply row k of the first N f rows of the determinant by exp(-N c m -k /2) and row k of the second N f rows of the determinant in (25) by exp(-N c m + k /2). In the microscopic limit we keep N c m ±,k fixed so that we can expand the masses that do not occur in this combination. By subtracting successive columns starting with the first one we obtain\n\nZ N f = 1 1≤k<l≤2N f (M l -M k ) ( 84\n\n) × e -Ncm-1 /2 m -1 e -Ncm-1/2 • • • m N f -1 -1 e -Ncm-1/2 m N f -1 e +Ncm-1/2 • • • m 2N f -1 -1\n\ne Ncm-1 /2 . . . . . . . . . . . . . . .\n\ne -Ncm-N f /2 m -N f e -Ncm-N f /2 • • • m N f -1 -N f e -Ncm-N f /2 m N f -N f e +Ncm-N f /2 • • • m 2N f -1 -N f e Ncm-N f /2 e -Ncm+ 1/2 m + 1 e -Ncm+ 1/2 • • • m N f -1 + 1 e -Ncm+ 1 /2 m N f + 1 e +Ncm+ 1 /2 • • • m 2N f -1 + 1\n\ne Ncm+ 1/2 . . . . . . . . . . . . . . .\n\ne -Ncm+ N f /2 m + N f e -Ncm+ N f /2 • • • m N f -1 + N f e -Ncm+ N f /2 m N f + N f e +Ncm+ N f /2 • • • m 2N f -1 + N f e Ncm+ N f /2 ,\n\nwhere the masses M K are defined below Eq. ( 25 ). By multiplying the columns\n\nl = N f + 1, • • • , 2N f by (-1) l-1 and\n\nintroducing microscopic masses, we obtain exactly the expression for the QCD 3 partition function given in (78) . For masses -(µ c + µ), -(µ c -µ), µ c -µ, µ c + µ corresponding to the microscopic limit of the two-flavor phase quenched partition function given in (34) the determinant in (84) is given by\n\n4nµ 2 (e 2nNcµc + e -2nNcµc ) -4nµ 2 c (e 2nNcµ + e -2nNcµ ) + 8n(µ 2 c -µ 2 ). ( 85\n\n)\n\nand the prefactor is equal to\n\n1≤k<l≤2N f (M l -M k ) = 64n 6 (µ 2 c -µ 2 )µ 2 c µ 2 . ( 86\n\n) 18\n\nFIG. 7: The number variance for n = 4 and Nc = 100.\n\nTheir ratio coincides with the microscopic phase quenched partition function given in (34) .\n\nAs second example, we consider the quenched average phase factor for U (N c ) given in (43) . For N f = 0 the partition function (82) simplifies to Nc(z1-z1-z2+z2) .\n\nZ 2|2 = (z 1 -z1 )(z 2 -z2 ) e Nc(z1+z2) N c (z 1 + z2 ) sinh N c (z 1 + z 2 ) N c (z 1 + z 2 ) + e -\n\n( 87\n\n)\n\nThe masses in this partition function corresponding to (43) are given by\n\nz 1 = nµ c + nµ, z 2 = nµ c -nµ, z2 = nµ -nµ c , z1 = -nµ -nµ c . ( 88\n\n)\n\nSubstituting them into (87) we obtain\n\nZ 2|2 = 1 - µ 2 µ 2 c + µ 2 µ 2 c e -4Ncnµ ( 89\n\n)\n\nwhich is exactly the microscopic limit of the quenched average phase factor (43) given in Eq. (44)."},{"section_type":"OTHER","section_title":"VIII. DIRAC SPECTRUM","text":"We consider the Dirac operator (4) in a gauge where all gauge fields except U n1 ≡ U and U 1,n = U † are equal to unity. If the eigenvalues of U are equal to exp(iθ k ) the eigenvalues of the Dirac operator are given by\n\nλ k,l = 1 2 (e 2πi(k+1/2)+iθ l n +µ -e -2πi(k+1/2)-iθ l n -µ ), k = 1, • • • , n, l = 1, • • • , N c . ( 90\n\n)\n\nContrary to QCD at µ = 0 in more than one dimension, where the eigenvalues are scattered in the complex plane [27, 30, 58, 59, 60, 61] , the eigenvalues are located on an ellipse in the complex plane with real and imaginary parts related by\n\nRe(λ k,l ) e µ -e -µ 2 + Im(λ k,l ) e µ + e -µ 2 = 1. ( 91\n\n) 19\n\nA. Universal Fluctuations In section II B we have shown that the continuum limit of the staggered Dirac operator in one dimension is in the same universality class as QCD in three dimensions. The continuum limit of the staggered three dimensional Dirac operator, though, is in the chiral symmetry universality class of QCD in four dimensions. For example, in [62] this was found for the distribution of the small Dirac eigenvalues. The reason for the chiral structure is that the staggered Dirac operator only couples even and odd lattice sites. This is also the case in one dimension, but the off-diagonal blocks, containing the gauge fields, are the same in the continuum limit (or occur in the combination U n + U † n ), resulting in a two flavor theory with opposite masses (see (12)). The secular equation is given by\n\ndet[(∂ 0 + iA 0 + µ + λ)(∂ 0 + iA 0 + µ -λ)] = 0. ( 92\n\n)\n\nThis corresponds to the superposition of the spectrum of\n\n∂ 0 + A 0 + µ and -∂ 0 -iA 0 -µ.\n\nIn the domain of the Dirac spectrum where ∂ 0 can be neglected, the eigenvalues on each of the lines ±µ are given by the Hermitean random matrix ensemble A 0 . For µ = 0 we have the superposition of the Hermitean random matrix ensembles of A 0 and -A 0 . From the eigenvalues (90) it is clear that the staggered lattice Dirac eigenvalues show a similar superposition. One ensemble is localized on the left half of the ellipse and the other ensemble on its right half. Superpostions of more ensembles arise for correlations on scales larger than N c level spacings. Eigenvalues of U (N c ) matrices are correlated according to the circular unitary ensemble. Therefore, the eigenvalues λ k,l are correlated according the Gaussian Unitary Ensemble (GUE) on a scale where the variations in the average spectral density can be neglected, i.e. on a scale much less than N c level spacings. By correcting for the variation of the average level spacing, the scale on which universal random matrix correlations are found can be extended, but this scale should always remain well below N c level spacings. Then we are necessarily within one of the n successive copies of the eigenvalues along the ellipse. At a scale of a finite number of average level spacings, we expect convergence to universal random matrix correlations in the large N c limit.\n\nAs measure of the correlations between eigenvalues, we use the number variance Σ 2 (n) defined as the variance on the number of eigenvalues in an interval that contains n eigenvalues on average. We calculate the number variance for intervals starting from zero. Because of the superposition of an enemble and its negative, this is equal to the number variance of a single ensemble for an interval that is symmetric about zero which is given by the GUE result. In Fig. 7 we show the number variance obtained from an average over 10 5 gauge field configurations for N c = 100 and µ = 0. The number variance is calculated for an interval starting at zero. The discrepancy between the analytical result and the GUE (solid curve) is only barely visible.\n\nThe increase of the domain of validity of chiral random matrix theory with increasing N c was also observed for Dirac operator of QCD in four dimensions [63] . This can be explained as follows: The number of eigenvalues that is described by random matrix theory is equal to ≈ F 2 π √ V (with V the space-time volume), but F 2 π scales with N c in the large N c limit."},{"section_type":"OTHER","section_title":"B. Chiral Symmetry Breaking at µ = 0","text":"We first discuss the case of µ = 0. Then the average spacing of the eigenvalues scales as 1/(nN c ). This means that the chiral condensate develops a discontinuity at m = 0 when nN c → ∞. Two cases of interest where a nonzero chiral condensate can be obtained are the limit n → ∞ for fixed N c , i.e. at zero temperature, or at fixed n in the limit N c → ∞. In both cases the chiral condensate is discontinuous for nN c → ∞ when the quark mass crosses the imaginary axis where the eigenvalues are located. However, Goldstone bosons and universal behavior is only found in the large N c limit. At fixed N c , chiral symmetry breaking is not associated with universal behavior.\n\nFor µ = 0 the Dirac eigenvalues are located on the ellipse (91). When the quark mass is inside the ellipse of eigenvalues the chiral condensate is zero for nN c → ∞ for the SU (N c ) partition function and the phase quenched U (N c ) partition function [24] . For gauge group U (N c ) with N f ≥ 1 flavors, though, the partition function is µindependent so that the chiral condensate is also nonzero if the mass is inside the eigenvalue ellipse. The Banks-Casher formula fails in this case which is known as the \"Silver Blaze Problem\" [64] . The resolution is the same as for QCD in four dimensions [31] : the re-weighted eigenvalue distribution defined as\n\nρ full (z) = 1 Z N f U(Nc) dU det N f D k δ 2 (z -λ k ) ( 93\n\n)\n\nshows oscillations with an amplitude that diverges exponentially with n and a period that is proportional to 1/n. Below we will illustrate this for U (1).\n\n20 For N c = 1 one easily derives that the spectral density is given by ρ full (z) = 4 e nµc + e -nµc -e n(iα+µ) -e -n(iα+µ) (e nµc + e -nµc )(e 2r + e -2r + e 2iα + e -2iα ) δ(r -µ), (\n\n) 94\n\nwhere z is parameterized as\n\nz = 1 2 (e r+iα -e -r-iα ), r > 0, α ∈ [0, 2π], ( 95\n\n)\n\nand we have used that Z N f =1 = 2 cosh(nµ c ) (see eq. ( 8 )). For µ > µ c this spectral density has oscillations with an amplitude that diverges exponentially with n and a period of order 1/n which are the essential properties of the spectral density of the Dirac operator of QCD in four dimensions [31, 65, 66] . It can be decomposed as\n\nρ full (z) = ρ q (z) + ρ osc (z). ( 96\n\n)\n\nThe average quenched spectral density is defined by\n\nρ q (z) = 1 2π dθ 1 n n k=1 δ 2 (z -λ k ) ( 97\n\n)\n\nwith the eigenvalues λ k given by Eq. ( 90 ). After changing variables according to (95) and integrating over θ we find\n\nρ q (z) = 1 2π 4 (e 2r + e -2r + e 2iα + e -2iα ) δ(r -µ). ( 98\n\n)\n\nThe oscillatory part of the spectral density, ρ osc (z) is equal to the difference ρ full (z) -ρ q (z). The chiral condensate is given by\n\nΣ full (m) = d 2 z ρ full (z) z + m , ( 99\n\n)\n\nand can also be decomposed as\n\nΣ full (m) = Σ q (m) + Σ osc (m). ( 100\n\n)\n\nThe Jacobian for the transformation of d 2 z to drdα is given by\n\nd 2 z = 1 4 (e 2r + e -2r + e 2iα + e -2iα )drdα, ( 101\n\n)\n\nso that the chiral condensate after integration over r can be simplified to\n\nΣ full (m) = 1 π 2π 0 dα e nµc + e -nµc -e n(iα+µ) -e -n(iα+µ) (e nµc + e -nµc )(e µ+iα -e -µ-iα -2m) = tanh(nµ c ) cosh µ c . ( 102\n\n)\n\nThis result remains finite for n → ∞. We remind the reader that the mass is parameterized as m = sinh µ c . This expression represents the resolvent at the quark mass which was evaluated in [49] and is in agreement with earlier work [6, 24] . Decomposing the spectral density and the chiral condensate according to (96) we obtain\n\nΣ osc (m) = θ(sinh(µ) -m) Σ full (m), Σ q (m) = θ(m -sinh(µ)) Σ full (m), ( 103\n\n)\n\nso that when the quark mass is inside the ellipse of eigenvalues, the entire chiral condensate is due to the oscillatory part of the spectral density. The discontinuity in the chiral condensate is reminiscent to a Stokes phenomenon. The alternative to the Banks-Casher relation proposed in [31] is also at work for QCD in one dimension. This solves the \"Silver Blaze Problem\" [64]. 21 IX. CONCLUSIONS We have studied QCD in one dimension at nonzero chemical potential. Both the full theory, its quenched and phase quenched versions and gauge groups U (N c ) and SU (N c ) have been considered. In one dimension, the QCD or QCD-like partition functions can be reduced to a single matrix integral, which because of recent advances by Conrey, Farmer and Zirnbauer, could be evaluated analytically. In this paper we have analyzed the small mass behavior of the partition function, the nature of the sign problem, and the relation between the Dirac spectrum and chiral symmetry breaking.\n\nTo put our results in perspective, we emphasize that QCD in one dimension, is quite different from QCD in more dimensions. In particular, we wish to mention the following three points. First, instead of being scattered in the complex plane, the eigenvalues of the staggered Dirac operator are located on an ellipse in the complex plane. Second, phase transitions can only take place for zero temperature or for N c → ∞. At the critical chemical potential, a transition from the vacuum state to a state of free quarks takes place. Third, color singlets are made out of noninteracting quarks so that the critical chemical potential is given by the quark mass. In particular, the critical chemical potential for the meson state and the baryon state is the same.\n\nBecause the Dirac eigenvalues are located on a curve, the large N c limit of staggered lattice QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. We have shown this both by an explicit evaluation of the partition function, and by analyzing the fluctuations of the Dirac eigenvalues for large N c . We have used this equivalence to explain the behavior of the partition functions and the average phase factor in the microscopic limit.\n\nContrary to QCD in four dimensions, the sign problem for QCD in one dimension is not severe in the case of gauge group SU (N c ). One reason is that the critical chemical potential for full QCD and phase quenched QCD is the same, so that the parameter domain m π /2 < µ < m N /3, where the sign problem becomes severe in four dimensions, is absent in one dimension. A second reason is that for µ > µ c both the full theory and the phase quenched theory become a theory of free quarks with the same free energy in the thermodynamic limit.\n\nFor gauge group U (N c ), on the other hand, the sign problem is severe when µ > m π /2 both in one dimension and in four dimensions. The reason is that the U (N c ) theory does not have charged excitations, whereas the phase quenched theory has charged mesons resulting in a phase transition at µ = m π /2. We have evaluated the average phase factor both by averaging with respect to the phase quenched partition function and the full partition function, and similar conclusions have been reached.\n\nThe condititon µ > m π /2 for having a severe sign problem can be rephrased as the quark mass being inside the ellipse of eigenvalues. In more dimensions this condition is that the quark mass is inside the support of the Dirac spectrum. It also applies to nonzero temperature as we have demonstrated explicitly in the framework of a chiral random matrix model. Also for U (N c ) QCD in one dimension, the chiral condensate is discontinuous across the imaginary axis in spite of the fact that there are no Dirac eigenvalues. This can only mean that the phase of the fermion determinant is responsible for the discontinuity. This is what happens in four dimensions where the discontinuity is due to a contribution to the spectral density that oscillates with a period of the inverse volume and amplitude that diverges exponentially with the volume. Exactly the same mechanism, where the discontinuity in the chiral condensate arises due to a Stokes like phenomenon, is at work in one dimension. This suggests that this is a universal mechanism for theories with a sign problem.\n\nWe end by repeating that QCD in one dimension with SU (N c ) as gauge group has no serious sign problem. Our hope is that part of this conclusion translates to four dimensions ameliorating the sign problem for µ > M N /3."},{"section_type":"OTHER","section_title":"Acknowledgments.","text":"We wish to thank K. Splittorff for numerous comments and criticism and P.H. Damgaard, H. Neuberger and R.Pisarski are thanked for valuable discussions. K. Splittorff is also thanked for a careful reading of the manuscript. This work was supported by U.S. DOE Grant No. DE-FG-88ER40388 (JV), the Angelo Della Riccia Foundation (LR), the Villum Kann Rassmussen Foundation (JV) and the Danish National Bank (JV).\n\nAPPENDIX A: AVERAGE U (Nc) PHASE FACTOR FOR N f = 1\n\nIn this appendix we give explicit results for the average phase factor for N f = 1 defined by\n\ne 2iθ N f =1 = Z 2|1 * (µ c , µ) Z N f =1 (µ c , µ) . ( A1\n\n)\n\nThe partition function Z N f =1 (µ c , µ) was already given in Eq. ( 8 ). The numerator can be obtained from the CFZ formula. After taking the limit of degenerate critical chemical potentials at the end of the calculation, we obtain for 22 µ < µ c Z 2|1 * (µ c , µ) = e n(Nc+1)µc e -4nµ (1 -e 2n(µ+µc) ) 2 (1 -e 2n(µ-µc) ) 2 (e nµc -e -nµc ) 5 + e -3n(Nc+1)µc (e nµ -e -nµ ) 4 (e nµc -e -nµc ) 5 +e -n(Nc+1)µc [f 0 + N c f 1 + N 2 c f 2 ]. (A2) with f 0 = 2e 2n(µ+µc) + 2e 2n(µ-µc) -e 4nµ -e -4nµ + 4e 2nµ + 4e -2nµ -e 4n(µ-µc) -e 4n(µ+µc) + 2e 6nµc-2nµ + 2e 6nµc+2nµ (e nµc -e -nµc ) 5 + -10e 2nµc + 6e 4nµc -2 -5e 6nµc -e -2nµc (e nµc -e -nµc ) 5 ,\n\nf 1 = (1 -e -2nµ ) 2 (3e 2nµc + e -2nµc -2e 2nµ -2e -2nµ ) (e nµc -e -nµc ) 3 , f 2 = - (1 -e -2nµ ) 2 (1 -e n(µ+µc ))(1 -e n(µ-µc )) (e nµc -e -nµc ) 3 . ( A3\n\n)\n\nFor µ > µ c a similar calculation results in\n\nZ 2|1 * = N c e nNc(µc-2µ) (1 -e -2n(µ+µc) )(1 -e -2nµ ) (1 -e -2nµc ) 2 (1 + g 1 N c ) + N c e -nNc(2µ+µc) (1 -e 2n(µc-µ) )(1 -e -2nµ ) (1 -e 2nµc ) 2 (1 + g 2 N c ) ( A4\n\n) with g 1 = 1 + 1 e 2nµ -1 + 1 e 2n(µ+µc) -1 - 2 e 2nµc -1 , g 2 = 1 + 1 e 2n(µ-µc) -1 + 1 e 2nµ -1 - 2 e -2nµc -1 . ( A5\n\n)\n\nWe have checked that the results in this appendix agree with the large N c limit of the average phase factor given in section IV B.\n\n[1] M. P. Lombardo, arXiv:hep-lat/0612017. [2] C. Schmidt, PoS LAT2006, 021 (2006) [arXiv:hep-lat/0610116]. [3] K. Splittorff, arXiv:hep-lat/0610072. [4] K. Splittorff and J. J. M. Verbaarschot, [arXiv:hep-lat/0702011]. [5] K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 98, 031601 (2007) [arXiv:hep-lat/0609076]. [6] P. E. Gibbs, Preprint PRINT-86-0389-GLASGOW, 1986; P. E. Gibbs, Phys. Lett. B 182 (1986) 369. [7] D. Toussaint, Nucl. Phys. Proc. Suppl. 17, 248 (1990). [8] P. de Forcrand and V. Laliena, Phys. Rev. D 61, 034502 (2000) [arXiv:hep-lat/9907004]. [9] Y. Sasai, A. Nakamura and T. Takaishi, Nucl. Phys. Proc. Suppl. 129, 539 (2004) [arXiv:hep-lat/0310046]. [10] S. Ejiri, Phys. Rev. D 69, 094506 (2004) [arXiv:hep-lat/0401012]; S. Ejiri, Phys. Rev. D 73, 054502 (2006) [arXiv:hep-lat/0506023]. [11] C. Schmidt, Z. Fodor and S. D. Katz, arXiv:hep-lat/0512032. [12] C. R. Allton et al., Phys. Rev. D 66, 074507 (2002) [arXiv:hep-lat/0204010]; C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]; C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [13] E. V. Shuryak and J. J. M. Verbaarschot, Nucl. Phys. A 560, 306 (1993) [arXiv:hep-th/9212088]. [14] J. J. M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994) [arXiv:hep-th/9401059]. [15] J.B. Conrey, D.W. Farmer and M.R. Zirnbauer, [arXiv:math-ph/0511024], 2005. [16] E. M. Ilgenfritz and J. Kripfganz, Z. Phys. C 29, 79 (1985). [17] P. H. Damgaard, N. Kawamoto and K. Shigemoto, Phys. Rev. Lett. 53 (1984) 2211; Nucl. Phys. B 264, 1 (1986). [18] A. Gocksch and M. Ogilvie, Phys. Rev. D 31 (1985) 877. [19] P. H. Damgaard, D. Hochberg and N. Kawamoto, Phys. Lett. B 158 (1985) 239. [20] N. Bilic, K. Demeterfi and B. Petersson, Nucl. Phys. B 377, 651 (1992). [21] P. de Forcrand and S. Kim, arXiv:hep-lat/0608012.\n\n[22] P. E. Gibbs, Phys. Lett. B 172, 53 (1986). [23] A. Gocksch, Phys. Rev. D 37, 1014 (1988). [24] N. Bilic and K. Demeterfi, Phys. Lett. B 212, 83 (1988). [25] S. Gupta, Phys. Lett. B 588, 136 (2004) [arXiv:hep-lat/0307007]. [26] M. P. Lombardo, Nucl. Phys. Proc. Suppl. 83, 375 (2000) [arXiv:hep-lat/9908006]. [27] I. Barbour, N. E. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H. W. Wyld, Nucl. Phys. B 275, 296 (1986). [28] J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 73, 2288 (1994) [arXiv:hep-th/9405005]. [29] A. J. McKane and M. Stone, Annals Phys. 131, 36 (1981). [30] M. A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996) [arXiv:hep-lat/9604003]. [31] J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 94, 202001 (2005) [arXiv:hep-th/0501210]. [32] P. Hasenfratz and F. Karsch, Phys. Lett. B 125, 308 (1983). [33] M.R. Zirnbauer, [arXiv:chao-dyn/9609007]. [34] B. Schlittgen and T. Wettig, Nucl. Phys. B 632, 155 (2002) [arXiv:hep-lat/0111039]. [35] J. Budczies, S. Nonnenmacher, Y. Shnir and M. R. Zirnbauer, Nucl. Phys. B 635, 309 (2002) [arXiv:hep-lat/0112018]. [36] J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein and N.C. Snaith, [arXiv:math-ph/0208007]. [37] E.L. Basor and P.J. Forrester, Mathematische Nachrichten, 170, 5 (1194). [38] Z. Fodor and S. D. Katz, JHEP 0203, 014 (2002) [arXiv:hep-lat/0106002]. [39] Z. Fodor and S. D. Katz, JHEP 0404, 050 (2004) [arXiv:hep-lat/0402006]. [40] P. de Forcrand and O. Philipsen, Nucl. Phys. B 642, 290 (2002) [arXiv:hep-lat/0205016]. [41] P. de Forcrand and O. Philipsen, Nucl. Phys. B 673, 170 (2003) [arXiv:hep-lat/0307020]. [42] M. D'Elia and M. P. Lombardo, Phys. Rev. D 67, 014505 (2003) [arXiv:hep-lat/0209146]. [43] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]. [44] C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [45] R. V. Gavai and S. Gupta, Phys. Rev. D 68, 034506 (2003) [arXiv:hep-lat/0303013]. [46] R. V. Gavai and S. Gupta, Phys. Rev. D 71, 114014 (2005) [arXiv:hep-lat/0412035]. [47] V. Azcoiti, G. di Carlo and A. F. Grillo, Phys. Rev. Lett. 65, 2239 (1990); J. Ambjorn, K. N. Anagnostopoulos, J. Nishimura and J. J. M. Verbaarschot, JHEP 0210, 062 (2002) [arXiv:hep-lat/0208025]. [48] A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 53, 7223 (1996) [arXiv:hep-ph/9509324]. [49] M. A. Halasz, A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 56 (1997) 5140 [arXiv:hep-lat/9703006]. [50] K. Splittorff, arXiv:hep-lat/0505001. [51] G. V. Dunne and S. M. Nishigaki, Nucl. Phys. B 670, 307 (2003) [arXiv:hep-ph/0306220]. [52] G. Akemann and P. H. Damgaard, Nucl. Phys. B 576, 597 (2000) [arXiv:hep-th/9910190]. [53] P. H. Damgaard and S. M. Nishigaki, Phys. Rev. D 57, 5299 (1998) [arXiv:hep-th/9711096]. [54] G. Akemann, D. Dalmazi, P. H. Damgaard and J. J. M. Verbaarschot, Nucl. Phys. B 601, 77 (2001) [arXiv:hep-th/0011072]. [55] T. Andersson, P. H. Damgaard and K. Splittorff, Nucl. Phys. B 707, 509 (2005) [arXiv:hep-th/0410163]. [56] G. Akemann and Y. V. Fyodorov, Nucl. Phys. B 664, 457 (2003) [arXiv:hep-th/0304095]. [57] R. J. Szabo, Nucl. Phys. B 598, 309 (2001) [arXiv:hep-th/0009237]. [58] D. Toublan and J. J. M. Verbaarschot, Int. J. Mod. Phys. B 15, 1404 (2001) [arXiv:hep-th/0001110]. [59] K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 683, 467 (2004) [arXiv:hep-th/0310271]. [60] G. Akemann and T. Wettig, Phys. Rev. Lett. 92, 102002 (2004) [Ibid. 96, 029902 (2006) ] [arXiv:hep-lat/0308003]. [61] J. Bloch and T. Wettig, Phys. Rev. Lett. 97, 012003 (2006) [arXiv:hep-lat/0604020]. [62] P. H. Damgaard, U. M. Heller, A. Krasnitz and T. Madsen, Phys. Lett. B 440, 129 (1998) [arXiv:hep-lat/9803012]. [63] R. Narayanan and H. Neuberger, Nucl. Phys. B 696, 107 (2004) [arXiv:hep-lat/0405025]. [64] T. D. Cohen, Phys. Rev. Lett. 91, 222001 (2003) [arXiv:hep-ph/0307089]. [65] J. C. Osborn, Phys. Rev. Lett. 93, 222001 (2004) [arXiv:hep-th/0403131]. [66] G. Akemann, J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 712, 287 (2005) [arXiv:hep-th/0411030]."}],"string":"[\n {\n \"section_type\": \"BACKGROUND\",\n \"section_title\": \"I. INTRODUCTION\",\n \"text\": \"In spite of significant recent progress, QCD at nonzero chemical potential remains a notoriously hard problem [1, 2, 3] . In particular, first principle nonperturbative results at low temperature are absent because the phase of the fermion determinant invalidates probabilistic methods to evaluate the partition function. This problem is known as the sign problem. The sign problem is particularly severe if the phase of the fermion determinant results in a different free energy, i.e.\\n\\ndet N f D |det N f D| = e V (Fpq-FN f ) with F N f > F pq . ( 1\\n\\n)\\n\\nIf the phase quenched free energy, F pq differs from the free energy of QCD with N f flavors, F N f , the number of required gauge field configurations grows exponentially with the volume. The ratio defined in equation (1) can also be interpreted as the average phase factor of the fermion determinant [4, 5]\\n\\ne iN f θ pq = e iN f θ |det N f D| |det N f D| . ( 2\\n\\n)\\n\\nAlternatively, one can define the average phase factor with respect to the full QCD partition function [4, 5]\\n\\ne 2iθ N f = e 2iθ det N f D det N f D , ( 3\\n\\n)\\n\\nwhich has also been used in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12] as a measure for the severity of the sign problem. When the chemical potential is sufficiently small, the average phase factor can be studied by means of chiral perturbation theory [4, 5] . Recently, the average phase factor was analyzed in the microscopic domain of QCD [13, 14] , where only the constant fields in the chiral Lagrangian contribute to the mass and chemical potential dependence of the partition function. It was found that the sign problem is not serious for µ < m π /2. For µ > m π /2 the chiral condensate of the phase quenched theory rotates into a pion condensate resulting in a free energy that is different from the full theory and a severe sign problem.\\n\\nIn this paper we study the sign problem for Euclidean QCD in one dimension. In one dimension, the only effect of the gauge field is in the boundary conditions, and lattice QCD simplifies to a matrix integral. A general formula for exactly this type of integrals was recently derived by Conrey, Farmer and Zirnbauer [15] . Using what we will call the CFZ formula, exact analytical results for the one dimensional QCD partition function and the average phase factor will be obtained.\\n\\nThe study of QCD in one dimension at finite chemical potential has had a long history. One reason to study this model is that it is an effective model for the large N c limit of strong coupling QCD [16, 17, 18, 19, 20, 21] . Among 2 others, it has been successfully used to explain [22, 23, 24] puzzling lattice results for quenched QCD at nonzero chemical potential. More recently, one dimensional QCD was used to study the complex zeros of the partition function [1, 25, 26] .\\n\\nThe sign problem of QCD at nonzero chemical potential arises because of the nonhermiticity of the Dirac operator. QCD in one dimension is not truly nonhermitean: Instead of being scattered in a two-dimensional domain of the complex plane [27] , the eigenvalues of the Dirac operator are localized on an ellipse in the complex plane [6, 22] . Another manifestation of the mild nonhermiticity is that the chemical potential can be absorbed into the quark masses. For each flavor in one dimension with quark mass m, we can associate flavors with mass m + µ and -m + µ. In the large N c limit, this results in spontaneous chiral symmetry breaking according to\\n\\nU (2N f ) → U (N f ) × U (N f ).\\n\\nTherefore, QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. The microscopic limit of QCD in one dimension is equivalent to the microscopic limit of QCD in three dimensions [28] . Another peculiarity of QCD in one dimension is that, already in the free theory, the eigenvalues near zero are spaced inversely proportional to the volume. Therefore, in the thermodynamic limit (i.e. at zero temperature) the chiral condensate is discontinuous across the imaginary axis. However, this type of symmetry breaking is not a collective phenomenon, and there are no associated Goldstone bosons. It is reminiscent to the alternative to Goldstone's theorem proposed by McKane and Stone [29] . Spontaneous symmetry breaking with Goldstone bosons takes place for a fixed number of lattice points in the limit N c → ∞.\\n\\nA chemical potential excites color singlet excitations with baryon charge q k and mass M k when q k µ > M k . This is also the case for one-dimensional QCD. For gauge group U (N c ), the only color singlet excitations are mesons that are uncharged with respect to the chemical potential, whereas for SU (N c ) gauge group we have both neutral mesonic and charged baryonic color singlet excitations. The complex conjugate of the fermion determinant can be interpreted in terms of conjugate quarks [23, 30] which have a baryon charge that is opposite to that of regular quarks. Therefore, both for U (N c ) and SU (N c ), the phase quenched partition function, where the fermion determinant has been replaced by its absolute value, has charged mesonic excitations made out of quarks and conjugate anti-quarks. This results in a phase transition at µ = µ c = m π /2. The SU (N c ) theory with N f flavors has only charged baryonic excitations and will have a phase transition at µ = µ c = m B /N c . In one dimension it turns out that the pion and the baryon have the same mass per quark number so that the critical chemical potential of the phase quenched U (N c ) theory and the SU (N c ) theory is the same.\\n\\nFor the U (N c ) partition function we expect the sign problem to be severe when µ > µ c because the phase quenched partition function has a phase transition at µ = µ c whereas the normal theory remains in the same phase. For SU (N c ), both the normal partition function and the phase quenched partition function do have a phase transition to a phase of free quarks at µ = µ c and have the same free energy not only for µ < µ c but also for µ > µ c . Therefore, in one dimension there is no severe sign problem for SU (N c ) QCD.\\n\\nBecause one dimensional QCD with gauge group U (N c ) does not have a phase transition, the chiral condensate in the thermodynamic limit or large N c limit is discontinuous at m = 0, independent of the value of the chemical potential. Because the Dirac eigenvalues are located on an ellipse for µ = 0, this discontinuity cannot be related to the Dirac spectrum by means of the Banks-Casher formula. In [31] a different mechanism to explain the discontinuity was discovered . It was found that an oscillating contribution to the spectral density with an amplitude that diverges exponentially with the volume, is responsible for the discontinuity of the chiral condensate. We will show that a similar mechanism is at work for U (N c ) gauge theory in one dimension.\\n\\nLattice QCD in one dimension will be introduced in section II, where we also discuss its continuum limit, mean field results and the Conrey-Farmer-Zirnbauer formula. In section III we will evaluate the U (N c ) partition function and its phase quenched version. The average phase factor for U (N c ) is calculated in section IV. In section V we study one dimensional QCD with SU (N c ) as gauge group and evaluate the regular and phase quenched partition functions and the average phase factor. The effect of temperature will be illustrated with results from a schematic random matrix model in section VI. The connection with the microscopic domain of QCD in three dimensions is discussed in section VII, and the Dirac spectrum and its relation with chiral symmetry breaking is analyzed in section VIII. Concluding remarks are made in section IX. 3 II. QCD IN ONE DIMENSION A. Lattice QCD The staggered lattice QCD Dirac operator in one dimension is given by D = mI e µ U 12 /2 . . . e -µ U † n1 /2 -e -µ U † 12 /2 mI • • • 0 . . . . . . 0 • • • mI e µ U n-1 n /2 -e µ U n1 /2 • • • -e -µ U † n-1 n /2 mI \\n\\n. ( 4\\n\\n)\\n\\nWe have used anti-periodic boundary conditions and the gauge fields on the links are taken to be in U (N c ) or SU (N c ). The chemical potential µ is an imaginary vector field introduced according to the Hasenfratz-Karsch prescription [32] .\\n\\nIn general, the lattice QCD partition function for N f flavors with mass m is given by\\n\\nZ N f (µ c , µ) = l dU l det N f D e -SYM/g 2 , ( 5\\n\\n)\\n\\nwhere the product is over the links and S YM is the Yang-Mills action. In one dimension, the partition function simplifies substantially. First, there is no Yang-Mills action, and second, by unitary transformations the integral over the links can be reduced to a single integral. In the gauge where all gauge fields except U n 1 ≡ U are equal the unity, the fermion determinant reduces to [24] det D = 2 -nNc det[e nµc + e -nµc + e nµ U + e -nµ U † ], (\\n\\n) 6\\n\\nwith µ c given by\\n\\nµ c = sinh -1 m. ( 7\\n\\n)\\n\\nThis value will turn out to be the critical value of the chemical potential. From now on, the overall factor 2 -nNc will be absorbed into the normalization of the partition function. To avoid sign factors, n is taken to be even throughout this paper. The standard method to evaluate the QCD partition function in one dimension is to use the eigenvalues of U as integration variables [6, 16, 17, 18, 24] . For example, one finds this way that the result for N f = 1 with U (N c ) as gauge group is given by [24]\\n\\nZ N f =1 (µ c , µ) = U(Nc) dU det D = sinh((N c + 1)nµ c ) sinh(nµ c ) . ( 8\\n\\n)\\n\\nFor most partition functions that are considered in this paper it is not possible to obtain analytical results by means of this method. Instead we will use powerful integration formulas for unitary integrals that were recently derived in [15] . These integrals are based on the color-flavor transformation [33] which has also been applied to lattice QCD with baryons in the canonical ensemble [34, 35] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Continuum Theory\",\n \"text\": \"The continuum limit of the staggered lattice action is given by\\n\\nD cont = m ∂ 0 + iA 0 + µ ∂ 0 + iA 0 + µ m , ( 9\\n\\n)\\n\\nwhere A 0 is a Hermitean N c × N c matrix and the off-diagonal blocks connect even and odd lattice sites. What is special in one dimension is that the off-diagonal blocks are identical. The eigenvalue equation (∂ 0 + iA 0 )ψ k = iu k ψ k is solved by ψ = P e -i R t\\n\\n0 A0dt χ 0 with ∂ 0 χ 0 = iEχ 0 , ( 10\\n\\n)\\n\\n4 with P e the path ordered exponent. Nontrivial eigenvalues are obtained by imposing boundary conditions on ψ. At nonzero chemical potential the eigenvalues of the Dirac operator are given by λ k = m ± (iu k + µ) with u k ∈ R. (11) This is very different from QCD with d ≥ 2 where the eigenvalues of the QCD Dirac operator at µ = 0 are scattered in the complex plane. Also the eigenvalues of the one dimensional staggered lattice Dirac at nonzero chemical potential are localized on a curve in the complex plane. Another consequence of the structure of the Dirac operator (9) is that the fermion determinant can be rewritten as\\n\\ndet D cont = det(∂ 0 + iA 0 + µ + m) det(∂ 0 + iA 0 + µ -m), ( 12\\n\\n)\\n\\nwhich can be interpreted as a two-flavor partition function with masses m + µ and -m + µ. This Dirac operator has the same structure as QCD in three dimensions: In the large N c limit, chiral symmetry is broken spontaneously according to U (2) → U (1) × U (1) with the squared Goldstone masses proportional to the difference of the positive and negative quark masses, i.e. m 2 π ∼ m -(-m) = 2m. At low energy, these Goldstone modes interact according to a chiral Lagrangian determined by the pattern of chiral symmetry breaking. In the large N c limit and m π β ≪ 1 (with β the length of the one-dimensional box) the QCD partition function in one dimension is therefore equivalent to the low-energy limit of QCD in three dimensions. In section VII this will be worked out explicitly for the microscopic limit of the partition function. This is the limit ρ(0) → ∞ with mπρ(0) = fixed and µπρ(0) = fixed, (\\n\\n) 13\\n\\nwhere ρ(0) is density of the eigenvalues, or their projections onto the imaginary axis, close to zero. Since ρ(0) = nN c /2π (see section VIII), this is the limit nN c → ∞ with nN c m and nN c µ fixed. When we use the term microscopic limit, we always mean the universal microscopic limit. This is the limit (13) associated with the formation of Goldstone bosons, i.e. the limit N c → ∞ at fixed n.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"C. Mean Field Limit for Large Nc\",\n \"text\": \"In the large N c limit, chiral symmetry is broken spontaneously even in one dimension, so that its low-energy limit is a theory of Goldstone bosons. In this section we give general arguments that determine the chemical potential dependence of the partition function in the microscopic domain where the Compton wavelength of the Goldstone bosons is much larger than β.\\n\\nAs was argued in [4, 5] , in the microscopic domain, the mean field limit of the partition function is given by\\n\\nZ = J k 1 m π (µ) e -V F , ( 14\\n\\n)\\n\\nwhere J is the value of the integration measure at the saddle point, F is the free energy, V is the space time volume, and m π are the masses of the Goldstone bosons. Let us apply this result to the average phase factor in one dimension. For µ < µ c , in the limit T → 0, only the vacuum state contributes to the partition function, so that the free energy is independent of µ. For quark mass m, the equivalent QCD 3 mass matrix of the Dirac operator D in (12) is given by diag(-m + µ, m + µ), whereas the hermitean conjugate Dirac operator D † , has diag(-m -µ, m -µ) as equivalent QCD 3 mass matrix. For N f flavors, the average phase factor is the partition function with N f + 1 fermionic quarks and one conjugate bosonic quark. We thus have 2(N f + 1) 2 Goldstone bosons made out of two fermionic quarks with squared mass m 2 π = 2mG (with G a constant), 4(N f + 1) fermionic Goldstone modes, half of them with squared mass m 2 π = 2(m -µ)G, and the other half with squared mass m 2 π = 2(m + µ)G. Finally, we have 2 Goldstone bosons composed out of two bosonic quarks with squared mass equal to m 2 π = 2mG. The fermionic partition function Z N f has 2N 2 f Goldstone bosons all with squared mass m 2 π = 2mG. Using (14) we thus find\\n\\ne 2iθ N f = Z N f +1|1 * (µ c , µ) Z N f (µ c , µ) = 1 - µ 2 m 2 N f +1 for µ < µ c . ( 15\\n\\n)\\n\\nFor phase quenched partition functions with N f flavors and N f conjugate flavors, the equivalent QCD 3 symmetry breaking pattern is\\n\\nU (4N f )/(U (2N f ) × U (2N f ))\\n\\n, so that the total number of Goldstone bosons is equal to 8N 2 f . Of these, 4N 2 f Goldstone bosons have squared mass m 2 π = 2mG, half made out of two quarks and the other half out of two 5 conjugate quarks. The other 4N 2 f Goldstone bosons are composed out of a quark and a conjugate anti-quark quark, half with squared mass m 2 π = 2(m -µ)G and the other half with squared mass m 2 π = 2(m + µ)G. All equivalent QCD 3 Goldstone bosons of the full QCD partition function with 2N f flavors have a squared mass equal to m 2 π = 2mG. For µ < µ c the free energy of the phase quenched partition function and the full QCD partition function is the same so that, using (14), the phase quenched average phase factor is given by\\n\\ne 2iθ N f +N * f = Z N f +1+(N f -1) * (µ c , µ) Z 2N f (µ c , µ) = 1 - µ 2 m 2 N 2 f for µ < µ c . ( 16\\n\\n)\\n\\nBelow we will show that the results derived in this section also follow from the zero temperature microscopic limit of the exact evaluation of the partition function.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"D. The Conrey-Farmer-Zirnbauer formula\",\n \"text\": \"Exactly the integrals that are required for the evaluation of the average phase factor of QCD at nonzero chemical potential in one dimension were studied in a recent paper by Conrey, Farmer and Zirnbauer [15] . They considered the partition function\\n\\nZ({ψ k , φ k }) = U(Nc) dU p j=1 det(1 -e ψj U ) det(1 -e φj U ) p+q l=p+1 det(1 -e -ψ l U † ) det(1 -e -φ l U † ) , ( 17\\n\\n)\\n\\nwith dU the Haar measure of U (N c ) and ψ k , φ k complex parameters with Re(φ j ) < 0 < Re(φ l ). Using the color flavor transformation [33] and Howe's theory of supersymmetric dual pairs, they derived the following formula\\n\\nZ({ψ k , φ k }) = π∈Sp+q/(Sp×Sq) p+q l+1 e Nc(π(ψ l )-ψ l ) p j=1 (1 -e φj -π(ψ l ) )(1 -e π(ψj)-φ l ) (1 -e π(ψj)-π(ψ l ) )(1 -e φj-φ l ) . ( 18\\n\\n)\\n\\nThe sum is over permutations in\\n\\nS p+q /(S p ×S q ) that interchange any of the ψ 1 , • • • , ψ p with any of the ψ p+1 , • • • , ψ p+q .\\n\\nPartition functions with an unequal number of bosonic and fermionic determinants can be obtained from special limits of (18) . In the case of only fermionic determinants, an equivalent expression was first obtained in [36, 37] . Expressions for degenerate parameters can be derived by carefully taking limits of the above formula.\\n\\nUnitary matrix integrals can also be calculated by using an eigenvalue representation of the unitary matrices. The integrals we are interested in are of the form\\n\\nZ = U(Nc) dU k F (e iθ k ), ( 19\\n\\n)\\n\\nwhere exp(iθ k ) are the eigenvalues of U . Using an eigenvalue representation of the unitary matrices, the orthogonal polynomial method can be used to express Z as [17]\\n\\nZ = det(B k-l ) k,l=0,•••,Nc-1 , ( 20\\n\\n)\\n\\nwhere\\n\\nB k = 1 2π π -π dθe ikθ F ({e iθ }). ( 21\\n\\n)\\n\\nThis is the method that was used in the literature on one-dimensional QCD prior to this paper [6, 16, 17, 18, 24] . In a few cases the determinant in (20) could be evaluated explicitly resulting in expressions that are similar to those derived directly from the CFZ-formula. We have used (20) to numerically check the results obtained by means of the CFZ-formula.\\n\\nIII. EXACT EVALUATION OF THE ONE-DIMENSIONAL U (Nc) PARTITION FUNCTION A. Partition Function for arbitrary N f\\n\\nIn this section we evaluate the one-dimensional U (N c ) QCD partition function for N f flavors.\\n\\n6 To apply the CFZ formula (18) we rewrite the determinant (6) as det D = e µcnNc det(1 + e n(µ-µc) U ) det(1 + e n(-µ-µc) U † ). (\\n\\n) 22\\n\\nFor arbitrary N f we then find the remarkably simple answer\\n\\nZ N f (µ c , µ) ≡ U(Nc) dU det N f D = σ∈S2N f /SN f ×SN f N f k=1 N f l=1 e Ncm σ(+ k) 1 -exp(m σ(-l) -m σ(+ k) ) , ( 23\\n\\n) with m -k = -nµ c k and m + k = nµ c k . ( 24\\n\\n)\\n\\nThe sum is over all permutations that interchange positive and negative masses. Notice that the µ-dependence has canceled from this expression. This also follows from an expansion of the determinant in powers of U and U † . Only terms with an equal number of factors U and factors U † are non-vanishing. Although the result for degenerate positive and negative masses can be obtained by carefully taking limits of (23), it is simpler to start from a different representation [36, 37] of partition function (23) given by\\n\\nZ N f (µ c , µ) = 1 1≤k<l≤2N f (e M l -e M k ) 1 e m-1 • • • e (N f -1)m-1 e (Nc+N f )m-1 • • • e (Nc+2N f -1)m-1 . . . . . . . . . . . . . . . 1 e m-N f • • • e (N f -1)m-N f e (Nc+N f )m-N f • • • e (Nc+2N f -1)m-N f 1 e m+ 1 • • • e (N f -1)m+ 1 e (Nc+N f )m+ 1 • • • e (Nc+2N f -1)m+ 1 . . . . . . . . . . . . . . . 1 e m+ N f • • • e (N f -1)m+ N f e (Nc+N f )m+ N f • • • e (Nc+2N f -1)m+ N f . ( 25\\n\\n)\\n\\nFor convenience we have introduced the mass matrix\\n\\nM k = (m 1 1 , • • • , m -N f , m + 1 , • • • , m + N f ).\\n\\nTo obtain an expression for degenerate masses, we Taylor expand the exponential functions exp(m + k ) and exp(m -k ) to order N f -1 about exp(-nµ c ) and exp(nµ c ), respectively, and write the resulting matrix as the product of two matrices, one containing the Taylor coefficients, and the other containing powers of the expansion parameters. The determinant of the second matrix can be written as a Vandermonde determinant which cancels against part of the prefactor in (25) . Our final expression for the partition function with degenerate masses is given by\\n\\nZ N f (µ c , µ) = 1 ( N f -1 k=0 k!) 2 1 (e nµc -e -nµc ) N 2 f × 1 • • • 0 1 • • • 0 e -nµc • • • e -nµc e nµc • • • e nµc e -2nµc • • • δ N f -1 - e -2nµc e 2nµc • • • δ N f -1 + e 2nµc . . . . . . . . . . . . e -n(N f -1)µc • • • δ N f -1 - e -n(N f -1)µc e n(N f -1)µc • • • δ N f -1 + e n(N f -1)µc e -n(Nc+N f )µc • • • δ N f -1 - e -n(Nc+N f )µc e n(Nc+N f )µc • • • δ N f -1 + e n(Nc+N f )µc . . . . . . . . . . . . e -n(Nc+2N f -1)µc • • • δ N f -1 - e -n(Nc+2N f -1)µc e n(Nc+2N f -1)µc • • • δ N f -1 + e n(Nc+2N f -1)µc , ( 26\\n\\n) with δ -= d d(-nµ c ) , δ + = d d(nµ c ) . ( 27\\n\\n)\\n\\nThe k'th column (with k ≤ N f ) is given by the δ\\n\\n(k-1) -\\n\\nderivative of the first column, and the N f + k'th column is given by the δ (k-1) + derivative of the N f 'th column. From this result one can easily derive explicit expressions for small values of N f . The partition function for N f = 2 reads\\n\\nZ N f =2 (µ c , µ) = (e n(Nc+2)µc -e -n(Nc+2)µc ) 2 (e nµc -e -nµc ) 4 - (N c + 2) 2 (e nµc -e -nµc ) 2 . ( 28\\n\\n)\\n\\n7 The microscopic limit of this partition function is given by\\n\\nZ micro N f =2 (µ c , µ) = (e Ncµc -e -Ncµc ) 2 16µ 4 c . ( 29\\n\\n)\\n\\nFor N f = 3 the partition function is given by:\\n\\nZ N f =3 (µ c , µ) = (e n(Nc+3)µc -e -n(Nc+3)µc ) 3 (e nµc -e -nµc ) 9 -(N c + 3) 3 (e n(Nc+1)µc -e -n(Nc+1)µc ) (e nµc -e -nµc ) 7 ( 30\\n\\n) +N c (N c + 2) 2 (e n(Nc+3)µc -e -n(Nc+3)µc ) (e nµc -e -nµc ) 7 - 1 4 (N c + 2) 2 (N c + 3) 2 (e n(Nc+3)µc -e -n(Nc+3)µc ) (e nµc -e -nµc ) 5 . ( 31\\n\\n)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. The Phase Quenched Partition Function\",\n \"text\": \"To calculate the average phase factor according to the definition (2) for N f = 2, we also need the one-dimensional phase quenched QCD partition function for two flavors which will be evaluated in this subsection. It is defined by\\n\\nZ 1+1 * (µ c , µ) = U∈U(Nc) dU det D det D † ( 32\\n\\n) = e 2µnNc U∈U(Nc) dU det(1 -e n(µ-µc) U ) det(1 -e n(-µ-µc) U † ) det(1 -e n(µ-µc) U † ) det(1 -e n(-µ-µc) U ).\\n\\nThis partition function is easily evaluated using the integration formulae of [15] . We find\\n\\nZ 1+1 * (µ c , µ) = cosh(2n(N c + 2)µ c ) 8 sinh(n(µ c -µ)) sinh(n(µ + µ c )) sinh 2 (nµ c ) + cosh(2n(N c + 2)µ)) 8 sinh(n(µ -µ c )) sinh(n(µ + µ c )) sinh 2 (nµ) + 1 8 sinh 2 (nµ) sinh 2 (nµ c ) . ( 33\\n\\n)\\n\\nFor µ = 0 this result agrees with the N f = 2 partition function given in (28) . Its microscopic limit simplifies to\\n\\nZ micro 1+1 * (µ c , µ) = e 2nNcµc + e -2nNcµc 16n 4 (µ 2 c -µ 2 )µ 2 c + 2 16n 4 µ 2 µ 2 c - e 2nNcµ + e -2nNcµ 16n 4 (µ 2 c -µ 2 )µ 2 . ( 34\\n\\n)\\n\\nIn section VII we will show that this result is equal to the microscopic limit of the\\n\\nQCD 3 partition function with masses -(µ c + µ), -(µ c -µ), µ c -µ, µ c + µ.\\n\\nThe large N c limit of the phase quenched partition function is given by\\n\\nZ 1+1 * (µ c , µ) = e 2n(Nc+2)µc 16 sinh(n(µ c -µ)) sinh(n(µ + µ c ))(sinh(nµ c )) 2 for µ < µ c , Z 1+1 * (µ c , µ) = e 2n(Nc+2)µ 16 sinh(n(µ -µ c )) sinh(n(µ + µ c ))(sinh(nµ)) 2 for µ > µ c , Z 1+1 * (µ c , µ) = N c e 2n(Nc+2)µc 8 sinh 2 (nµ c ) sinh(2nµ c ) 1 + O(N -1 c ) for µ = µ c , ( 35\\n\\n)\\n\\nwhich will be used to calculate the phase quenched average phase factor in this limit.\\n\\nIV. AVERAGE PHASE FACTOR FOR U (Nc)\\n\\nIn this section we will evaluate the average phase factor, first from the ratio of the full QCD partition function and the phase quenched partition function and in the next subsection starting from the definition in Eq. (3). 8 A. Phase Quenched Average Phase Factor For N f = 2, the average phase factor with the absolute value of the fermion determinant as weight is given by\\n\\ne 2iθ pq = e 2iθ | det(D)| 2 | det(D)| 2 = Z N f =2 (µ c , µ) Z 1+1 * (µ c , µ) . ( 36\\n\\n)\\n\\nThe two-flavor partition function was evaluated in subsection III A, and the phase quenched partition function was calculated in subsection III B. The large N c limit of the phase quenched average phase factor is given by\\n\\ne 2iθ pq = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc)\\n\\nfor µ < µ c , sinh 2 (nµ) sinh(n(µ-µc)) sinh(n(µ+µc)) sinh 4 (nµc) e -2n(Nc+2)(µ-µc) for µ > µ c , 2 cosh(nµc) Nc sinh(nµc)\\n\\nfor µ = µ c . ( 37\\n\\n)\\n\\nThe large µ c N c , µN c limit of the microscopic phase quenched partition function (34) for µ < µ c is given by\\n\\nZ pq (µ c , µ) = e 2nNcµc 16n 4 (µ 2 c -µ 2 )µ 2 c , ( 38\\n\\n)\\n\\nwhereas the two-flavor partition function in this limit reads\\n\\nZ N f =2 (µ c , µ) = e 2nNcµc 16n 4 µ 4 c , ( 39\\n\\n)\\n\\nso that the average phase factor simplifies to\\n\\ne 2iθ pq = 1 - µ 2 µ 2 c for µ < µ c . ( 40\\n\\n)\\n\\nThis result is in agreement with the mean field result (16) . Indeed, the denominator of (38) is the product of the Goldstone masses µ c -µ, µ c + µ, µ c and µ c , whereas the denominator of (40) is the product of four Goldstone masses µ c . The free energy of both partition functions is equal to 2nN c µ c . For µ > µ c , the large N c limit of the two flavor partition function is still given by (39), but the large N c µ limit of the microscopic phase quenched partition function is now given by\\n\\nZ pq = e 2nNcµ 16n 4 (µ 2 -µ 2 c )µ 2 , ( 41\\n\\n)\\n\\nresulting in the average phase factor\\n\\ne 2iθ pq = Z N f =2 Z pq = µ 2 µ 2 c ( µ 2 µ 2 c -1)e -2nNc(µ-µc) for µ > µ c . ( 42\\n\\n)\\n\\nThis result can also be obtained from the small nµ, nµ c expansion of the large N c limit of the average phase factor (see Eq. ( 37 )) at fixed nN c µ c . We conclude that the sign problem becomes exponentially hard for µ > µ c . In Fig. 1 we show the average phase factor for different values of n and N c and a critical chemical potential of µ c = 0.1. In the thermodynamic limit at fixed N c , the average phase factor is one for µ < µ c and jumps to zero for µ > µ c . In the right figure we observe a rapid convergence to the mean field result discussed in section II C.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Average phase factor for full QCD\",\n \"text\": \"The average phase factor with the N f -flavor quark determinant as weight is defined by\\n\\ne 2iθ N f = 1 Z N f (µ c , µ) dU det D det D † det N f D = e nN f Ncµc Z N f (µ c , µ) dU det(1 -U e nµ-nµc ) det(1 -U † e -nµ-nµc ) det(1 -U e -nµ-nµc ) det(1 -U † e nµ-nµc ) det N f (1 -U e nµ-nµc )det N f (1 -U † e -nµ-nµc ). 9\\n\\nFIG. 1: The average phase factor for one-dimensional QCD with gauge group U (Nc) calculated as the ratio of the two flavor flavor partition function and the phase quenched partition function. In the left figure the number of lattice points varies as indicated for fixed Nc = 3, and in the right figure the number of colors varies as indicated for fixed n = 4.\\n\\nWe evaluate this integral exactly for N f = 0 and N f = 1 but only give its large N c limit for other values of N f . Using the CFZ-formula (18) [15] we obtain in the quenched case\\n\\ne 2iθ N f =0 = (1 -e -2n(µ+µc) )(1 -e 2n(µ-µc) ) (1 -e -2nµc ) 2 + e -2nNcµc (1 -e -2nµ )(1 -e 2nµ ) (1 -e 2nµc )(1 -e -2nµc ) for µ < µ c . ( 43\\n\\n)\\n\\nThe large N c limit of this result coincides with the large N c limit of the phase quenched average phase factor. In the microscopic limit where µ c nN c and µnN c remain fixed for N c → ∞ we obtain\\n\\ne 2iθ N f =0 = 1 - µ 2 µ 2 c - µ 2 µ 2 c e -2nNcµc . ( 44\\n\\n)\\n\\nIn the large nN c µ c limit the second term does not contribute resulting in\\n\\ne 2iθ N f =0 = 1 - µ 2 µ 2 c , ( 45\\n\\n)\\n\\nin agreement with the results obtained in [4] (see subsection II C). In the thermodynamic limit at fixed N c the average phase factor converges to one for µ < µ c . After rewriting the determinants in (43) as\\n\\ndet(1 -U † e -nµ-nµc ) det(1 -U † e nµ-nµc ) → e -2nNcµ det(1 -U e nµ+nµc ) det(1 -U e -nµ+nµc ) , ( 46\\n\\n)\\n\\nthe denominator of (43) can be expanded in powers of U for µ > µ c . For N f = 0 only the constant term in the integrand yields a non-vanishing result. We thus find\\n\\ne 2iθ N f =0 = e -2nNcµ for µ > µ c , ( 47\\n\\n)\\n\\nso that the average phase factor vanishes in the large nN c µ limit. In Fig. 2 we show the quenched average phase factor for various values of n and N c . Also in this case the average phase factor jumps from 1 to 0 at µ = µ c in the thermodynamic limit. The convergence to the mean field result for increasing N c (right figure) is very rapid. This can be understood from the expansion of the microscopic result given by 1\\n\\n-(µ 2 /µ 2 c )(1 -1 3 (nµ) 2 + • • •.\\n\\n10 FIG. 2: The quenched average phase factor for one-dimensional QCD with gauge group U (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\\n\\nFinally we calculate the average phase factor for an arbitrary number of flavors in the large N c limit. The CFZ formula can again be applied after inserting the factor\\n\\ndet N f (1 -U e -α )det N f (1 -U † e -β )\\n\\nin the denominator and taking the limit α → ∞ and β → ∞ at the end of the calculation.\\n\\nFor µ < µ c , the leading order large N c result is given by the identity permutation in the sum over permutations in the CFZ formula. This results in\\n\\ne 2iθ N f = (1 -e -2n(µ+µc) )(1 -e 2n(µ-µc) ) (1 -e -2nµc ) 2 N f +1 for µ < µ c . ( 48\\n\\n)\\n\\nThe microscopic limit of this result is given by\\n\\ne 2iθ N f = 1 - µ 2 µ 2 c N f +1 for µ < µ c , ( 49\\n\\n)\\n\\nin agreement with the discussion given in section II C. For µ > µ c the CFZ formula can again be applied after rewriting the partition function according to (46) and the insertion (48) . Because of degenerate prefactors, all critical chemical potentials that occur in the combination µ + µ c have to be taken different. After carefully taking limits we find\\n\\nZ N f +1|1 * (µ c , µ) = N N f c N f ! e -2µnNc e µcnNcN f (1 -e -2µcn ) N 2 f (1 -e -2n(µ+µc) ) N f (1 -e -2nµ ) N f (1 -e -2nµc ) N f . ( 50\\n\\n)\\n\\nThe average phase factor given by\\n\\ne 2iθ N f = Z N f +1|1 * Z N f ∼ e -2nNcµ for µ > µ c ( 51\\n\\n)\\n\\nvanishes in the thermodynamic limit. The exact result becomes rather cumbersome even for small values of N f . As an illustration we give the result for Z 2|1 * (µ c , µ) in Appendix A . 11 V. AVERAGE PHASE FACTOR FOR SU (Nc)\\n\\nTo calculate integrals over SU (N c ) we use the identity SU(Nc)\\n\\ndU • • • = ∞ p=-∞ U(Nc) dU det p (U ) • • • . ( 52\\n\\n)\\n\\nFor the partition functions discussed below, with a few exceptions the sum truncates to a small number of terms.\\n\\nA. SU (Nc) QCD Partition Function\\n\\nLet us first calculate the 1d SU (N c ) QCD partition function for N f = 1. The partition function is defined by\\n\\nZ SU(Nc) N f (µ c , µ) = SU(Nc) dU det N f D = ∞ p=-∞ U(Nc) dU det p U e nNcµc det N f (1 + U e nµ-nµc )det N f (1 + U † e -nµ-nµc ). ( 53\\n\\n)\\n\\nFor p ≥ N f the integrand can be rewritten in terms of determinant of matrices U only\\n\\nZ N f ,p (µ c , µ) = U(Nc) dU det p U det N f D = U(Nc) dU e nNcN f µc det p-N f (U )det N f (1 + U e nµ-nµc )det N f (U + e -nµ-nµc ) . ( 54\\n\\n)\\n\\nFor p > N f it follows immediately that the integral vanishes. For p = N f only the constant term inside the determinants contributes to integral resulting in\\n\\nZ N f ,p=N f (µ c , µ) = e -nN f Ncµ , Z N f ,p>N f (µ c , µ) = 0. ( 55\\n\\n)\\n\\nFor p < 0 we combine det p U = det -p U † with the factor det N f (1 -U e nµ-nµc ). We then find\\n\\nZ N f ,p=-N f (µ c , µ) = e nN f Ncµ , Z N f ,p<-N f (µ c , µ) = 0. ( 56\\n\\n)\\n\\nUsing that the hermitean conjugate of a unitary matrix is also unitary we obtain\\n\\nZ N f ,-p (µ c , µ) = Z N f ,p (µ c , -µ). ( 57\\n\\n)\\n\\nFor |p| < N f the integral can be calculated by means of the CFZ formula by choosing all µ c different and introducing the limit\\n\\ndet U = lim α→∞ det(U -e -α ). ( 58\\n\\n)\\n\\nThe limit of degenerate µ c and α → ∞ is taken at the end of the calculation. We will only give exact results for N f = 1 and N f = 2. Using (55,56) and the result for U (N c ) we obtain for N f = 1 Z SU(Nc)\\n\\nN f =1 (µ c , µ) = e -nN f N f µ + e nN f N f µ + sinh(n(N c + 1)µ c ) sinh(nµ c ) , ( 59\\n\\n)\\n\\nin agreement with earlier work by Bilic and Demeterfi [24] . For N f = 2, the only integrals that have not yet been calculated are those for p = ±1 which are related by (57). For the p = 1 contribution we find\\n\\nZ N f =2,p=1 (µ c , µ) = e nNc(µc-µ) N c e -2n(Nc+1)µc + e 2nµc (e nµc -e -nµc ) 2 - e -2nNcµc (e -3nµc -3e -nµc ) -e 3nµc + 3e nµc (e nµc -e -nµc ) 3 . ( 60\\n\\n)\\n\\n12 For µ < µ c the large nN c µ limit of the N f = 2 partition function is dominated by the p = 0 term, whereas for µ > µ c the large nN c µ c limit is given by the p = -2 term. For µ = µ c the large N c limit is dominated by the p = -1 with 1/N c corrections from the p = 0 and p = -2 terms. We thus find as leading large N c result Z SU(Nc)\\n\\nN f =2 (µ c , µ) = e 2n(Nc+2)µc 16 sinh 4 (nµ c ) for µ < µ c , Z SU(Nc) N f =2 (µ c , µ) = e 2nNcµ for µ > µ c , Z SU(Nc) N f =2 (µ c , µ) = N c e 2n(Nc+1)µc 4 sinh 2 (nµ c ) 1 + O(N -1 c ) for µ = µ c . ( 61\\n\\n)\\n\\nBoth for µ < µ c and µ > µ c the terms that are canceled by the integration over the unitary group are subleading in the thermodynamic limit. In other words there is no serious sign problem. By inspection one can easily show that the dominance of the p = 0 terms for µ < µ c and the p = -N f term for µ > µ c is a feature of the large N c limit that is valid for any number of flavors. Therefore, for µ < µ c , the chiral condensate is the same as for the U (N c ) theory. For µ > µ c though, the dominant p = -N f term is mass-independent. This results in a vanishing chiral condensate so that the large N c limit of the U (N c ) theory and the SU (N c ) theory is different.\\n\\nB. The Phase Quenched Partition Function for SU (Nc)\\n\\nA second ingredient for the average phase factor is the phase quenched SU (N c ) partition function. Using the same arguments as for the full QCD partition function, one easily obtains\\n\\nZ 1+1 * ,|p|>2 (µ c , µ) = 0, Z 1+1 * ,|p|=2 (µ c , µ) = 1, Z 1+1 * ,-p (µ c , µ) = Z 1+1 * ,p (µ c , -µ). ( 62\\n\\n)\\n\\nThe partition function for p = 0 is the U (N c ) partition function which was already given in (33) . What remains to be calculated is the partition function for p = +1. Using the CFZ formula one easily arrives at\\n\\nZ 1+1 * ,p=1 (µ c , µ) = 1 4 sinh(nµ) sinh(nµ c ) sinh(n(N c + 2)(µ + µ c )) sinh(n(µ + µ c )) - sinh(n(N c + 2)(µ -µ c )) sinh(n(µ -µ c ))) . ( 63\\n\\n)\\n\\nBoth for µ < µ c and µ > µ c , the leading large N c result of the phase quenched partition function resides in the p = 0 term given in Eq. (35).\\n\\nC. Average Phase Factor for SU (Nc)\\n\\nThe average phase factor can be either obtained from the ratio of the full QCD partition function and the phase quenched partition function or can be calculated with the full QCD partition function as weight (see Eqs. (2) and (3)). In the first case we find for large N c ,\\n\\ne 2iθ SU(Nc) pq = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc) for µ < µ c , 16 sinh(n(µ -µ c )) sinh(n(µ c + µ)) sinh 2 (nµ) for µ > µ c , 1 -e -4nµc for µ = µ c . ( 64\\n\\n)\\n\\nThe microscopic limit of the average phase factor is obtained by expanding this result for small µ c , µ. We again find the usual mean field result\\n\\ne 2iθ SU(Nc) pq = 1 - µ 2 µ 2 c θ(µ c -µ). ( 65\\n\\n)\\n\\nIn the thermodynamic limit, the average phase factor converges to one in both for µ < µ c and µ > µ c . 13 FIG. 3: The phase quenched average phase factor for one-dimensional QCD with gauge group SU (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\\n\\nExact results for the average phase factor are displayed in Fig. 3 . At finite N c , we observe a rapid convergence to the asymptotic value of 1. At fixed n, the approach to the large N c limit is slow. Also clearly visible is that for µ ≈ µ c the corrections to the microscopic limit are large. As was already discussed before, this is due to the p = -1 contribution.\\n\\nIn the quenched case the average phase factor for SU (N c ) defined according to Eq. ( 3 ) can be written as\\n\\ne 2iθ SU(Nc) N f =0 = ∞ p=-∞ U∈U(Nc) dU det p (U ) det(1 + U e nµ-nµc ) det(1 + U † e -nµ-nµc ) det(1 + U e -nµ-nµc ) det(1 + U † e nµ-nµc ) . ( 66\\n\\n)\\n\\nThe integrals can again be calculated by means of the CFZ formula. For N c ≥ 3 one can easily show that the limit introduced in (58) gives vanishing results for |p| > 2. However, for N c = 1 and N c = 2 additional terms may contribute to this limit. For N c = 2, where the average phase factor is equal to one, contributions for |p| > 2 vanish for µ < µ c , but terms with p < -2 are nonzero for µ > µ c . The formulae for |p| ≤ 2 given below are valid for N c = 2. For N c = 1 the integrals in (66) only vanish for p ≥ 2 and µ > µ c . Also the general answer for p = -2 is not correct in this case. For p = -1 we obtain Ncn(µc-µ) (1-e -2n(µc +µ) )(1-e -2nµ ) 1-e -2nµc for µ < µ c ,\\n\\ne 2iθ N f =0 = e -\\n\\ne nNc(µc -µ) (1-e -2nµ ) 1-e -2nµc [(1 -e -2nµc-2nµ ) -e -2nNcµc (e -2nµc -e -2nµ )] for µ > µ c . ( 67\\n\\n)\\n\\nFor p = 1 we find e 2iθ N f =0 = e -Ncn(µc+µ) (1-e -2n(µc -µ) )(1-e 2nµ ) 1-e -2nµc for µ < µ c ,\\n\\n0 for µ > µ c . ( 68\\n\\n)\\n\\nFinally, for p = -2 the result is\\n\\ne 2iθ N f =0 = 0 for µ < µ c , (1 -e -2nµ )(1 -e -2n(µ+µc) )(1 -e 2n(µc-µ) )(1 -e -2nµ ) for µ > µ c . ( 69\\n\\n)\\n\\nFor p = 2 the average phase factor vanishes. Notice that the relation\\n\\ne 2iθ N f =0,-p (µ c , µ) = e 2iθ N f =0,p (µ c , -µ) ( 70\\n\\n) 14 FIG. 4: Average phase factor for one-dimensional QCD with gauge group SU (Nc). In the left figure Nc = 3 and the number of lattice points is as indicated. In the right figure the number of colors varies whereas n = 4.\\n\\nis also valid in this case. It can be used to derive results for negative values of µ. In the large N c limit, the average phase factor is dominated by the p = 0 contribution for µ < µ c and by the p = -2 contribution for µ > µ c . For µ = µ c , the p = 0, p = -1 and p = -2 contributions are of equal order in N c , but only the p = -1 term is nonvanishing. As large N c limit of the average phase factor we thus find e 2iθ SU(Nc) N f =0 = sinh(n(µc-µ)) sinh(n(µc+µ)) sinh 2 (nµc) for µ < µ c , 16 sinh(n(µ -µ c )) sinh(n(µ c + µ)) sinh 2 (nµ) for µ > µ c ,\\n\\n1 -e -4nµc for µ = µ c , ( 71\\n\\n)\\n\\nwhich is the same expression as obtained in (64) for the phase quenched average phase factor. The microscopic limit of the average phase factor is again given by the small µ-µ c expansion of this result which is equal to the usual mean field result\\n\\ne 2iθ SU(Nc) N f =0 = 1 - µ 2 µ 2 c θ(µ c -µ). ( 72\\n\\n)\\n\\nIn Fig. 4 we show the exact result for the SU (N c ) average phase factor. We observe a rapid approach to the thermodynamic limit at fixed N c just like in Fig. 3 . In the large N c limit the result for the quenched average phase calculated according to (66) is also given by (64). As already was argued in the introduction, we conclude that QCD in one dimension with SU (N c ) as gauge group does not have a serious sign problem, not only for µ < µ c , but also for µ > µ c . Indeed one-dimensional QCD at nonzero chemical potential could be simulated reliably by the Glasgow method [26] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"VI. THE SIGN PROBLEM AT NONZERO TEMPERATURE\",\n \"text\": \"The results of the previous two sections show that the sign problem is severe for QCD in one dimension with gauge group U (N c ) and µ > µ c . Although this theory does not have a critical temperature, it is clear from Fig. 1 that the average phase factor for µ > µ c increases significantly for higher temperatures (i.e. for lower values of n). However, this is a geometric effect due to the Boltzmann factor. Recent lattice simulations [10, 11, 12, 26, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47] suggest that the sign problem becomes much milder for T around T c . Because of the absence of a critical temperature, this temperature dependence of the average phase factor cannot be investigated in 15 FIG. 5: Scatter plot of the eigenvalues of the random matrix Dirac operator for µ =0.3 and T = 0, 0.7, 0.9, 1.0, 1.1 . The quark mass at m = 0.1 is indicated by the black dot.\\n\\none dimensional QCD. Instead we study this problem in a schematic random matrix model at nonzero temperature and chemical potential [30, 48, 49] . This model has a severe sign problem as well as a critical temperature and its eigenvalues are scattered in a finite domain of the complex plane, just like in QCD. The quenched average phase factor in this model is given by the matrix integral\\n\\ne 2iθ (m, µ, T ) = dCe -N TrCC † det(D + m) det(D † + m) , ( 73\\n\\n)\\n\\nwith Dirac operator given by\\n\\nD = 0 iC + µ + it iC † + µ + it 0 , ( 74\\n\\n)\\n\\nand t is the traceless diagonal matrix t = diag(-T, • • • , -T, T, • • • , T ). The integral is over the real and imaginary parts of the matrix elements of the complex N × N matrices C. The phase quenched average phase factor is defined by\\n\\ne 2iθ pq = dCdet 2 (D + m)e -N TrCC † dCdet(D + m) det(D † + m)e -N TrCC † ( 75\\n\\n)\\n\\nIf at all possible, it certainly takes a significant effort to evaluate these integrals analytically. Therefore we have studied the quenched (73) and phase quenched (75) average phase factor numerically. In Fig. 5 , we give scatter plots of the eigenvalues of the random matrix Dirac operator. For each of the temperatures T = 0, 0.7, 0.9, 1, 1.1, the figures show results of 40 matrices with N = 100 and µ = 0.3. The black dot represents the quark mass for which the average phase factor is calculated. In Fig. 6 we show the µ-dependence of the phase quenched (left) and quenched (right) average phase factor obtained by averaging over 10,000 to 80,000 matrices C with N = 100. The results for the phase quenched average phase factor for T = 0 and T = 0.7 and µ > µ c are not displayed because they do not converge because of the severity of the sign problem. The average phase factor depends only weakly on the size of 16 FIG. 6: Phase quenched (left) and quenched (right) average phase factor versus chemical potential. Results have been obtained from the chiral random matrix model (74).\\n\\nthe matrices. The conclusion of Figs. 5 and 6 is that the sign problem is not serious when the quark mass is outside the domain of the eigenvalues. This confirms the conjecture that was made in [4, 5, 50] . It also agrees with the observation in [30] that the free energy of the full theory and the phase quenched theory are the same for T = 0 if the quark mass is outside the domain of the eigenvalues (this is the case for µ < µ c ). The µ-dependence is due to the curvatures at the saddle point.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"VII. COMPARISON WITH THE QCD3 PARTITION FUNCTION\",\n \"text\": \"According to the general arguments given in section II B, the microscopic limit of the one dimensional QCD partition function at nonzero chemical potential is equal to the microscopic limit of the QCD 3 partition function at zero chemical potential, but with shifted quark masses. In this section we show this explicitly for some of the results derived before. We first discuss the microscopic limit of the QCD 3 partition function which can be derived from either a chiral Lagrangian or a random matrix model with the same global symmetries [28] . For a discussion of QCD 3 in terms of chiral Lagrangians we refer to [51] .\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"A. Random Matrix Model\",\n \"text\": \"From the continuum Dirac operator (9) it follows that the random matrix model that describes the fluctuations of the low-lying Dirac eigenvalues is given by\\n\\nZ = dHP (H) det(iH + µ + m) det(iH -m + µ), ( 76\\n\\n)\\n\\nwhere the probability distribution can conveniently be taken to be the Gaussian distribution\\n\\nP (H) = e -2N Σ 2 TrH 2 . ( 77\\n\\n)\\n\\nSuch random matrix partition functions have been studied elaborately in the literature to analyze the microscopic limit of QCD in three dimensions [28, 52, 53, 54, 55] . The microscopic limit of the partition function (76)\\n\\nwith quark masses given by M = diag(-m 1 , • • • , -m N f , m 1 , • • • , m N f ) is equal to [53, 54] Z QCD3 = 1 ∆(M ) det A(m) A(-m) A(-m) A(m) , ( 78\\n\\n)\\n\\n17 where the matrix elements of the N f × N f matrix A(m) are given by\\n\\nA(m) kl = m l k e -m k , ( 79\\n\\n)\\n\\nand ∆ is the Vandermonde determinant ∆(M ) =\\n\\n2N f k>l (M kk -M ll ). ( 80\\n\\n)\\n\\nAlthough general expressions for QCD 3 partition functions with an arbitrary number of bosonic and fermionic determinants are also known [56] , for our purposes we only need the partition function\\n\\nZ N f +2|2 = N f k=1 det(D + m k ) det(D -m k ) det(D + z 1 ) det(D -z 2 ) det(D + z1 ) det(D -z2 ) . ( 81\\n\\n)\\n\\nThis partition function was evaluated in [57] by means of the supersymmetric method and is given by\\n\\nZ N f +2|2 = (z 1 -z1 )(z 2 -z2 ) ∆ 2 N f (m) exp N c (z 1 + z2 ) N c (z 1 + z2 ) N f k=1 (m k -z1 )(m k -z2 ) (m k -z 1 )(m k -z 2 ) det sinh Nc(m k +m l ) Nc(m k +m l ) sinh Nc(m k +z1) Nc(m k +z1) sinh Nc(z2+m l ) Nc(z2+m l )\\n\\nsinh Nc(z1+z2) Nc(z1+z2) + e -Nc(z1-z1)+Nc(z2-z2)\\n\\n∆ 2 N f (m) det sinh N c (m k + m l ) N c (m k + m l ) . ( 82\\n\\n)\\n\\nThe last term is a so-called Efetov-Wegner term, and the Vandermonde determinant is over positive masses only\\n\\n∆(m) = N f k>l (m k -m l ). ( 83\\n\\n)\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Microscopic Limit\",\n \"text\": \"We will now show that the microscopic limit of the partition function (25) is equal to the microscopic limit of the QCD 3 partition function. To this end we multiply row k of the first N f rows of the determinant by exp(-N c m -k /2) and row k of the second N f rows of the determinant in (25) by exp(-N c m + k /2). In the microscopic limit we keep N c m ±,k fixed so that we can expand the masses that do not occur in this combination. By subtracting successive columns starting with the first one we obtain\\n\\nZ N f = 1 1≤k<l≤2N f (M l -M k ) ( 84\\n\\n) × e -Ncm-1 /2 m -1 e -Ncm-1/2 • • • m N f -1 -1 e -Ncm-1/2 m N f -1 e +Ncm-1/2 • • • m 2N f -1 -1\\n\\ne Ncm-1 /2 . . . . . . . . . . . . . . .\\n\\ne -Ncm-N f /2 m -N f e -Ncm-N f /2 • • • m N f -1 -N f e -Ncm-N f /2 m N f -N f e +Ncm-N f /2 • • • m 2N f -1 -N f e Ncm-N f /2 e -Ncm+ 1/2 m + 1 e -Ncm+ 1/2 • • • m N f -1 + 1 e -Ncm+ 1 /2 m N f + 1 e +Ncm+ 1 /2 • • • m 2N f -1 + 1\\n\\ne Ncm+ 1/2 . . . . . . . . . . . . . . .\\n\\ne -Ncm+ N f /2 m + N f e -Ncm+ N f /2 • • • m N f -1 + N f e -Ncm+ N f /2 m N f + N f e +Ncm+ N f /2 • • • m 2N f -1 + N f e Ncm+ N f /2 ,\\n\\nwhere the masses M K are defined below Eq. ( 25 ). By multiplying the columns\\n\\nl = N f + 1, • • • , 2N f by (-1) l-1 and\\n\\nintroducing microscopic masses, we obtain exactly the expression for the QCD 3 partition function given in (78) . For masses -(µ c + µ), -(µ c -µ), µ c -µ, µ c + µ corresponding to the microscopic limit of the two-flavor phase quenched partition function given in (34) the determinant in (84) is given by\\n\\n4nµ 2 (e 2nNcµc + e -2nNcµc ) -4nµ 2 c (e 2nNcµ + e -2nNcµ ) + 8n(µ 2 c -µ 2 ). ( 85\\n\\n)\\n\\nand the prefactor is equal to\\n\\n1≤k<l≤2N f (M l -M k ) = 64n 6 (µ 2 c -µ 2 )µ 2 c µ 2 . ( 86\\n\\n) 18\\n\\nFIG. 7: The number variance for n = 4 and Nc = 100.\\n\\nTheir ratio coincides with the microscopic phase quenched partition function given in (34) .\\n\\nAs second example, we consider the quenched average phase factor for U (N c ) given in (43) . For N f = 0 the partition function (82) simplifies to Nc(z1-z1-z2+z2) .\\n\\nZ 2|2 = (z 1 -z1 )(z 2 -z2 ) e Nc(z1+z2) N c (z 1 + z2 ) sinh N c (z 1 + z 2 ) N c (z 1 + z 2 ) + e -\\n\\n( 87\\n\\n)\\n\\nThe masses in this partition function corresponding to (43) are given by\\n\\nz 1 = nµ c + nµ, z 2 = nµ c -nµ, z2 = nµ -nµ c , z1 = -nµ -nµ c . ( 88\\n\\n)\\n\\nSubstituting them into (87) we obtain\\n\\nZ 2|2 = 1 - µ 2 µ 2 c + µ 2 µ 2 c e -4Ncnµ ( 89\\n\\n)\\n\\nwhich is exactly the microscopic limit of the quenched average phase factor (43) given in Eq. (44).\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"VIII. DIRAC SPECTRUM\",\n \"text\": \"We consider the Dirac operator (4) in a gauge where all gauge fields except U n1 ≡ U and U 1,n = U † are equal to unity. If the eigenvalues of U are equal to exp(iθ k ) the eigenvalues of the Dirac operator are given by\\n\\nλ k,l = 1 2 (e 2πi(k+1/2)+iθ l n +µ -e -2πi(k+1/2)-iθ l n -µ ), k = 1, • • • , n, l = 1, • • • , N c . ( 90\\n\\n)\\n\\nContrary to QCD at µ = 0 in more than one dimension, where the eigenvalues are scattered in the complex plane [27, 30, 58, 59, 60, 61] , the eigenvalues are located on an ellipse in the complex plane with real and imaginary parts related by\\n\\nRe(λ k,l ) e µ -e -µ 2 + Im(λ k,l ) e µ + e -µ 2 = 1. ( 91\\n\\n) 19\\n\\nA. Universal Fluctuations In section II B we have shown that the continuum limit of the staggered Dirac operator in one dimension is in the same universality class as QCD in three dimensions. The continuum limit of the staggered three dimensional Dirac operator, though, is in the chiral symmetry universality class of QCD in four dimensions. For example, in [62] this was found for the distribution of the small Dirac eigenvalues. The reason for the chiral structure is that the staggered Dirac operator only couples even and odd lattice sites. This is also the case in one dimension, but the off-diagonal blocks, containing the gauge fields, are the same in the continuum limit (or occur in the combination U n + U † n ), resulting in a two flavor theory with opposite masses (see (12)). The secular equation is given by\\n\\ndet[(∂ 0 + iA 0 + µ + λ)(∂ 0 + iA 0 + µ -λ)] = 0. ( 92\\n\\n)\\n\\nThis corresponds to the superposition of the spectrum of\\n\\n∂ 0 + A 0 + µ and -∂ 0 -iA 0 -µ.\\n\\nIn the domain of the Dirac spectrum where ∂ 0 can be neglected, the eigenvalues on each of the lines ±µ are given by the Hermitean random matrix ensemble A 0 . For µ = 0 we have the superposition of the Hermitean random matrix ensembles of A 0 and -A 0 . From the eigenvalues (90) it is clear that the staggered lattice Dirac eigenvalues show a similar superposition. One ensemble is localized on the left half of the ellipse and the other ensemble on its right half. Superpostions of more ensembles arise for correlations on scales larger than N c level spacings. Eigenvalues of U (N c ) matrices are correlated according to the circular unitary ensemble. Therefore, the eigenvalues λ k,l are correlated according the Gaussian Unitary Ensemble (GUE) on a scale where the variations in the average spectral density can be neglected, i.e. on a scale much less than N c level spacings. By correcting for the variation of the average level spacing, the scale on which universal random matrix correlations are found can be extended, but this scale should always remain well below N c level spacings. Then we are necessarily within one of the n successive copies of the eigenvalues along the ellipse. At a scale of a finite number of average level spacings, we expect convergence to universal random matrix correlations in the large N c limit.\\n\\nAs measure of the correlations between eigenvalues, we use the number variance Σ 2 (n) defined as the variance on the number of eigenvalues in an interval that contains n eigenvalues on average. We calculate the number variance for intervals starting from zero. Because of the superposition of an enemble and its negative, this is equal to the number variance of a single ensemble for an interval that is symmetric about zero which is given by the GUE result. In Fig. 7 we show the number variance obtained from an average over 10 5 gauge field configurations for N c = 100 and µ = 0. The number variance is calculated for an interval starting at zero. The discrepancy between the analytical result and the GUE (solid curve) is only barely visible.\\n\\nThe increase of the domain of validity of chiral random matrix theory with increasing N c was also observed for Dirac operator of QCD in four dimensions [63] . This can be explained as follows: The number of eigenvalues that is described by random matrix theory is equal to ≈ F 2 π √ V (with V the space-time volume), but F 2 π scales with N c in the large N c limit.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"B. Chiral Symmetry Breaking at µ = 0\",\n \"text\": \"We first discuss the case of µ = 0. Then the average spacing of the eigenvalues scales as 1/(nN c ). This means that the chiral condensate develops a discontinuity at m = 0 when nN c → ∞. Two cases of interest where a nonzero chiral condensate can be obtained are the limit n → ∞ for fixed N c , i.e. at zero temperature, or at fixed n in the limit N c → ∞. In both cases the chiral condensate is discontinuous for nN c → ∞ when the quark mass crosses the imaginary axis where the eigenvalues are located. However, Goldstone bosons and universal behavior is only found in the large N c limit. At fixed N c , chiral symmetry breaking is not associated with universal behavior.\\n\\nFor µ = 0 the Dirac eigenvalues are located on the ellipse (91). When the quark mass is inside the ellipse of eigenvalues the chiral condensate is zero for nN c → ∞ for the SU (N c ) partition function and the phase quenched U (N c ) partition function [24] . For gauge group U (N c ) with N f ≥ 1 flavors, though, the partition function is µindependent so that the chiral condensate is also nonzero if the mass is inside the eigenvalue ellipse. The Banks-Casher formula fails in this case which is known as the \\\"Silver Blaze Problem\\\" [64] . The resolution is the same as for QCD in four dimensions [31] : the re-weighted eigenvalue distribution defined as\\n\\nρ full (z) = 1 Z N f U(Nc) dU det N f D k δ 2 (z -λ k ) ( 93\\n\\n)\\n\\nshows oscillations with an amplitude that diverges exponentially with n and a period that is proportional to 1/n. Below we will illustrate this for U (1).\\n\\n20 For N c = 1 one easily derives that the spectral density is given by ρ full (z) = 4 e nµc + e -nµc -e n(iα+µ) -e -n(iα+µ) (e nµc + e -nµc )(e 2r + e -2r + e 2iα + e -2iα ) δ(r -µ), (\\n\\n) 94\\n\\nwhere z is parameterized as\\n\\nz = 1 2 (e r+iα -e -r-iα ), r > 0, α ∈ [0, 2π], ( 95\\n\\n)\\n\\nand we have used that Z N f =1 = 2 cosh(nµ c ) (see eq. ( 8 )). For µ > µ c this spectral density has oscillations with an amplitude that diverges exponentially with n and a period of order 1/n which are the essential properties of the spectral density of the Dirac operator of QCD in four dimensions [31, 65, 66] . It can be decomposed as\\n\\nρ full (z) = ρ q (z) + ρ osc (z). ( 96\\n\\n)\\n\\nThe average quenched spectral density is defined by\\n\\nρ q (z) = 1 2π dθ 1 n n k=1 δ 2 (z -λ k ) ( 97\\n\\n)\\n\\nwith the eigenvalues λ k given by Eq. ( 90 ). After changing variables according to (95) and integrating over θ we find\\n\\nρ q (z) = 1 2π 4 (e 2r + e -2r + e 2iα + e -2iα ) δ(r -µ). ( 98\\n\\n)\\n\\nThe oscillatory part of the spectral density, ρ osc (z) is equal to the difference ρ full (z) -ρ q (z). The chiral condensate is given by\\n\\nΣ full (m) = d 2 z ρ full (z) z + m , ( 99\\n\\n)\\n\\nand can also be decomposed as\\n\\nΣ full (m) = Σ q (m) + Σ osc (m). ( 100\\n\\n)\\n\\nThe Jacobian for the transformation of d 2 z to drdα is given by\\n\\nd 2 z = 1 4 (e 2r + e -2r + e 2iα + e -2iα )drdα, ( 101\\n\\n)\\n\\nso that the chiral condensate after integration over r can be simplified to\\n\\nΣ full (m) = 1 π 2π 0 dα e nµc + e -nµc -e n(iα+µ) -e -n(iα+µ) (e nµc + e -nµc )(e µ+iα -e -µ-iα -2m) = tanh(nµ c ) cosh µ c . ( 102\\n\\n)\\n\\nThis result remains finite for n → ∞. We remind the reader that the mass is parameterized as m = sinh µ c . This expression represents the resolvent at the quark mass which was evaluated in [49] and is in agreement with earlier work [6, 24] . Decomposing the spectral density and the chiral condensate according to (96) we obtain\\n\\nΣ osc (m) = θ(sinh(µ) -m) Σ full (m), Σ q (m) = θ(m -sinh(µ)) Σ full (m), ( 103\\n\\n)\\n\\nso that when the quark mass is inside the ellipse of eigenvalues, the entire chiral condensate is due to the oscillatory part of the spectral density. The discontinuity in the chiral condensate is reminiscent to a Stokes phenomenon. The alternative to the Banks-Casher relation proposed in [31] is also at work for QCD in one dimension. This solves the \\\"Silver Blaze Problem\\\" [64]. 21 IX. CONCLUSIONS We have studied QCD in one dimension at nonzero chemical potential. Both the full theory, its quenched and phase quenched versions and gauge groups U (N c ) and SU (N c ) have been considered. In one dimension, the QCD or QCD-like partition functions can be reduced to a single matrix integral, which because of recent advances by Conrey, Farmer and Zirnbauer, could be evaluated analytically. In this paper we have analyzed the small mass behavior of the partition function, the nature of the sign problem, and the relation between the Dirac spectrum and chiral symmetry breaking.\\n\\nTo put our results in perspective, we emphasize that QCD in one dimension, is quite different from QCD in more dimensions. In particular, we wish to mention the following three points. First, instead of being scattered in the complex plane, the eigenvalues of the staggered Dirac operator are located on an ellipse in the complex plane. Second, phase transitions can only take place for zero temperature or for N c → ∞. At the critical chemical potential, a transition from the vacuum state to a state of free quarks takes place. Third, color singlets are made out of noninteracting quarks so that the critical chemical potential is given by the quark mass. In particular, the critical chemical potential for the meson state and the baryon state is the same.\\n\\nBecause the Dirac eigenvalues are located on a curve, the large N c limit of staggered lattice QCD in one dimension is in the same chiral symmetry class as QCD in three dimensions. We have shown this both by an explicit evaluation of the partition function, and by analyzing the fluctuations of the Dirac eigenvalues for large N c . We have used this equivalence to explain the behavior of the partition functions and the average phase factor in the microscopic limit.\\n\\nContrary to QCD in four dimensions, the sign problem for QCD in one dimension is not severe in the case of gauge group SU (N c ). One reason is that the critical chemical potential for full QCD and phase quenched QCD is the same, so that the parameter domain m π /2 < µ < m N /3, where the sign problem becomes severe in four dimensions, is absent in one dimension. A second reason is that for µ > µ c both the full theory and the phase quenched theory become a theory of free quarks with the same free energy in the thermodynamic limit.\\n\\nFor gauge group U (N c ), on the other hand, the sign problem is severe when µ > m π /2 both in one dimension and in four dimensions. The reason is that the U (N c ) theory does not have charged excitations, whereas the phase quenched theory has charged mesons resulting in a phase transition at µ = m π /2. We have evaluated the average phase factor both by averaging with respect to the phase quenched partition function and the full partition function, and similar conclusions have been reached.\\n\\nThe condititon µ > m π /2 for having a severe sign problem can be rephrased as the quark mass being inside the ellipse of eigenvalues. In more dimensions this condition is that the quark mass is inside the support of the Dirac spectrum. It also applies to nonzero temperature as we have demonstrated explicitly in the framework of a chiral random matrix model. Also for U (N c ) QCD in one dimension, the chiral condensate is discontinuous across the imaginary axis in spite of the fact that there are no Dirac eigenvalues. This can only mean that the phase of the fermion determinant is responsible for the discontinuity. This is what happens in four dimensions where the discontinuity is due to a contribution to the spectral density that oscillates with a period of the inverse volume and amplitude that diverges exponentially with the volume. Exactly the same mechanism, where the discontinuity in the chiral condensate arises due to a Stokes like phenomenon, is at work in one dimension. This suggests that this is a universal mechanism for theories with a sign problem.\\n\\nWe end by repeating that QCD in one dimension with SU (N c ) as gauge group has no serious sign problem. Our hope is that part of this conclusion translates to four dimensions ameliorating the sign problem for µ > M N /3.\"\n },\n {\n \"section_type\": \"OTHER\",\n \"section_title\": \"Acknowledgments.\",\n \"text\": \"We wish to thank K. Splittorff for numerous comments and criticism and P.H. Damgaard, H. Neuberger and R.Pisarski are thanked for valuable discussions. K. Splittorff is also thanked for a careful reading of the manuscript. This work was supported by U.S. DOE Grant No. DE-FG-88ER40388 (JV), the Angelo Della Riccia Foundation (LR), the Villum Kann Rassmussen Foundation (JV) and the Danish National Bank (JV).\\n\\nAPPENDIX A: AVERAGE U (Nc) PHASE FACTOR FOR N f = 1\\n\\nIn this appendix we give explicit results for the average phase factor for N f = 1 defined by\\n\\ne 2iθ N f =1 = Z 2|1 * (µ c , µ) Z N f =1 (µ c , µ) . ( A1\\n\\n)\\n\\nThe partition function Z N f =1 (µ c , µ) was already given in Eq. ( 8 ). The numerator can be obtained from the CFZ formula. After taking the limit of degenerate critical chemical potentials at the end of the calculation, we obtain for 22 µ < µ c Z 2|1 * (µ c , µ) = e n(Nc+1)µc e -4nµ (1 -e 2n(µ+µc) ) 2 (1 -e 2n(µ-µc) ) 2 (e nµc -e -nµc ) 5 + e -3n(Nc+1)µc (e nµ -e -nµ ) 4 (e nµc -e -nµc ) 5 +e -n(Nc+1)µc [f 0 + N c f 1 + N 2 c f 2 ]. (A2) with f 0 = 2e 2n(µ+µc) + 2e 2n(µ-µc) -e 4nµ -e -4nµ + 4e 2nµ + 4e -2nµ -e 4n(µ-µc) -e 4n(µ+µc) + 2e 6nµc-2nµ + 2e 6nµc+2nµ (e nµc -e -nµc ) 5 + -10e 2nµc + 6e 4nµc -2 -5e 6nµc -e -2nµc (e nµc -e -nµc ) 5 ,\\n\\nf 1 = (1 -e -2nµ ) 2 (3e 2nµc + e -2nµc -2e 2nµ -2e -2nµ ) (e nµc -e -nµc ) 3 , f 2 = - (1 -e -2nµ ) 2 (1 -e n(µ+µc ))(1 -e n(µ-µc )) (e nµc -e -nµc ) 3 . ( A3\\n\\n)\\n\\nFor µ > µ c a similar calculation results in\\n\\nZ 2|1 * = N c e nNc(µc-2µ) (1 -e -2n(µ+µc) )(1 -e -2nµ ) (1 -e -2nµc ) 2 (1 + g 1 N c ) + N c e -nNc(2µ+µc) (1 -e 2n(µc-µ) )(1 -e -2nµ ) (1 -e 2nµc ) 2 (1 + g 2 N c ) ( A4\\n\\n) with g 1 = 1 + 1 e 2nµ -1 + 1 e 2n(µ+µc) -1 - 2 e 2nµc -1 , g 2 = 1 + 1 e 2n(µ-µc) -1 + 1 e 2nµ -1 - 2 e -2nµc -1 . ( A5\\n\\n)\\n\\nWe have checked that the results in this appendix agree with the large N c limit of the average phase factor given in section IV B.\\n\\n[1] M. P. Lombardo, arXiv:hep-lat/0612017. [2] C. Schmidt, PoS LAT2006, 021 (2006) [arXiv:hep-lat/0610116]. [3] K. Splittorff, arXiv:hep-lat/0610072. [4] K. Splittorff and J. J. M. Verbaarschot, [arXiv:hep-lat/0702011]. [5] K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 98, 031601 (2007) [arXiv:hep-lat/0609076]. [6] P. E. Gibbs, Preprint PRINT-86-0389-GLASGOW, 1986; P. E. Gibbs, Phys. Lett. B 182 (1986) 369. [7] D. Toussaint, Nucl. Phys. Proc. Suppl. 17, 248 (1990). [8] P. de Forcrand and V. Laliena, Phys. Rev. D 61, 034502 (2000) [arXiv:hep-lat/9907004]. [9] Y. Sasai, A. Nakamura and T. Takaishi, Nucl. Phys. Proc. Suppl. 129, 539 (2004) [arXiv:hep-lat/0310046]. [10] S. Ejiri, Phys. Rev. D 69, 094506 (2004) [arXiv:hep-lat/0401012]; S. Ejiri, Phys. Rev. D 73, 054502 (2006) [arXiv:hep-lat/0506023]. [11] C. Schmidt, Z. Fodor and S. D. Katz, arXiv:hep-lat/0512032. [12] C. R. Allton et al., Phys. Rev. D 66, 074507 (2002) [arXiv:hep-lat/0204010]; C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]; C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [13] E. V. Shuryak and J. J. M. Verbaarschot, Nucl. Phys. A 560, 306 (1993) [arXiv:hep-th/9212088]. [14] J. J. M. Verbaarschot, Phys. Rev. Lett. 72, 2531 (1994) [arXiv:hep-th/9401059]. [15] J.B. Conrey, D.W. Farmer and M.R. Zirnbauer, [arXiv:math-ph/0511024], 2005. [16] E. M. Ilgenfritz and J. Kripfganz, Z. Phys. C 29, 79 (1985). [17] P. H. Damgaard, N. Kawamoto and K. Shigemoto, Phys. Rev. Lett. 53 (1984) 2211; Nucl. Phys. B 264, 1 (1986). [18] A. Gocksch and M. Ogilvie, Phys. Rev. D 31 (1985) 877. [19] P. H. Damgaard, D. Hochberg and N. Kawamoto, Phys. Lett. B 158 (1985) 239. [20] N. Bilic, K. Demeterfi and B. Petersson, Nucl. Phys. B 377, 651 (1992). [21] P. de Forcrand and S. Kim, arXiv:hep-lat/0608012.\\n\\n[22] P. E. Gibbs, Phys. Lett. B 172, 53 (1986). [23] A. Gocksch, Phys. Rev. D 37, 1014 (1988). [24] N. Bilic and K. Demeterfi, Phys. Lett. B 212, 83 (1988). [25] S. Gupta, Phys. Lett. B 588, 136 (2004) [arXiv:hep-lat/0307007]. [26] M. P. Lombardo, Nucl. Phys. Proc. Suppl. 83, 375 (2000) [arXiv:hep-lat/9908006]. [27] I. Barbour, N. E. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H. W. Wyld, Nucl. Phys. B 275, 296 (1986). [28] J. J. M. Verbaarschot and I. Zahed, Phys. Rev. Lett. 73, 2288 (1994) [arXiv:hep-th/9405005]. [29] A. J. McKane and M. Stone, Annals Phys. 131, 36 (1981). [30] M. A. Stephanov, Phys. Rev. Lett. 76, 4472 (1996) [arXiv:hep-lat/9604003]. [31] J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Phys. Rev. Lett. 94, 202001 (2005) [arXiv:hep-th/0501210]. [32] P. Hasenfratz and F. Karsch, Phys. Lett. B 125, 308 (1983). [33] M.R. Zirnbauer, [arXiv:chao-dyn/9609007]. [34] B. Schlittgen and T. Wettig, Nucl. Phys. B 632, 155 (2002) [arXiv:hep-lat/0111039]. [35] J. Budczies, S. Nonnenmacher, Y. Shnir and M. R. Zirnbauer, Nucl. Phys. B 635, 309 (2002) [arXiv:hep-lat/0112018]. [36] J.B. Conrey, D.W. Farmer, J.P. Keating, M.O. Rubinstein and N.C. Snaith, [arXiv:math-ph/0208007]. [37] E.L. Basor and P.J. Forrester, Mathematische Nachrichten, 170, 5 (1194). [38] Z. Fodor and S. D. Katz, JHEP 0203, 014 (2002) [arXiv:hep-lat/0106002]. [39] Z. Fodor and S. D. Katz, JHEP 0404, 050 (2004) [arXiv:hep-lat/0402006]. [40] P. de Forcrand and O. Philipsen, Nucl. Phys. B 642, 290 (2002) [arXiv:hep-lat/0205016]. [41] P. de Forcrand and O. Philipsen, Nucl. Phys. B 673, 170 (2003) [arXiv:hep-lat/0307020]. [42] M. D'Elia and M. P. Lombardo, Phys. Rev. D 67, 014505 (2003) [arXiv:hep-lat/0209146]. [43] C. R. Allton, S. Ejiri, S. J. Hands, O. Kaczmarek, F. Karsch, E. Laermann and C. Schmidt, Phys. Rev. D 68, 014507 (2003) [arXiv:hep-lat/0305007]. [44] C. R. Allton et al., Phys. Rev. D 71, 054508 (2005) [arXiv:hep-lat/0501030]. [45] R. V. Gavai and S. Gupta, Phys. Rev. D 68, 034506 (2003) [arXiv:hep-lat/0303013]. [46] R. V. Gavai and S. Gupta, Phys. Rev. D 71, 114014 (2005) [arXiv:hep-lat/0412035]. [47] V. Azcoiti, G. di Carlo and A. F. Grillo, Phys. Rev. Lett. 65, 2239 (1990); J. Ambjorn, K. N. Anagnostopoulos, J. Nishimura and J. J. M. Verbaarschot, JHEP 0210, 062 (2002) [arXiv:hep-lat/0208025]. [48] A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 53, 7223 (1996) [arXiv:hep-ph/9509324]. [49] M. A. Halasz, A. D. Jackson and J. J. M. Verbaarschot, Phys. Rev. D 56 (1997) 5140 [arXiv:hep-lat/9703006]. [50] K. Splittorff, arXiv:hep-lat/0505001. [51] G. V. Dunne and S. M. Nishigaki, Nucl. Phys. B 670, 307 (2003) [arXiv:hep-ph/0306220]. [52] G. Akemann and P. H. Damgaard, Nucl. Phys. B 576, 597 (2000) [arXiv:hep-th/9910190]. [53] P. H. Damgaard and S. M. Nishigaki, Phys. Rev. D 57, 5299 (1998) [arXiv:hep-th/9711096]. [54] G. Akemann, D. Dalmazi, P. H. Damgaard and J. J. M. Verbaarschot, Nucl. Phys. B 601, 77 (2001) [arXiv:hep-th/0011072]. [55] T. Andersson, P. H. Damgaard and K. Splittorff, Nucl. Phys. B 707, 509 (2005) [arXiv:hep-th/0410163]. [56] G. Akemann and Y. V. Fyodorov, Nucl. Phys. B 664, 457 (2003) [arXiv:hep-th/0304095]. [57] R. J. Szabo, Nucl. Phys. B 598, 309 (2001) [arXiv:hep-th/0009237]. [58] D. Toublan and J. J. M. Verbaarschot, Int. J. Mod. Phys. B 15, 1404 (2001) [arXiv:hep-th/0001110]. [59] K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 683, 467 (2004) [arXiv:hep-th/0310271]. [60] G. Akemann and T. Wettig, Phys. Rev. Lett. 92, 102002 (2004) [Ibid. 96, 029902 (2006) ] [arXiv:hep-lat/0308003]. [61] J. Bloch and T. Wettig, Phys. Rev. Lett. 97, 012003 (2006) [arXiv:hep-lat/0604020]. [62] P. H. Damgaard, U. M. Heller, A. Krasnitz and T. Madsen, Phys. Lett. B 440, 129 (1998) [arXiv:hep-lat/9803012]. [63] R. Narayanan and H. Neuberger, Nucl. Phys. B 696, 107 (2004) [arXiv:hep-lat/0405025]. [64] T. D. Cohen, Phys. Rev. Lett. 91, 222001 (2003) [arXiv:hep-ph/0307089]. [65] J. C. Osborn, Phys. Rev. Lett. 93, 222001 (2004) [arXiv:hep-th/0403131]. [66] G. Akemann, J. C. Osborn, K. Splittorff and J. J. M. Verbaarschot, Nucl. Phys. B 712, 287 (2005) [arXiv:hep-th/0411030].\"\n }\n]","isConversational":false}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":1,"numItemsPerPage":100,"numTotalItems":343056,"offset":100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc5MDcyNDk3MCwianRpIjoiYTI5NDhhMzctMjY4ZS00ZmViLWFlMDktNzM0ZDJlN2M0ZjVlIiwic3ViIjoiL2RhdGFzZXRzL2tvbnNtYW4vcGh5c2ljcy1jb3JwdXMiLCJleHAiOjE3OTA3Mjg1NzAsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.uaFYHEjr4gbNYrzMpSl_0KeN2jgpxzBbqtj9jGgvcT64bvvs3fcoWvPxtNi029yW5-P6MJKedZM1MzEk_v9hBg","displayUrls":true,"splitSizeSummaries":[{"config":"quantum-physics","split":"train","numRows":343056,"numBytesParquet":15067827363}]},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
paper_id stringlengths 15 15 | arxiv_id stringlengths 9 9 | arxiv_version int32 1 1 | doi stringlengths 0 94 | content_hash stringlengths 64 64 | title stringlengths 5 243 | abstract stringlengths 13 2.11k | authors listlengths 1 816 | arxiv_categories listlengths 1 13 | primary_category stringclasses 116
values | journal stringclasses 1
value | keywords listlengths 0 0 | submission_date stringdate 2007-03-31 00:00:00 2012-12-06 00:00:00 | processed_date stringdate 2026-02-26 00:00:00 2026-02-26 00:00:00 | full_text stringlengths 29 1.72M | sections listlengths 1 507 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
arxiv:0704.0549 | 0704.0549 | 1 | 571e1b1abaad561dace7da6ce4ba21baac8a4f9197e5c34637727e1fc293aad6 | Kinks and Particles in Non-integrable Quantum Field Theories | In this talk we discuss an elementary derivation of the semi-classical
spectrum of neutral particles in two field theories with kink excitations. We
also show that, in the non-integrable cases, each vacuum state cannot
generically support more than two stable particles, since all other neutral
exitations are resonances... | [
"Giuseppe Mussardo"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | Two-dimensional massive Integrable Quantum Field Theories (IQFTs) have proven to be one of the most successful topics of relativistic field theory, with a large variety of applications to statistical mechanical models. The main reason for this success consists of their simplified on-shell dynamics which is encoded into... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Two-dimensional massive Integrable Quantum Field Theories (IQFTs) have proven to be one of the most successful topics of relativistic field theory, with a large variety of applications to statistical mechanical models. The main re... | ||
arxiv:0704.0553 | 0704.0553 | 1 | 10.1016/j.physletb.2007.11.073 | 629f50de3c15170135a05e4956a1178c736d7c4757e189038d713448f716d8c8 | Spontaneous Lorentz Violation: Non-Abelian Gauge Fields as
Pseudo-Goldstone Vector Bosons | We argue that non-Abelian gauge fields can be treated as the pseudo-Goldstone
vector bosons caused by spontaneous Lorentz invariance violation (SLIV). To
this end, the SLIV which evolves in a general Yang-Mills type theory with the
nonlinear vector field constraint $Tr(% \boldsymbol{A}_{\mu
}\boldsymbol{A}^{\mu})=\pm M... | [
"J.L. Chkareuli and J.G. Jejelava"
] | [
"hep-th",
"hep-ph"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | The old idea[1] that spontaneous Lorentz invariance violation (SLIV) may lead to an alternative theory of quantum electrodynamics still remains extremely attractive in numerous theoretical contexts [2] (for some later developments, see the papers [3] ). The SLIV could generally cause the appearance of massless vector N... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The old idea[1] that spontaneous Lorentz invariance violation (SLIV) may lead to an alternative theory of quantum electrodynamics still remains extremely attractive in numerous theoretical contexts [2] (for some later developments... | |
arxiv:0704.0564 | 0704.0564 | 1 | 669120ea371220fa010dc9f22c2f7822a4b8c64e91a83fe4ab880af382eedfa7 | Spectral action on noncommutative torus | The spectral action on noncommutative torus is obtained, using a
Chamseddine--Connes formula via computations of zeta functions. The importance
of a Diophantine condition is outlined. Several results on holomorphic
continuation of series of holomorphic functions are obtained in this context. | [
"D. Essouabri",
"B. Iochum",
"C. Levy and A. Sitarz"
] | [
"hep-th",
"math-ph",
"math.MP"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | The spectral action on noncommutative torus is obtained, using a Chamseddine-Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.
The spectral action introdu... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The spectral action on noncommutative torus is obtained, using a Chamseddine-Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series... | ||
arxiv:0704.0573 | 0704.0573 | 1 | 05a07c71e5ae114221137d0f5aa9063589e159ec0dabb4c42451345ca0534803 | Elativistic treatment in}$D$ - Dimensions to a spin-zero particle with
noncentral equal scalar and vector ring-shaped Kratzer potential | The Klein-Gordon equation in D-dimensions for a recently proposed Kratzer
potential plus ring-shaped potential is solved analytically by means of the
conventional Nikiforov-Uvarov method. The exact energy bound-states and the
corresponding wave functions of the Klein-Gordon are obtained in the presence
of the noncentra... | [
"Sameer M. Ikhdair and Ramazan Sever"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equations for mixed vector and scalar potential. The Klein-Gordon and Dirac wave equations are frequ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In various physical applications including those in nuclear physics and high energy physics [1, 2] , one of the interesting problems is to obtain exact solutions of the relativistic equations like Klein-Gordon and Dirac equatio... | ||
arxiv:0704.0577 | 0704.0577 | 1 | 10.1088/1126-6708/2007/05/095 | ce0ab2b4b46f25a981791d5bad84502afb6e5d08ab4eb45027cf6589060de736 | Membrane in M5-branes Background | In this paper, we investigate the properties of a membrane in the M5-brane
background. Through solving the classical equations of motion of the membrane,
we can understand the classical dynamics of the membrane in this background. | [
"Wei-shui Xu and Ding-fang Zeng"
] | [
"hep-th"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | In this paper, we investigate the properties of a membrane in the M5-brane background. Through solving the classical equations of motion of the membrane, we can understand the classical dynamics of the membrane in this background.
In eleven-dimensional M theory, there exists two extended brane solutions, i.e membrane ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In this paper, we investigate the properties of a membrane in the M5-brane background. Through solving the classical equations of motion of the membrane, we can understand the classical dynamics of the membrane in this background."
},... | |
arxiv:0704.0593 | 0704.0593 | 1 | de0479bac9dd00afc88d9cbf4bf73aac9108046c5cdb147fc66f45978ad8df2f | Local-field effects in radiatively broadened magneto-dielectric media:
negative refraction and absorption reduction | We give a microscopic derivation of the Clausius-Mossotti relations for a
homogeneous and isotropic magneto-dielectric medium consisting of radiatively
broadened atomic oscillators. To this end the diagram series of electromagnetic
propagators is calculated exactly for an infinite bi-cubic lattice of
dielectric and mag... | [
"J\\\"urgen K\\\"astel",
"Michael Fleischhauer",
"Gediminas Juzeli\\=unas"
] | [
"cond-mat.other",
"physics.optics",
"quant-ph"
] | cond-mat.other | [] | 2007-04-04 | 2026-02-26 | It is well known that in dense dielectric materials the induced polarization P alters the field strength E loc acting on the constituents (i.e. the local field) compared to the average macroscopic field E m . Macroscopic considerations show that in systems with high symmetry such as a cubic lattice the two fields are r... | [
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "It is well known that in dense dielectric materials the induced polarization P alters the field strength E loc acting on the constituents (i.e. the local field) compared to the average macroscopic field E m . Macroscopic considera... | ||
arxiv:0704.0595 | 0704.0595 | 1 | 10.1088/1751-8113/40/46/006 | afbc3979cbfd1b052e026145eb8850b5cf3046501ce8f39984275d91e374e1e4 | About curvature, conformal metrics and warped products | We consider the curvature of a family of warped products of two
pseduo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ furnished with metrics of
the form $c^{2}g_B \oplus w^2 g_F$ and, in particular, of the type $w^{2
\mu}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth
functions and $\mu$ is a real pa... | [
"Fernando Dobarro",
"Bulent Unal"
] | [
"math.DG",
"gr-qc",
"hep-th",
"math-ph",
"math.AP",
"math.MP"
] | math.DG | [] | 2007-04-04 | 2026-02-26 | The main concern of this paper is the curvature of a special family of warped pseudo-metrics on product manifolds. We introduce a suitable form for the relations among the involved curvatures in such metrics and apply them to the existence and/or construction of Einstein and constant scalar curvature metrics in this fa... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The main concern of this paper is the curvature of a special family of warped pseudo-metrics on product manifolds. We introduce a suitable form for the relations among the involved curvatures in such metrics and apply them to the ... | |
arxiv:0704.0602 | 0704.0602 | 1 | 10.1103/PhysRevD.76.014014 | 36559ff38669c2be9e10c467814b06c4ccf59644512eaaa798cb6a3967020d6b | Alternative Large Nc Schemes and Chiral Dynamics | We compare the dependences on the number of colors of the leading pion pion
scattering amplitudes using the single index quark field and two index quark
fields. These are seen to have different relationships to the scattering
amplitudes suggested by chiral dynamics which can explain the long puzzling
pion pion s wave s... | [
"Francesco Sannino (CERN and University of Southern Denmark)",
"Joseph\n Schechter (Syracuse University)"
] | [
"hep-ph",
"hep-lat",
"hep-th",
"nucl-th"
] | hep-ph | [] | 2007-04-04 | 2026-02-26 | Gaining control of QCD in its strongly interacting (low energy) regime constitutes a real challenge. One very attractive approach is based on studying the theory in the large number of colors (N c ) limit [1, 2] . At the same time one may obtain more information by requiring the theory to model the (almost) spontaneous... | [
{
"section_type": "BACKGROUND",
"section_title": "BACKGROUND",
"text": "Gaining control of QCD in its strongly interacting (low energy) regime constitutes a real challenge. One very attractive approach is based on studying the theory in the large number of colors (N c ) limit [1, 2] . At the same time o... | |
arxiv:0704.0606 | 0704.0606 | 1 | 10.1103/PhysRevA.76.013619 | 029c732780e50f74b1b1c2b9a6a8087a7e0b3f7ef9cdd99340991bbbe7d0e048 | Three-dimensional effects in "atom diodes": atom-optical devices for
one-way motion | The ``atom diode'' is a laser device that lets the ground state atom pass in
one direction but not in the opposite direction. We examine three-dimensional
effects of that device for arbitrary atomic incidence angles on flat laser
sheets and set breakdown limiting angles and velocities. It is found that a
correct diodic... | [
"A. Ruschhaupt and J. G. Muga"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Atom optics is "basically the development and perfection of techniques and devices which can manipulate the atom waves coherently" [1] . Many of these devices (lenses, mirrors, beam splitters) have their origin in standard light optics, and others are influenced by electronics: this has lead to the idea of guiding the ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Atom optics is \"basically the development and perfection of techniques and devices which can manipulate the atom waves coherently\" [1] . Many of these devices (lenses, mirrors, beam splitters) have their origin in standard li... | |
arxiv:0704.0609 | 0704.0609 | 1 | 10.1088/1742-6596/70/1/012011 | cc0bcd9ba60eb0644c5237a75521b2d4a3ea80bae2247d760e1a8aabf80d51af | The effectiveness of quantum operations for eavesdropping on sealed
messages | A quantum protocol is described which enables a user to send sealed messages
and that allows for the detection of active eavesdroppers. We examine a class
of eavesdropping strategies, those that make use of quantum operations, and we
determine the information gain versus disturbance caused by these strategies.
We demon... | [
"Paul A Lopata",
"Thomas B Bahder"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | A quantum protocol is described which enables a user to send sealed messages and that allows for the detection of active eavesdroppers. We examine a class of eavesdropping strategies, those that make use of quantum operations, and we determine the information gain versus disturbance caused by these strategies. We demon... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "A quantum protocol is described which enables a user to send sealed messages and that allows for the detection of active eavesdroppers. We examine a class of eavesdropping strategies, those that make use of quantum operations, and we de... | |
arxiv:0704.0628 | 0704.0628 | 1 | 10.1103/PhysRevD.76.024008 | c803622105bb67f1aa73c2feddf14f30b7d4ef99ffda534d82606e76cc3e0f82 | Black hole puncture initial data with realistic gravitational wave
content | We present improved post-Newtonian-inspired initial data for non-spinning
black-hole binaries, suitable for numerical evolution with punctures. We
revisit the work of Tichy et al. [W. Tichy, B. Bruegmann, M. Campanelli, and P.
Diener, Phys. Rev. D 67, 064008 (2003)], explicitly calculating the remaining
integral terms.... | [
"Bernard J Kelly",
"Wolfgang Tichy",
"Manuela Campanelli",
"Bernard F Whiting"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-04 | 2026-02-26 | Post-Newtonian (PN) methods have played a fundamental role in our understanding of the astrophysical implications of Einstein's theory of general relativity. Most importantly, they have been used to confirm that the radiation of gravitational waves accounts for energy loss in known binary pulsar configurations. They ha... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Post-Newtonian (PN) methods have played a fundamental role in our understanding of the astrophysical implications of Einstein's theory of general relativity. Most importantly, they have been used to confirm that the radiation o... | |
arxiv:0704.0631 | 0704.0631 | 1 | 10.1016/j.physleta.2007.05.006 | 3967806fd08953aae6749e0b2718ae633b8aec2200bc2524a06c82b8287d194f | Orthogonality criterion for banishing hydrino states from standard
quantum mechanics | Orthogonality criterion is used to shown in a very simple and general way
that anomalous bound-state solutions for the Coulomb potential (hydrino states)
do not exist as bona fide solutions of the Schr\"{o}dinger, Klein-Gordon and
Dirac equations. | [
"Antonio S. de Castro"
] | [
"hep-th",
"quant-ph"
] | hep-th | [] | 2007-04-04 | 2026-02-26 | Orthogonality criterion is used to shown in a very simple and general way that anomalous bound-state solutions for the Coulomb potential (hydrino states) do not exist as bona fide solutions of the Schrödinger, Klein-Gordon and Dirac equations.
An alleged tightly-bound state of hydrogen with strong singularity of the e... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Orthogonality criterion is used to shown in a very simple and general way that anomalous bound-state solutions for the Coulomb potential (hydrino states) do not exist as bona fide solutions of the Schrödinger, Klein-Gordon and Dirac equ... | |
arxiv:0704.0639 | 0704.0639 | 1 | 10.1103/PhysRevA.76.042327 | e26bdfad2367d7b6c67c93fe8c8dd0945ad3f4869607c69832d84af9600ac298 | A measure of the non-Gaussian character of a quantum state | We address the issue of quantifying the non-Gaussian character of a bosonic
quantum state and introduce a non-Gaussianity measure based on the
Hilbert-Schmidt distance between the state under examination and a reference
Gaussian state. We analyze in details the properties of the proposed measure
and exploit it to evalu... | [
"Marco G. Genoni",
"Matteo G. A. Paris",
"Konrad Banaszek"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | Gaussian states play a crucial role in quantum information processing with continuous variables. This is especially true for quantum optical implementations since radiation at thermal equilibrium, including the vacuum state, is itself a Gaussian state and most of the Hamiltonians achievable within the current technolog... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Gaussian states play a crucial role in quantum information processing with continuous variables. This is especially true for quantum optical implementations since radiation at thermal equilibrium, including the vacuum state, is... | |
arxiv:0704.0641 | 0704.0641 | 1 | ca21011e2b7bd2073cb4f57d5443458dc7c52e0ad59d7230f5e656bc3d2dbba5 | Quantum engineering of photon states with entangled atomic ensembles | We propose and analyze a new method to produce single and entangled photons
which does not require cavities. It relies on the collective enhancement of
light emission as a consequence of the presence of entanglement in atomic
ensembles. Light emission is triggered by a laser pulse, and therefore our
scheme is determini... | [
"D. Porras",
"J.I. Cirac"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-04 | 2026-02-26 | The deterministic generation of collimated single and entangled photons is of crucial importance in Quantum Information, like in quantum cryptography [1], quantum computation [2] , quantum lithography [3] or quantum interferometry [4, 5] . Most of the methods tested so far require high-Q cavities, something which is ve... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "The deterministic generation of collimated single and entangled photons is of crucial importance in Quantum Information, like in quantum cryptography [1], quantum computation [2] , quantum lithography [3] or quantum interferometry ... | ||
arxiv:0704.0646 | 0704.0646 | 1 | 10.1007/s10701-007-9186-9 | 34b23905706f4d5243f6afeeb3f5830a109f702b95687d9e20cce4b56b4ef250 | The Mathematical Universe | I explore physics implications of the External Reality Hypothesis (ERH) that
there exists an external physical reality completely independent of us humans.
I argue that with a sufficiently broad definition of mathematics, it implies
the Mathematical Universe Hypothesis (MUH) that our physical world is an
abstract mathe... | [
"Max Tegmark (MIT)"
] | [
"gr-qc",
"astro-ph",
"hep-th"
] | gr-qc | [] | 2007-04-05 | 2026-02-26 | The idea that our universe is in some sense mathematical goes back at least to the Pythagoreans, and has been extensively discussed in the literature (see, e.g., ). Galileo Galilei stated that the Universe is a grand book written in the language of mathematics, and Wigner reflected on the "unreasonable effectiveness of... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The idea that our universe is in some sense mathematical goes back at least to the Pythagoreans, and has been extensively discussed in the literature (see, e.g., ). Galileo Galilei stated that the Universe is a grand book writt... | |
arxiv:0704.0647 | 0704.0647 | 1 | 10.1088/1475-7516/2007/04/017 | 56d8248106834f181dcadea6aa8f8e642ae87448dd12bf46c016a4eec4b1bb4f | Testing String Theory with CMB | Future detection/non-detection of tensor modes from inflation in CMB
observations presents a unique way to test certain features of string theory.
Current limit on the ratio of tensor to scalar perturbations, r=T/S, is r <
0.3, future detection may take place for r > 10^{-2}-10^{-3}. At present all
known string theory ... | [
"Renata Kallosh and Andrei Linde"
] | [
"hep-th",
"astro-ph",
"gr-qc",
"hep-ph"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | Cosmological observations may provide several interesting ways of testing string theory, which is important for its further development. For example, a discovery of the cosmological acceleration corresponding to the existence of the cosmological constant Λ ∼ 10 -120 (in Planck units M p = 1, where M p = 2.435 × 10 18 G... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Cosmological observations may provide several interesting ways of testing string theory, which is important for its further development. For example, a discovery of the cosmological acceleration corresponding to the existence of t... | |
arxiv:0704.0663 | 0704.0663 | 1 | 10.1103/PhysRevA.75.063809 | 88ce938963665a016a381b8d1719fb2ecdffb319a51d288ea6688fd4b2dcdab0 | Decoherence of Quantum-Enhanced Timing Accuracy | Quantum enhancement of optical pulse timing accuracy is investigated in the
Heisenberg picture. Effects of optical loss, group-velocity dispersion, and
Kerr nonlinearity on the position and momentum of an optical pulse are studied
via Heisenberg equations of motion. Using the developed formalism, the impact
of decohere... | [
"Mankei Tsang (California Institute of Technology)"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | It has been suggested that the use of correlated photons is able to enhance the position accuracy of an optical pulse beyond the standard quantum limit, and the enhancement can be useful for positioning and clock synchronization applications [1] . Generation of two photons with the requisite correlation has been demons... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "It has been suggested that the use of correlated photons is able to enhance the position accuracy of an optical pulse beyond the standard quantum limit, and the enhancement can be useful for positioning and clock synchronizatio... | |
arxiv:0704.0666 | 0704.0666 | 1 | 10.1103/PhysRevD.75.123513 | b1e8d7fb470b5fdd8b363257916d2fc82ac45c9d0606862483fcede44a802265 | String inspired explanation for the super-acceleration of our universe | We investigate the effect of the bulk content in the general Gauss-Bonnet
braneworld on the evolution of the universe. We find that the Gauss-Bonnet term
and the combination of the dark radiation and the matter content of the bulk
play a crucial role in the universe evolution. We show that our model can
describe the su... | [
"Ahmad Sheykhi",
"Bin Wang and Nematollah Riazi"
] | [
"hep-th",
"astro-ph",
"gr-qc"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | One of the most dramatic discoveries of the modern cosmology in the past decade is that our universe is currently accelerating [1] . A component that causes the accelerated expansion of the universe is referred to as "dark energy". Although the nature of such dark energy is still speculative, an overwhelming flood of p... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "One of the most dramatic discoveries of the modern cosmology in the past decade is that our universe is currently accelerating [1] . A component that causes the accelerated expansion of the universe is referred to as \"dark ene... | |
arxiv:0704.0672 | 0704.0672 | 1 | 10.1103/PhysRevD.76.043521 | ef58ce26aaac7dc1840505f446655452ae0509b615362f7cc644beaeffd2e3f5 | Hamiltonian formalism in Friedmann cosmology and its quantization | We propose a Hamiltonian formalism for a generalized
Friedmann-Roberson-Walker cosmology model in the presence of both a variable
equation of state (EOS) parameter $w(a)$ and a variable cosmological constant
$\Lambda(a)$, where $a$ is the scale factor. This Hamiltonian system containing
1 degree of freedom and without ... | [
"Jie Ren",
"Xin-He Meng",
"Liu Zhao"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | We propose a Hamiltonian formalism for a generalized Friedmann-Roberson-Walker cosmology model in the presence of both a variable equation of state (EOS) parameter w(a) and a variable cosmological constant Λ(a), where a is the scale factor. This Hamiltonian system containing 1 degree of freedom and without constraint, ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We propose a Hamiltonian formalism for a generalized Friedmann-Roberson-Walker cosmology model in the presence of both a variable equation of state (EOS) parameter w(a) and a variable cosmological constant Λ(a), where a is the scale fac... | |
arxiv:0704.0678 | 0704.0678 | 1 | 10.1103/PhysRevLett.99.163604 | 45652cc91b4089f7b0604742eff76b93e13e1e50aaefec63777bfdfc7ccb6f5a | Generation of Large Number-Path Entanglement Using Linear Optics and
Feed-Forward | We show how an idealised measurement procedure can condense photons from two
modes into one, and how, by feeding forward the results of the measurement, it
is possible to generate efficiently superpositions of components for which only
one mode is populated, commonly called ``N00N states''. For the basic
procedure, sou... | [
"Hugo Cable and Jonathan P. Dowling"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | We show how an idealised measurement procedure can condense photons from two modes into one, and how, by feeding forward the results of the measurement, it is possible to generate efficiently superpositions of components for which only one mode is populated, commonly called "N00N states". For the basic procedure, sourc... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We show how an idealised measurement procedure can condense photons from two modes into one, and how, by feeding forward the results of the measurement, it is possible to generate efficiently superpositions of components for which only ... | |
arxiv:0704.0680 | 0704.0680 | 1 | 10.1142/S0217732308026042 | 54a0d18bbf13539b6efab0128bd9ef20444ee6c5c4e9b778b06c111e44ceae31 | An Inverse $f(R)$ Gravitation for Cosmic Speed up, and Dark Energy
Equivalent | To explain the cosmic speed up, brought to light by the recent SNIa and CMB
observations, we propose the following: a) In a spacetime endowed with a FRW
metric, we choose an empirical scale factor that best explains the
observations. b) We assume a modified gravity, generated by an unspecified
field lagrangian, $f(R)$.... | [
"Sohrab Rahvar",
"Yousef Sobouti"
] | [
"astro-ph",
"gr-qc",
"hep-th"
] | astro-ph | [] | 2007-04-05 | 2026-02-26 | To explain the cosmic speed up, brought to light by the recent SNIa and CMB observations, we propose the following: a) In a spacetime endowed with a FRW metric, we choose an empirical scale factor that best explains the observations. b) We assume a modified gravity, generated by an unspecified field lagrangian, f (R). ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "To explain the cosmic speed up, brought to light by the recent SNIa and CMB observations, we propose the following: a) In a spacetime endowed with a FRW metric, we choose an empirical scale factor that best explains the observations. b)... | |
arxiv:0704.0690 | 0704.0690 | 1 | 10.1088/1126-6708/2007/06/056 | d8a0cd5ec9110b059024b4d1ea3c5a297c40a04007da56c2e3d991c33239a399 | Fuzzballs with internal excitations | We construct general 2-charge D1-D5 horizon-free non-singular solutions of
IIB supergravity on T^4 and K3 describing fuzzballs with excitations in the
internal manifold; these excitations are characterized by arbitrary curves. The
solutions are obtained via dualities from F1-P solutions of heterotic and type
IIB on T^4... | [
"Ingmar Kanitscheider",
"Kostas Skenderis and Marika Taylor"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | We construct general 2-charge D1-D5 horizon-free non-singular solutions of IIB supergravity on T 4 and K3 describing fuzzballs with excitations in the internal manifold; these excitations are characterized by arbitrary curves. The solutions are obtained via dualities from F1-P solutions of heterotic and type IIB on T 4... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We construct general 2-charge D1-D5 horizon-free non-singular solutions of IIB supergravity on T 4 and K3 describing fuzzballs with excitations in the internal manifold; these excitations are characterized by arbitrary curves. The solut... | |
arxiv:0704.0709 | 0704.0709 | 1 | 10.1103/PhysRevA.76.042324 | 433cfe6aa24b2a4b91fb9412579e33ce404bf5adc7610cddb903a050968b765b | Braiding transformation, entanglement swapping and Berry phase in
entanglement space | We show that braiding transformation is a natural approach to describe
quantum entanglement, by using the unitary braiding operators to realize
entanglement swapping and generate the GHZ states as well as the linear cluster
states. A Hamiltonian is constructed from the unitary
$\check{R}_{i,i+1}(\theta,\phi)$-matrix, w... | [
"Jing-Ling Chen",
"Kang Xue",
"Mo-Lin Ge"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | Quantum entanglement is the most surprising nonclassical property of composite quantum systems that Schrödinger singled out many decades ago as "the characteristic trait of quantum mechanics". Recently entanglement has become one of the most fascinating topics in quantum information, because it has been shown that enta... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Quantum entanglement is the most surprising nonclassical property of composite quantum systems that Schrödinger singled out many decades ago as \"the characteristic trait of quantum mechanics\". Recently entanglement has become... | |
arxiv:0704.0719 | 0704.0719 | 1 | 10.1103/PhysRevD.76.024018 | 68d29a838103153ef8ea346d5d466d636d1558b25e20face62aae00048cbb631 | Gravitational Wave Signals from Chaotic System: A Point Mass with A Disk | We study gravitational waves from a particle moving around a system of a
point mass with a disk in Newtonian gravitational theory. A particle motion in
this system can be chaotic when the gravitational contribution from a surface
density of a disk is comparable with that from a point mass. In such an orbit,
we sometime... | [
"Kenta Kiuchi",
"Hiroko Koyama",
"Kei-ichi Maeda"
] | [
"gr-qc",
"astro-ph"
] | gr-qc | [] | 2007-04-05 | 2026-02-26 | Chaos appears universally in nature and it is expected to be a fundamental tool to understand various nonlinear phenomena. Historically, research on chaos started from the famous three-body problem by Poincaré, and so far many attempts to reveal the character of chaos have been done in dynamical systems. Following such... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Chaos appears universally in nature and it is expected to be a fundamental tool to understand various nonlinear phenomena. Historically, research on chaos started from the famous three-body problem by Poincaré, and so far many ... | |
arxiv:0704.0722 | 0704.0722 | 1 | 10.1103/PhysRevD.75.116004 | 1b7f2105205a4efb9ab4b5d87edeb88d38f04ed3dd93bced15720bf4c49645c5 | Two- and three-point Green's functions in two-dimensional Landau-gauge
Yang-Mills theory | The ghost and gluon propagator and the ghost-gluon and three-gluon vertex of
two-dimensional SU(2) Yang-Mills theory in (minimal) Landau gauge are studied
using lattice gauge theory. It is found that the results are qualitatively
similar to the ones in three and four dimensions. The propagators and the
Faddeev-Popov op... | [
"Axel Maas"
] | [
"hep-lat",
"hep-ph",
"hep-th"
] | hep-lat | [] | 2007-04-05 | 2026-02-26 | Two-dimensional Yang-Mills theory turns out to be a very fascinating topic. Quite a number of quantities, e. g. the string tension [1] , can be calculated exactly, although not all quantities are (yet) known analytically. In particular, up to now it was not possible to calculate the Green's functions in Landau gauge. H... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Two-dimensional Yang-Mills theory turns out to be a very fascinating topic. Quite a number of quantities, e. g. the string tension [1] , can be calculated exactly, although not all quantities are (yet) known analytically. In pa... | |
arxiv:0704.0729 | 0704.0729 | 1 | 10.1088/1126-6708/2007/06/026 | e6789dbd885501b91f2ab22129779c9b3d471f1e506e69bbdc922446610436a2 | The First Law for Boosted Kaluza-Klein Black Holes | We study the thermodynamics of Kaluza-Klein black holes with momentum along
the compact dimension, but vanishing angular momentum. These black holes are
stationary, but non-rotating. We derive the first law for these spacetimes and
find that the parameter conjugate to variations in the length of the compact
direction i... | [
"David Kastor",
"Sourya Ray and Jennie Traschen"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | We study the thermodynamics of Kaluza-Klein black holes with momentum along the compact dimension, but vanishing angular momentum. These black holes are stationary, but non-rotating. We derive the first law for these spacetimes and find that the parameter conjugate to variations in the length of the compact direction i... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study the thermodynamics of Kaluza-Klein black holes with momentum along the compact dimension, but vanishing angular momentum. These black holes are stationary, but non-rotating. We derive the first law for these spacetimes and find... | |
arxiv:0704.0732 | 0704.0732 | 1 | 71637f3679c583d00c6ed049c4b1a13801523ce9fa7728ca3d8d531d8cae8b0f | Cosmological Singularities and a Conjectured Gravity/Coset
Correspondence | We review the recently discovered connection between the
Belinsky-Khalatnikov-Lifshitz-like ``chaotic'' structure of generic
cosmological singularities in eleven-dimensional supergravity and the ``last''
hyperbolic Kac-Moody algebra E(10). This intriguing connection suggests the
existence of a hidden ``correspondence''... | [
"Thibault Damour"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | We review the recently discovered connection between the Belinsky-Khalatnikov-Lifshitz-like "chaotic" structure of generic cosmological singularities in eleven-dimensional supergravity and the "last" hyperbolic Kac-Moody algebra E 10 . This intriguing connection suggests the existence of a hidden "correspondence" betwe... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We review the recently discovered connection between the Belinsky-Khalatnikov-Lifshitz-like \"chaotic\" structure of generic cosmological singularities in eleven-dimensional supergravity and the \"last\" hyperbolic Kac-Moody algebra E 1... | ||
arxiv:0704.0737 | 0704.0737 | 1 | 10.1088/1126-6708/2007/09/031 | 2d9684cfed8958cfe84f7f91e7aacb22c76ea83845908dcdeccc3b10f6173ebc | Jumping Through Loops: On Soft Terms from Large Volume Compactifications | We subject the phenomenologically successful large volume scenario of
hep-th/0502058 to a first consistency check in string theory. In particular, we
consider whether the expansion of the string effective action is consistent in
the presence of D-branes and O-planes. Due to the no-scale structure at
tree-level, the sce... | [
"Marcus Berg",
"Michael Haack",
"Enrico Pajer"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | Contents 1 Introduction 2 Review 2.1 KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Consistency of KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Large volume scenario (LVS) . . . . . . . . . . . . . . . . . . . . . . . .
3.1 From toroidal orientifolds to Calabi-Yau... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Contents 1 Introduction 2 Review 2.1 KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Consistency of KKLT . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n\n2.3 Large volume scenario (LVS) . ... | |
arxiv:0704.0748 | 0704.0748 | 1 | b53610d65c52244ab91c8355413ba92ce2afbcb686f6e795755ef81e0c9a31e9 | Experimental Challenges Involved in Searches for Axion-Like Particles
and Nonlinear Quantum Electrodynamic Effects by Sensitive Optical Techniques | We discuss the experimental techniques used to date for measuring the changes
in polarization state of a laser produced by a strong transverse magnetic field
acting in a vacuum. We point out the likely artifacts that can arise in such
experiments, with particular reference to the recent PVLAS observations and the
previ... | [
"Christopher C. Davis",
"Joseph Harris",
"Robert W. Gammon",
"Igor I.\n Smolyaninov and Kyuman Cho"
] | [
"hep-th",
"hep-ph"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | We discuss the experimental techniques used to date for measuring the changes in polarization state of a laser produced by a strong transverse magnetic field acting in a vacuum. We point out the likely artifacts that can arise in such experiments, with particular reference to the recent PVLAS observations and the previ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We discuss the experimental techniques used to date for measuring the changes in polarization state of a laser produced by a strong transverse magnetic field acting in a vacuum. We point out the likely artifacts that can arise in such e... | ||
arxiv:0704.0749 | 0704.0749 | 1 | 6eb1e94edb8ade09da3a7a84537985883ba324f7cda0734b7227df28af06b1b7 | Binary Systems as Test-beds of Gravity Theories | We review the general relativistic theory of the motion, and of the timing,
of binary systems containing compact objects (neutron stars or black holes).
Then we indicate the various ways one can use binary pulsar data to test the
strong-field and/or radiative aspects of General Relativity, and of general
classes of alt... | [
"Thibault Damour"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-05 | 2026-02-26 | The discovery of binary pulsars in 1974 [1] opened up a new testing ground for relativistic gravity. Before this discovery, the only available testing ground for relativistic gravity was the solar system. As Einstein's theory of General Relativity (GR) is one of the basic pillars of modern science, it deserves to be te... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The discovery of binary pulsars in 1974 [1] opened up a new testing ground for relativistic gravity. Before this discovery, the only available testing ground for relativistic gravity was the solar system. As Einstein's theory of G... | ||
arxiv:0704.0752 | 0704.0752 | 1 | 10.1590/S0103-97332008000200010 | cd2c8efb55708a8f82254f534d87d85a574a5fb630bb36a06188d36fd2083ebc | Actions for the Bosonic String with the Curved Worldsheet | At first we introduce an action for the string, which leads to a worldsheet
that always is curved. For this action we study the Poincar\'e symmetry and the
associated conserved currents. Then, a generalization of the above action,
which contains an arbitrary function of the two-dimensional scalar curvature,
will be int... | [
"Davoud Kamani"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | At first we introduce an action for the string, which leads to a worldsheet that always is curved. For this action we study the Poincaré symmetry and the associated conserved currents. Then, a generalization of the above action, which contains an arbitrary function of the two-dimensional scalar curvature, will be intro... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "At first we introduce an action for the string, which leads to a worldsheet that always is curved. For this action we study the Poincaré symmetry and the associated conserved currents. Then, a generalization of the above action, which c... | |
arxiv:0704.0753 | 0704.0753 | 1 | 10.1088/0256-307X/24/8/082 | c5c4ce3121fda24e472afae7466f5479b43951059787e449326237c08bfc3f73 | Bianchi Type I Massive String Magnetized Barotropic Perfect Fluid
Cosmological Model in General Relativity | Bianchi type I massive string cosmological model with magnetic field of
barotropic perfect fluid distribution through the techniques used by Latelier
and Stachel, is investigated. To get the deterministic model of the universe,
it is assumed that the universe is filled with barotropic perfect fluid
distribution. The ma... | [
"Raj Bali",
"Umesh Kumar Pareek and Anirudh Pradhan"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-05 | 2026-02-26 | The cosmic strings play an important role in the study of the early universe. These strings arise during the phase transition after the big bang explosion as the temperature drops down below some critical temperature as predicted by grand unified theories [1] [2] [3] [4] [5] . It is thought that cosmic strings cause de... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The cosmic strings play an important role in the study of the early universe. These strings arise during the phase transition after the big bang explosion as the temperature drops down below some critical temperature as predicted ... | |
arxiv:0704.0754 | 0704.0754 | 1 | 10.1007/978-3-7643-8524-8_1 | 2b6ae25471d1ef8fd716fcc9eda87970002282368287839b25d63376554e1c14 | General Relativity Today | After recalling the conceptual foundations and the basic structure of general
relativity, we review some of its main modern developments (apart from
cosmology) : (i) the post-Newtonian limit and weak-field tests in the solar
system, (ii) strong gravitational fields and black holes, (iii) strong-field
and radiative test... | [
"Thibault Damour"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-05 | 2026-02-26 | The theory of general relativity was developed by Einstein in work that extended from 1907 to 1915. The starting point for Einstein's thinking was the composition of a review article in 1907 on what we today call the theory of special relativity. Recall that the latter theory sprang from a new kinematics governing leng... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The theory of general relativity was developed by Einstein in work that extended from 1907 to 1915. The starting point for Einstein's thinking was the composition of a review article in 1907 on what we today call the theory of spe... | |
arxiv:0704.0756 | 0704.0756 | 1 | 10.1103/PhysRevLett.99.071601 | baf3c9ed87da97a62969cf1a713b5082874cef84c6fd7267a2e6e84c77a91587 | The dynamical Casimir effect in braneworlds | In braneworld cosmology the expanding Universe is realized as a brane moving
through a warped higher-dimensional spacetime. Like a moving mirror causes the
creation of photons out of vacuum fluctuations, a moving brane leads to
graviton production. We show that, very generically, Kaluza-Klein (KK)
particles scale like ... | [
"Ruth Durrer and Marcus Ruser"
] | [
"hep-th",
"astro-ph",
"hep-ph"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | In braneworld cosmology the expanding Universe is realized as a brane moving through a warped higher-dimensional spacetime. Like a moving mirror causes the creation of photons out of vacuum fluctuations, a moving brane leads to graviton production. We show that, very generically, KKparticles scale like stiff matter wit... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In braneworld cosmology the expanding Universe is realized as a brane moving through a warped higher-dimensional spacetime. Like a moving mirror causes the creation of photons out of vacuum fluctuations, a moving brane leads to graviton... | |
arxiv:0704.0757 | 0704.0757 | 1 | 10.1103/PhysRevA.76.022320 | 2b8ffa29123d493677ad112a0e60c5c9a5b2b994d222f05a4a97d67648efd311 | Bounds on Negativity of Superpositions | The entanglement quantified by negativity of pure bipartite superposed states
is studied. We show that if the entanglement is quantified by the concurrence
two pure states of high fidelity to one another still have nearly the same
entanglement. Furthermore this conclusion can be guaranteed by our obtained
inequality, a... | [
"Yong-Cheng Ou and Heng Fan"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | The entanglement quantified by negativity of pure bipartite superposed states is studied. We show that if the entanglement is quantified by the concurrence two pure states of high fidelity to one another still have nearly the same entanglement. Furthermore this conclusion can be guaranteed by our obtained inequality, a... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The entanglement quantified by negativity of pure bipartite superposed states is studied. We show that if the entanglement is quantified by the concurrence two pure states of high fidelity to one another still have nearly the same entan... | |
arxiv:0704.0758 | 0704.0758 | 1 | 10.1038/nphys700 | 4a2d5e9775d730fd45b88c04ce08fad0c40253fdb248e08951141b280c02ad88 | Entangling Independent Photons by Time Measurement | A quantum system composed of two or more subsystems can be in an entangled
state, i.e. a state in which the properties of the global system are well
defined but the properties of each subsystem are not. Entanglement is at the
heart of quantum physics, both for its conceptual foundations and for
applications in informat... | [
"Matthaeus Halder",
"Alexios Beveratos",
"Nicolas Gisin",
"Valerio Scarani",
"Christoph Simon and Hugo Zbinden"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | A quantum system composed of two or more subsystems can be in an entangled state, i.e. a state in which the properties of the global system are well defined but the properties of each subsystem are not. Entanglement is at the heart of quantum physics, both for its conceptual foundations and for applications in informat... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "A quantum system composed of two or more subsystems can be in an entangled state, i.e. a state in which the properties of the global system are well defined but the properties of each subsystem are not. Entanglement is at the heart of q... | |
arxiv:0704.0763 | 0704.0763 | 1 | 10.1088/0953-4075/41/11/115502 | 63bfb120446ce7cb65de2c3b967b36331be17234235779c3bffa058dfcaae70a | Coherent control of atomic tunneling | We study the tunneling of a two-level atom in a double well potential while
the atom is coupled to a single electromagnetic field mode of a cavity. The
coupling between internal and external degrees of freedom, due to the
mechanical effect on the atom from photon emission into the cavity mode, can
dramatically change t... | [
"John Martin and Daniel Braun"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | The tunneling effect is considered one of the hallmarks of quantum mechanical behavior. Historically, tunneling was first examined for single particles (e.g. α particles [1], electrons in field emission [2] and later in mesoscopic circuits [3]), for Cooper pairs [4] , and for molecular groups [5, 6, 7] . Recently the t... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The tunneling effect is considered one of the hallmarks of quantum mechanical behavior. Historically, tunneling was first examined for single particles (e.g. α particles [1], electrons in field emission [2] and later in mesosco... | |
arxiv:0704.0766 | 0704.0766 | 1 | 59c26136306dc8b8ff895bf0ed0dffc651c036163b26ba532c6310f6184f2606 | Local de Broglie-Bohm Trajectories from Entangled Wavefunctions | We present a local interpretation of what is usually considered to be a
nonlocal de Broglie-Bohm trajectory prescription for an entangled singlet state
of massive particles. After reviewing various meanings of the term
``nonlocal'', we show that by using appropriately retarded wavefunctions (i.e.,
the locality loophole... | [
"Michael Clover"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | In the EPR literature, it is usually considered that the only local models that can explain EPR experiments are contrived intellectual diversions 1, 2, 3 , bearing no relation to physical reality, and that additionally require some "unfair" sampling mechanism to generate the correct experimental results 4, 5, 6, 7, 8 .... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In the EPR literature, it is usually considered that the only local models that can explain EPR experiments are contrived intellectual diversions 1, 2, 3 , bearing no relation to physical reality, and that additionally require ... | ||
arxiv:0704.0777 | 0704.0777 | 1 | 10.1103/PhysRevLett.99.041601 | b5b33ec059ad72aefa8addbaa26576f9447417dece1405312c9c26bb1f9795ca | Decoupling Supergravity from the Superstring | We consider the conditions necessary for obtaining perturbative maximal
supergravity in d dimensions as a decoupling limit of type II superstring
theory compactified on a (10 -- d)-torus. For dimensions d = 2 and d = 3 it is
possible to define a limit in which the only finite-mass states are the 256
massless states of ... | [
"Michael B. Green",
"Hirosi Ooguri",
"John H. Schwarz"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | We consider the conditions necessary for obtaining perturbative maximal supergravity in d dimensions as a decoupling limit of type II superstring theory compactified on a (10d)-torus. For dimensions d = 2 and d = 3 it is possible to define a limit in which the only finite-mass states are the 256 massless states of maxi... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider the conditions necessary for obtaining perturbative maximal supergravity in d dimensions as a decoupling limit of type II superstring theory compactified on a (10d)-torus. For dimensions d = 2 and d = 3 it is possible to def... | |
arxiv:0704.0782 | 0704.0782 | 1 | 10.1088/0264-9381/24/16/006 | fed111466e2d981b81451159efae46e960b6d2e53a0acc663790bad850827532 | Testing outer boundary treatments for the Einstein equations | Various methods of treating outer boundaries in numerical relativity are
compared using a simple test problem: a Schwarzschild black hole with an
outgoing gravitational wave perturbation. Numerical solutions computed using
different boundary treatments are compared to a `reference' numerical solution
obtained by placin... | [
"Oliver Rinne",
"Lee Lindblom and Mark A. Scheel"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-05 | 2026-02-26 | A fundamental problem in numerical relativity is the need to solve Einstein's equations on spatially unbounded domains with finite computer resources. There are various ways of addressing this issue. Most often, the spatial domain is truncated at a finite distance and suitable boundary conditions are imposed at the art... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "A fundamental problem in numerical relativity is the need to solve Einstein's equations on spatially unbounded domains with finite computer resources. There are various ways of addressing this issue. Most often, the spatial domain... | |
arxiv:0704.0784 | 0704.0784 | 1 | 10.1088/1126-6708/2007/07/038 | d9813ccac1f6dadc299b4ac09d99f1065b8a62faadcc0fa0b9663ec5de86d81d | D-brane Instantons on the T^6/Z_3 orientifold | We give a detailed microscopic derivation of gauge and stringy instanton
generated superpotentials for gauge theories living on D3-branes at
Z_3-orientifold singularities. Gauge instantons are generated by D(-1)-branes
and lead to Affleck, Dine and Seiberg (ADS) like superpotentials in the
effective N=1 gauge theories ... | [
"Massimo Bianchi",
"Francesco Fucito and Jose F. Morales"
] | [
"hep-th"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | 1 Introduction 1 2 The Gauge Theory 4 3 D(-1) Instantons 6 3.1 D3-D(-1) in flat space . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 D(-1)-D3 at the 3 / 3 -orientifold . . . . . . . . . . . . . . . 7 4 ADS-like superpotential 9 4.1 D3-D(-1) one-loop vacuum amplitudes . . . . . . . . . . . . . 9 4.2 Sp(6) × U(2) su... | [
{
"section_type": "OTHER",
"section_title": "Contents",
"text": "1 Introduction 1 2 The Gauge Theory 4 3 D(-1) Instantons 6 3.1 D3-D(-1) in flat space . . . . . . . . . . . . . . . . . . . . . . . 6 3.2 D(-1)-D3 at the 3 / 3 -orientifold . . . . . . . . . . . . . . . 7 4 ADS-like superpotential 9 4.1 D3... | |
arxiv:0704.0790 | 0704.0790 | 1 | 10.1103/PhysRevD.76.104014 | 48ec209ef01b1ec4fff2c03c60b206679cd2240d75e8ffd40605655c7d426769 | Dynamical Casimir effect for gravitons in bouncing braneworlds | We consider a two-brane system in a five-dimensional anti-de Sitter
spacetime. We study particle creation due to the motion of the physical brane
which first approaches the second static brane (contraction) and then recedes
from it(expansion). The spectrum and the energy density of the generated
gravitons are calculate... | [
"Marcus Ruser and Ruth Durrer"
] | [
"hep-th",
"astro-ph",
"hep-ph"
] | hep-th | [] | 2007-04-05 | 2026-02-26 | In recent times, the possibility that our observed Universe might represent a hypersurface in a higherdimensional space-time has received considerable attention. The main motivation for this idea is the fact, that string theory [1, 2] , which is consistent only in ten spac-etime dimensions (or 11 for M-theory) allows f... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In recent times, the possibility that our observed Universe might represent a hypersurface in a higherdimensional space-time has received considerable attention. The main motivation for this idea is the fact, that string theory... | |
arxiv:0704.0793 | 0704.0793 | 1 | 10.1103/PhysRevLett.99.211301 | fab0bebe53d58ed22c96fcd24c6a61dcaf142956979fdb6c068b6a3b98e44fd9 | Friedmann Equations and Thermodynamics of Apparent Horizons | With the help of a masslike function which has dimension of energy and equals
to the Misner-Sharp mass at the apparent horizon, we show that the first law of
thermodynamics of the apparent horizon $dE=T_AdS_A$ can be derived from the
Friedmann equation in various theories of gravity, including the Einstein,
Lovelock, n... | [
"Yungui Gong and Anzhong Wang"
] | [
"hep-th",
"astro-ph",
"gr-qc"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | With the help of a masslike function which has dimension of energy and equals to the Misner-Sharp mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon dE = TAdSA can be derived from the Friedmann equation in various theories of gravity, including the Einstein, Lovelock, non... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "With the help of a masslike function which has dimension of energy and equals to the Misner-Sharp mass at the apparent horizon, we show that the first law of thermodynamics of the apparent horizon dE = TAdSA can be derived from the Frie... | |
arxiv:0704.0796 | 0704.0796 | 1 | 10.1088/1751-8113/40/30/020 | f7d127078936deeee1718e06ca52c9d24cd93e4e83008e5ef600a4989a6fbae2 | A density tensor hierarchy for open system dynamics: retrieving the
noise | We introduce a density tensor hierarchy for open system dynamics, that
recovers information about fluctuations lost in passing to the reduced density
matrix. For the case of fluctuations arising from a classical probability
distribution, the hierarchy is formed from expectations of products of pure
state density matrix... | [
"Stephen L. Adler"
] | [
"quant-ph",
"astro-ph",
"cond-mat.stat-mech",
"hep-ph",
"math-ph",
"math.MP"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | We develop a density tensor hierarchy for open system dynamics, that recovers information about fluctuations (or "noise") lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability distribution, the hierarchy is formed from expectations of products of pure state den... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We develop a density tensor hierarchy for open system dynamics, that recovers information about fluctuations (or \"noise\") lost in passing to the reduced density matrix. For the case of fluctuations arising from a classical probability... | |
arxiv:0704.0797 | 0704.0797 | 1 | 10.1103/PhysRevD.75.124011 | e30cbaf9bf81495e5613d930fa607cf3125f06c86c03e42e276fde680e448b0d | Scalar self-force on eccentric geodesics in Schwarzschild spacetime: a
time-domain computation | We calculate the self-force acting on a particle with scalar charge moving on
a generic geodesic around a Schwarzschild black hole. This calculation requires
an accurate computation of the retarded scalar field produced by the moving
charge; this is done numerically with the help of a fourth-order convergent
finite-dif... | [
"Roland Haas"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-05 | 2026-02-26 | The inspiral and capture of solar-mass compact objects by supermassive black holes is one of the most promising and interesting sources of gravitational radiation to be detected by the future space-based gravitational-wave antenna LISA [1] . For these extreme mass-ratio inspirals, one can treat the compact object as a ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The inspiral and capture of solar-mass compact objects by supermassive black holes is one of the most promising and interesting sources of gravitational radiation to be detected by the future space-based gravitational-wave ante... | |
arxiv:0704.0799 | 0704.0799 | 1 | 10.1103/PhysRevD.76.064019 | 2290bda37eaead54abe4e17993bc6e25e8fd8bf8ac8102c0c39943a706904369 | Spin Evolution of Accreting Neutron Stars: Nonlinear Development of the
R-mode Instability | The nonlinear saturation of the r-mode instability and its effects on the
spin evolution of Low Mass X-ray Binaries (LMXBs) are modeled using the triplet
of modes at the lowest parametric instability threshold. We solve numerically
the coupled equations for the three mode amplitudes in conjunction with the
spin and tem... | [
"Ruxandra Bondarescu",
"Saul A. Teukolsky and Ira Wasserman (Cornell\n University)"
] | [
"astro-ph",
"gr-qc"
] | astro-ph | [] | 2007-04-05 | 2026-02-26 | R-modes are oscillations in rotating fluids that are due to the Coriolis effect. They are subject to the classical Chandrashekar-Friedman-Shutz (CFS) instability [2, 3] , which is driven by the gravitational radiation backreaction force. Andersson [4] and Friedman and Morsink [5] showed that, in the absence of fluid di... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "R-modes are oscillations in rotating fluids that are due to the Coriolis effect. They are subject to the classical Chandrashekar-Friedman-Shutz (CFS) instability [2, 3] , which is driven by the gravitational radiation backreact... | |
arxiv:0704.0800 | 0704.0800 | 1 | cf12a7a00cfd3e77301ea851518da96bf76c020a99ada3c1d773bb68b18883bc | Quantum Auctions | We present a quantum auction protocol using superpositions to represent bids
and distributed search to identify the winner(s). Measuring the final quantum
state gives the auction outcome while simultaneously destroying the
superposition. Thus non-winning bids are never revealed. Participants can use
entanglement to arr... | [
"Tad Hogg",
"Pavithra Harsha and Kay-Yut Chen"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | Quantum information processing [23] offers potential improvements in a variety of applications. Computational advantages [26, 14] of quantum computers with many qubits have received the most attention but are difficult to implement physically. On the other hand, technology for manipulating and communicating just a few ... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Quantum information processing [23] offers potential improvements in a variety of applications. Computational advantages [26, 14] of quantum computers with many qubits have received the most attention but are difficult to implemen... | ||
arxiv:0704.0803 | 0704.0803 | 1 | ac0468843a196db71d77bb6a9f949f2bf4a76c21292efe806a33aa7092783f26 | Geometric Phase and Superconducting Flux Quantization | In a ring of s-wave superconducting material the magnetic flux is quantized
in units of $\Phi_0 = \frac{h}{2e}$. It is well known from the theory of
Josephson junctions that if the ring is interrupted with a piece of d-wave
material, then the flux is quantized in one-half of those units due to a
additional phase shift ... | [
"Walter A. Simmons and Sandip S. Pakvasa"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | In a ring of s-wave superconducting material the magnetic flux is quantized in units of 0 2 . It is well known from the theory of Josephson junctions that if the ring is interrupted with a piece of dwave material, then the flux is quantized in one-half of those units due to a additional phase shift of π . We reinterpre... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In a ring of s-wave superconducting material the magnetic flux is quantized in units of 0 2 . It is well known from the theory of Josephson junctions that if the ring is interrupted with a piece of dwave material, then the flux is quant... | ||
arxiv:0704.0814 | 0704.0814 | 1 | 10.1103/PhysRevA.77.043813 | 7d00190419eeedb31fa8885fc7d157b6a17f2356e3a1671b0fc694804c789911 | Photons as quasi-charged particles | The Schrodinger motion of a charged quantum particle in an electromagnetic
potential can be simulated by the paraxial dynamics of photons propagating
through a spatially inhomogeneous medium. The inhomogeneity induces geometric
effects that generate an artificial vector potential to which signal photons
are coupled. Th... | [
"K.-P. Marzlin",
"Juergen Appel",
"A. I. Lvovsky"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | -It is known since the ground-breaking work of Berry on geometric phases [1] that artificial gauge potentials can be induced if the spatial dynamics of a system that obeys a wave equation is confined in a certain way. For instance, if the internal Hamiltonian of neutral atoms contains an energy barrier but the spin eig... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "-It is known since the ground-breaking work of Berry on geometric phases [1] that artificial gauge potentials can be induced if the spatial dynamics of a system that obeys a wave equation is confined in a certain way. For instance... | |
arxiv:0704.0815 | 0704.0815 | 1 | 05e60d3660e39206f744715367579b54d7710614a5e0c2357548ea624b63a6a4 | Exchange of quantum states between coupled oscillators | Exchange of quantum states between two interacting harmonic oscillator along
their evolution time is discussed. It is analyzed the conditions for such
exchange starting from a generic initial state and demonstrating that the
effect occurs exactly only for the particular states C0|0>+Cn|N>, which
includes the interestin... | [
"D. Portes Jr.",
"H. Rodrigues",
"S. B. Duarte and B. Baseia"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-05 | 2026-02-26 | Exchange of quantum states between two interacting harmonic oscillator along their evolution time is discussed. It is analyzed the conditions for such exchange starting from a generic initial state and demonstrating that the effect occurs exactly only for the particular states C0 |0 > +CN |N >, which includes the inter... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Exchange of quantum states between two interacting harmonic oscillator along their evolution time is discussed. It is analyzed the conditions for such exchange starting from a generic initial state and demonstrating that the effect occu... | ||
arxiv:0704.0822 | 0704.0822 | 1 | 10.1143/PTP.118.785 | accf12b73c15c060c1736d6b91281a9bb59ac3090fc3dc7883d6b64baa142f5d | Braided quantum field theories and their symmetries | Braided quantum field theories proposed by Oeckl can provide a framework for
defining quantum field theories having Hopf algebra symmetries. In quantum
field theories, symmetries lead to non-perturbative relations among correlation
functions. We discuss Hopf algebra symmetries and such relations in braided
quantum fiel... | [
"Yuya Sasai",
"Naoki Sasakura"
] | [
"hep-th"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | Symmetry is one of the most important notions in quantum field theory. In many examples, it is useful in investigating properties of quantum field theories non-perturbatively, is a guiding principle in constructing field theories for various purposes such as grand unification, or gives powerful methods in finding exact... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Symmetry is one of the most important notions in quantum field theory. In many examples, it is useful in investigating properties of quantum field theories non-perturbatively, is a guiding principle in constructing field theories ... | |
arxiv:0704.0829 | 0704.0829 | 1 | 10.1088/0253-6102/47/4/027 | 8dd49c1981379fe3f3f52f27e72d352676fb912915f53dd383d13c50f2a3b51f | Quark-Antiquark and Diquark Condensates in Vacuum in a 3D Two-Flavor
Gross-Neveu Model | The effective potential analysis indicates that, in a 3D two-flavor
Gross-Neveu model in vacuum, depending on less or bigger than the critical
value 2/3 of $G_S/H_P$, where $G_S$ and $H_P$ are respectively the coupling
constants of scalar quark-antiquark channel and pseudoscalar diquark channel,
the system will have th... | [
"Bang-Rong Zhou (Graduate School of Chinese Academy of Sciences)"
] | [
"hep-th"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | The effective potential analysis indicates that, in a 3D two-flavor Gross-Neveu model in vacuum, depending on less or bigger than the critical value 2/3 of GS/HP , where GS and HP are respectively the coupling constants of scalar quark-antiquark channel and pseudoscalar diquark channel, the system will have the ground ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The effective potential analysis indicates that, in a 3D two-flavor Gross-Neveu model in vacuum, depending on less or bigger than the critical value 2/3 of GS/HP , where GS and HP are respectively the coupling constants of scalar quark-... | |
arxiv:0704.0837 | 0704.0837 | 1 | 10.3938/jkps.50.1229 | 23415e7ed6f052040e50db6ee73496784e811b44f8e4571469f9ce1106f052c6 | Bremsstrahlung Radiation At a Vacuum Bubble Wall | When charged particles collide with a vacuum bubble, they can radiate strong
electromagnetic waves due to rapid deceleration. Owing to the energy loss of
the particles by this bremsstrahlung radiation, there is a non-negligible
damping pressure acting on the bubble wall even when thermal equilibrium is
maintained. In t... | [
"Jae-Weon Lee",
"Kyungsub Kim",
"Chul H. Lee",
"Ji-ho Jang"
] | [
"hep-th"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | 2 This approximation is good for the usual tanh profile of the wall. The radiation power of an accelerated particle is given by a relativistic version of the Larmor's formula [16]
:
dE rad dt = A m 2 d k dt 2 , ( 1
)
where A = 2e 2 /3c 3 ≃ 0.0611 in the natural units ( = c = k = 1) and k = (k x , k y , k z ) is the... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "2 This approximation is good for the usual tanh profile of the wall. The radiation power of an accelerated particle is given by a relativistic version of the Larmor's formula [16]\n\n:\n\ndE rad dt = A m 2 d k dt 2 , ( 1\n\n)\n\nwh... | |
arxiv:0704.0841 | 0704.0841 | 1 | 812e649607eb79733b627fc9fb8acdb62895caa1b6926ea8413812375ad6497f | Difermion condensates in vacuum in 2-4D four-fermion interaction models | Theoretical analysis of interplay between the condensates $<\bar{q}q>$ and
$<qq>$ in vacuum is generally made by relativistic effective potentials in the
mean field approximation in 2D, 3D and 4D models with two flavor and $N_c$
color massless fermions. It is found that in ground states of these models,
interplay betwe... | [
"Bang-Rong Zhou (Graduate School of Chinese Academy of Sciences)"
] | [
"hep-th"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | In any four fermion (denoted by q) interaction models, the couplings of (qq) 2 -form can always coexist with the ones of (qq) 2 -form via the Fierz transformations. Hence, even in vacuum, there could be interplay between the condensates qq and qq . Theoretical analysis of this problem is generally made by relativistic ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In any four fermion (denoted by q) interaction models, the couplings of (qq) 2 -form can always coexist with the ones of (qq) 2 -form via the Fierz transformations. Hence, even in vacuum, there could be interplay between the condensates... | ||
arxiv:0704.0849 | 0704.0849 | 1 | 15479003afd208910be80d6ea72df6e6ba25203ffadf1114a463f1c8445b35d4 | LRS Bianchi Type-V Viscous Fluid Universe With a Time Dependent
Cosmological Term $\Lambda$ | An LRS Bianchi type-V cosmological models representing a viscous fluid
distribution with a time dependent cosmological term $\Lambda$ is investigated.
To get a determinate solution, the viscosity coefficient of bulk viscous fluid
is assumed to be a power function of mass density. It turns out that the
cosmological term... | [
"Anirudh Pradhan",
"J. P. Shahi and C. V. Singh"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-06 | 2026-02-26 | An LRS Bianchi type-V cosmological models representing a viscous fluid distribution with a time dependent cosmological term Λ is investigated. To get a determinate solution, the viscosity coefficient of bulk viscous fluid is assumed to be a power function of mass density. It turns out that the cosmological term Λ(t) is... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "An LRS Bianchi type-V cosmological models representing a viscous fluid distribution with a time dependent cosmological term Λ is investigated. To get a determinate solution, the viscosity coefficient of bulk viscous fluid is assumed to ... | ||
arxiv:0704.0857 | 0704.0857 | 1 | a578be21736c068809d16e0f7007e7fd15735dcdc461142f3857c56eb49e1cf5 | Extrasolar scale change in Newton's Law from 5D `plain' R^2-gravity | Galactic rotation curves and lack of direct observations of Dark Matter may
indicate that General Relativity is not valid (on galactic scale) and should be
replaced with another theory. There is the only variant of Absolute Parallelism
which solutions are free of arising singularities, if D=5 (there is no room for
chan... | [
"I.L. Zhogin"
] | [
"astro-ph",
"gr-qc",
"hep-ph"
] | astro-ph | [] | 2007-04-06 | 2026-02-26 | Being a 'close relative' of General Relativity (GR), Absolute Parallelism (AP) has many interesting features: larger symmetry group of equations; field irreducibility with respect to this group; vast list of compatible second order equations (discovered by Einstein and Mayer [1] ) not restricted to Lagrangian ones.
Th... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Being a 'close relative' of General Relativity (GR), Absolute Parallelism (AP) has many interesting features: larger symmetry group of equations; field irreducibility with respect to this group; vast list of compatible second orde... | ||
arxiv:0704.0893 | 0704.0893 | 1 | 10.1103/PhysRevLett.99.160401 | 332b37aad7c0889901efb6457624a119f0378599c0dbc037fb8f823d2cd27209 | Topological phase for spin-orbit transformations on a laser beam | We investigate the topological phase associated with the double connectedness
of the SO(3) representation in terms of maximally entangled states. An
experimental demonstration is provided in the context of polarization and
spatial mode transformations of a laser beam carrying orbital angular momentum.
The topological p... | [
"C.E.R. Souza",
"J.A.O. Huguenin",
"P. Milman",
"A.Z. Khoury"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-06 | 2026-02-26 | We investigate the topological phase associated with the double connectedness of the SO(3) representation in terms of maximally entangled states. An experimental demonstration is provided in the context of polarization and spatial mode transformations of a laser beam carrying orbital angular momentum. The topological p... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We investigate the topological phase associated with the double connectedness of the SO(3) representation in terms of maximally entangled states. An experimental demonstration is provided in the context of polarization and spatial mode ... | |
arxiv:0704.0898 | 0704.0898 | 1 | 6a243a5737258ade4867729fa8e6919dcf452bda726e7bdf544cf286a274cabd | Higher spin algebras as higher symmetries | The exhaustive study of the rigid symmetries of arbitrary free field theories
is motivated, along several lines, as a preliminary step in the completion of
the higher-spin interaction problem in full generality. Some results for the
simplest example (a scalar field) are reviewed and commented along these lines. | [
"Xavier Bekaert"
] | [
"hep-th"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | The exhaustive study of the rigid symmetries of arbitrary free field theories is motivated, along several lines, as a preliminary step in the completion of the higher-spin interaction problem in full generality. Some results for the simplest example (a scalar field) are reviewed and commented along these lines.
Wherea... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The exhaustive study of the rigid symmetries of arbitrary free field theories is motivated, along several lines, as a preliminary step in the completion of the higher-spin interaction problem in full generality. Some results for the sim... | ||
arxiv:0704.0921 | 0704.0921 | 1 | 10.1142/S0217751X07038426 | f3ff4e17bde31c292eade1a3a65006e9dd509ff773b9c130829cba5792877d86 | Z^0 \to 2\gamma and the Twisted Coproduct of the Poincar\'{e} Group | Yang's theorem forbids the process $Z^0 \to 2\gamma$ in any Poincar\'{e}
invariant theory if photons are bosons and their two-particle states transform
under the Poincar\'{e} group in the standard way (under the standard coproduct
of the Poincar\'{e} group). This is an important result as it does not depend
on the assu... | [
"A. P. Balachandran",
"S. G. Jo"
] | [
"hep-th",
"gr-qc",
"hep-ph",
"math-ph",
"math.MP",
"math.QA"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | Yang's theorem forbids the process Z 0 → 2γ in any Poincaré invariant theory if photons are bosons and their two-particle states transform under the Poincaré group in the standard way (under the standard coproduct of the Poincaré group). This is an important result as it does not depend on the assumptions of quantum fi... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "Yang's theorem forbids the process Z 0 → 2γ in any Poincaré invariant theory if photons are bosons and their two-particle states transform under the Poincaré group in the standard way (under the standard coproduct of the Poincaré group)... | |
arxiv:0704.0928 | 0704.0928 | 1 | 10.1103/PhysRevD.76.126005 | 8784aca8a80493bc1c2b78ecd3f8114b734783dd5a1f30341ad8a0bc2bfa9c33 | Cosmology from String Theory | We explore the cosmological content of Salam-Sezgin six dimensional
supergravity, and find a solution to the field equations in qualitative
agreement with observation of distant supernovae, primordial nucleosynthesis
abundances, and recent measurements of the cosmic microwave background. The
carrier of the acceleration... | [
"Luis Anchordoqui",
"Haim Goldberg",
"Satoshi Nawata",
"Carlos Nunez"
] | [
"hep-ph",
"astro-ph",
"hep-th"
] | hep-ph | [] | 2007-04-06 | 2026-02-26 | The mechanism involved in generating a very small cosmological constant that satisfies 't Hooft naturalness is one of the most pressing questions in contemporary physics. Recent observations of distant Type Ia supernovae [1] strongly indicate that the universe is expanding in an accelerating phase, with an effective de... | [
{
"section_type": "OTHER",
"section_title": "I. GENERAL IDEA",
"text": "The mechanism involved in generating a very small cosmological constant that satisfies 't Hooft naturalness is one of the most pressing questions in contemporary physics. Recent observations of distant Type Ia supernovae [1] strongl... | |
arxiv:0704.0929 | 0704.0929 | 1 | 10.1140/epjc/s10052-009-1117-9 | e3c3b2a8fc3d36404b2decfd967f93bae2d6021ca6b5c0016b395ab121f1a75c | Noncommutative Electromagnetism As A Large N Gauge Theory | We map noncommutative (NC) U(1) gauge theory on R^d_C X R^{2n}_{NC} to U(N ->
\infty) Yang-Mills theory on R^d_C, where R^d_C is a d-dimensional commutative
spacetime while R^{2n}_{NC} is a 2n-dimensional NC space. The resulting U(N)
Yang-Mills theory on R^d_C is equivalent to that obtained by the dimensional
reduction... | [
"Hyun Seok Yang"
] | [
"hep-th",
"gr-qc",
"hep-ph"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | We map noncommutative (NC) U(1) gauge theory on C is equivalent to that obtained by the dimensional reduction of (d + 2n)-dimensional U(N) Yang-Mills theory onto IR d C . We show that the gauge-Higgs system (A µ , Φ a ) in the U(N → ∞) Yang-Mills theory on IR d C leads to an emergent geometry in the (d + 2n)-dimensiona... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We map noncommutative (NC) U(1) gauge theory on C is equivalent to that obtained by the dimensional reduction of (d + 2n)-dimensional U(N) Yang-Mills theory onto IR d C . We show that the gauge-Higgs system (A µ , Φ a ) in the U(N → ∞) ... | |
arxiv:0704.0930 | 0704.0930 | 1 | 10.1088/1126-6708/2007/07/050 | 4bc974d4316e5831c556f29fe34524f347a8135f7a5389013c90b6998d4c51cf | Marginal Solutions for the Superstring | We construct a class of analytic solutions of WZW-type open superstring field
theory describing marginal deformations of a reference D-brane background. The
deformations we consider are generated by on-shell vertex operators with
vanishing operator products. The superstring solution exhibits an intriguing
duality with ... | [
"Theodore Erler"
] | [
"hep-th"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | We construct a class of analytic solutions of WZW-type open superstring field theory describing marginal deformations of a reference D-brane background. The deformations we consider are generated by on-shell vertex operators with vanishing operator products. The superstring solution exhibits an intriguing duality with ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We construct a class of analytic solutions of WZW-type open superstring field theory describing marginal deformations of a reference D-brane background. The deformations we consider are generated by on-shell vertex operators with vanish... | |
arxiv:0704.0936 | 0704.0936 | 1 | 10.1088/1126-6708/2007/09/084 | 945d76b2177c785af7654bfc3c67b14c11426abda6c346cf62c87e22ec91f66a | Analytic solutions for marginal deformations in open superstring field
theory | We extend the calculable analytic approach to marginal deformations recently
developed in open bosonic string field theory to open superstring field theory
formulated by Berkovits. We construct analytic solutions to all orders in the
deformation parameter when operator products made of the marginal operator and
the ass... | [
"Yuji Okawa (DESY)"
] | [
"hep-th"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | We extend the calculable analytic approach to marginal deformations recently developed in open bosonic string field theory to open superstring field theory formulated by Berkovits. We construct analytic solutions to all orders in the deformation parameter when operator products made of the marginal operator and the ass... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We extend the calculable analytic approach to marginal deformations recently developed in open bosonic string field theory to open superstring field theory formulated by Berkovits. We construct analytic solutions to all orders in the de... | |
arxiv:0704.0940 | 0704.0940 | 1 | 10.1103/PhysRevD.75.101702 | 542db58b77f2f282bfd108324e481b9bacc2c08b0e88806cc2c172c66e151aa0 | Glueball Masses in (2+1)-Dimensional Anisotropic Weakly-Coupled
Yang-Mills Theory | The confinement problem has been solved in the anisotropic (2+1)-dimensional
SU(N) Yang-Mills theory at weak coupling. In this paper, we find the low-lying
spectrum for N=2. The lightest excitations are pairs of fundamental particles
of the (1+1)-dimensional SU(2)XSU(2) principal chiral sigma model bound in a
linear po... | [
"Peter Orland"
] | [
"hep-th",
"cond-mat.str-el",
"math-ph",
"math.MP"
] | hep-th | [] | 2007-04-06 | 2026-02-26 | The confinement problem has been solved in the anisotropic (2+1)-dimensional SU(N) Yang-Mills theory at weak coupling. In this paper, we find the low-lying spectrum for N = 2. The lightest excitations are pairs of fundamental particles of the (1 + 1)dimensional SU(2) × SU(2) principal chiral nonlinear sigma model bound... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The confinement problem has been solved in the anisotropic (2+1)-dimensional SU(N) Yang-Mills theory at weak coupling. In this paper, we find the low-lying spectrum for N = 2. The lightest excitations are pairs of fundamental particles ... | |
arxiv:0704.0943 | 0704.0943 | 1 | 10.1088/0264-9381/24/22/002 10.1088/0264-9381/25/3/039801 | 0a88e0d172c6c371eee3eecd22f35d371577a72efdef61fa0ed9e044970c7e87 | Search for gravitational-wave bursts in LIGO data from the fourth
science run | The fourth science run of the LIGO and GEO 600 gravitational-wave detectors,
carried out in early 2005, collected data with significantly lower noise than
previous science runs. We report on a search for short-duration
gravitational-wave bursts with arbitrary waveform in the 64-1600 Hz frequency
range appearing in all ... | [
"LIGO Scientific Collaboration"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-06 | 2026-02-26 | Search for gravitational-wave bursts in LIGO data 3 Submitted to Classical and Quantum Gravity 1. Introduction Large interferometers are now being used to search for gravitational waves with sufficient sensitivity to be able to detect signals from distant astrophysical sources. At present, the three detectors of the La... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "Search for gravitational-wave bursts in LIGO data 3 Submitted to Classical and Quantum Gravity 1. Introduction Large interferometers are now being used to search for gravitational waves with sufficient sensitivity to be able to det... | |
arxiv:0704.0951 | 0704.0951 | 1 | 10.1103/PhysRevB.76.174425 | e5371da64587926088819d6f027201b5eead4a64c317055b0aa061f8e3231ef3 | Correlation amplitude and entanglement entropy in random spin chains | Using strong-disorder renormalization group, numerical exact diagonalization,
and quantum Monte Carlo methods, we revisit the random antiferromagnetic XXZ
spin-1/2 chain focusing on the long-length and ground-state behavior of the
average time-independent spin-spin correlation function C(l)=\upsilon
l^{-\eta}. In addit... | [
"J. A. Hoyos",
"Andr\\'e P. Vieira",
"Nicolas Laflorencie",
"E. Miranda"
] | [
"cond-mat.dis-nn",
"cond-mat.str-el",
"quant-ph"
] | cond-mat.dis-nn | [] | 2007-04-08 | 2026-02-26 | Random low-dimensional quantum spin systems have been intensively investigated recently. The interplay between disorder, quantum fluctuations, and correlations generates lowtemperature phase diagrams with exotic phases. 1 In this context, one of the most investigated systems is the random antiferromagnetic (AF) quantum... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Random low-dimensional quantum spin systems have been intensively investigated recently. The interplay between disorder, quantum fluctuations, and correlations generates lowtemperature phase diagrams with exotic phases. 1 In th... | |
arxiv:0704.0955 | 0704.0955 | 1 | 10.1088/0264-9381/24/20/009 | cbae61325c024009ffe766991ef9f26411e0d88936868b2b75ebcc3c1129f423 | Charges from Attractors | We describe how to recover the quantum numbers of extremal black holes from
their near horizon geometries. This is achieved by constructing the
gravitational Noether-Wald charges which can be used for non-extremal black
holes as well. These charges are shown to be equivalent to the U(1) charges of
appropriately dimensi... | [
"Nemani V. Suryanarayana and Matthias C. Wapler"
] | [
"hep-th",
"gr-qc"
] | hep-th | [] | 2007-04-07 | 2026-02-26 | We describe how to recover the quantum numbers of extremal black holes from their near horizon geometries. This is achieved by constructing the gravitational Noether-Wald charges which can be used for non-extremal black holes as well. These charges are shown to be equivalent to the U(1) charges of appropriately dimensi... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We describe how to recover the quantum numbers of extremal black holes from their near horizon geometries. This is achieved by constructing the gravitational Noether-Wald charges which can be used for non-extremal black holes as well. T... | |
arxiv:0704.0960 | 0704.0960 | 1 | 7f7204caf001394d170d7838ed60363e714c9cf1faf2586c0ade2e8b64084288 | Generating Squeezed States of Nanomechanical Resonator | We propose a scheme for generating squeezed states in solid state circuits
consisting of a nanomechanical resonator (NMR), a superconducting Cooper-pair
box (CPB) and a superconducting transmission line resonator (STLR). The
nonlinear interaction between the NMR and the STLR can be implemented by
setting the external b... | [
"Wen Yi Huo and Gui Lu Long"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-07 | 2026-02-26 | Mechanical harmonic oscillator plays an important role in the historical development of quantum mechanics. The harmonic oscillator problem was one of the few completely solvable problems when one began to learn quantum mechanics. Due to the macroscopic nature, the experiments of mechanical harmonic oscillators didn't a... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Mechanical harmonic oscillator plays an important role in the historical development of quantum mechanics. The harmonic oscillator problem was one of the few completely solvable problems when one began to learn quantum mechanic... | ||
arxiv:0704.0964 | 0704.0964 | 1 | 10.1103/PhysRevA.76.052319 | 85910f0c7fe6fb44f00caa186f7ccefc4f26ecef54b84f80a66435e8f137d10d | Upper bounds on entangling rates of bipartite Hamiltonians | We discuss upper bounds on the rate at which unitary evolution governed by a
non-local Hamiltonian can generate entanglement in a bipartite system. Given a
bipartite Hamiltonian H coupling two finite dimensional particles A and B, the
entangling rate is shown to be upper bounded by c*log(d)*norm(H), where d is
the smal... | [
"Sergey Bravyi"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-09 | 2026-02-26 | Consider two remote parties Alice and Bob controlling finite-dimensional quantum systems A and B. Suppose A and B interact with each other according to a time-independent Hamiltonian H. Unless H is a sum of local Hamiltonians, a unitary evolution e iHt is a non-local operation capable of creating entanglement between A... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "Consider two remote parties Alice and Bob controlling finite-dimensional quantum systems A and B. Suppose A and B interact with each other according to a time-independent Hamiltonian H. Unless H is a sum of local Hamiltonians, a u... | |
arxiv:0704.0965 | 0704.0965 | 1 | 426bce66540e7ec6c7bc9fcbf071a7b01b95b1447169df929a73e850dd5a73c3 | Separability Criterion for Multipartite Pure States | In this paper, we give out some effective criterions which can be used to
judge the separability of multipartite pure states. We obtain the relationship
between separability and Schmidt decomposable of multipartite pure states in
Theorem1. The first criterion derived from Theorem2 dose not need the Schmidt
decompositio... | [
"Zongwen Yu and Su Hu"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-07 | 2026-02-26 | In this letter, we give out some effective criterions which can be used to judge the separability of multipartite pure states. We obtain the relationship between separability and Schmidt decomposable of multipartite pure states in Theorem 1. The first criterion derived from Theorem 2 dose not need the Schmidt decomposi... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In this letter, we give out some effective criterions which can be used to judge the separability of multipartite pure states. We obtain the relationship between separability and Schmidt decomposable of multipartite pure states in Theor... | ||
arxiv:0704.0969 | 0704.0969 | 1 | 1610b706fa3b0b87e572910ed41a5e6fe43aa6dca2f22e21a965b82bed95d29a | The Necessary and Sufficient Conditions of Separability for Bipartite
Pure States in Infinite Dimensional Hilbert Spaces | In this paper, we present the necessary and sufficient conditions of
separability for bipartite pure states in infinite dimensional Hilbert spaces.
Let $M$ be the matrix of the amplitudes of $\ket\psi$, we prove $M$ is a
compact operator. We also prove $\ket\psi$ is separable if and only if $M$ is a
bounded linear oper... | [
"Su Hu and Zongwen Yu"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-07 | 2026-02-26 | In this letter, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let M be the matrix of the amplitudes of |ψ , we prove M is a compact operator. We also prove |ψ is separable if and only if M is a bounded linear operator with rank 1, th... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "In this letter, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let M be the matrix of the amplitudes of |ψ , we prove M is a compact operator. We also... | ||
arxiv:0704.0984 | 0704.0984 | 1 | 10.1080/09500340701515120 | 36e5c4e93bf61f4c1adf53fdccc23d3e58c419591f5aa1b26ba85499ca8280c0 | Transfer of a Polaritonic Qubit through a Coupled Cavity Array | We demonstrate a scheme for quantum communication between the ends of an
array of coupled cavities. Each cavity is doped with a single two level system
(atoms or quantum dots) and the detuning of the atomic level spacing and
photonic frequency is appropriately tuned to achieve photon blockade in the
array. We show that... | [
"Sougato Bose",
"Dimitris G. Angelakis",
"Daniel Burgarth"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-09 | 2026-02-26 | We demonstrate a scheme for quantum communication between the ends of an array of coupled cavities. Each cavity is doped with a single two level system (atoms or quantum dots) and the detuning of the atomic level spacing and photonic frequency is appropriately tuned to achieve photon blockade in the array. We show that... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We demonstrate a scheme for quantum communication between the ends of an array of coupled cavities. Each cavity is doped with a single two level system (atoms or quantum dots) and the detuning of the atomic level spacing and photonic fr... | |
arxiv:0704.0986 | 0704.0986 | 1 | 10.1088/0264-9381/24/10/017 | 2fc1e0647e75aa3d45ec64cbb2839bb96882ad612cf83ef86fb70a681256811a | On reference frames in spacetime and gravitational energy in freely
falling frames | We consider the interpretation of tetrad fields as reference frames in
spacetime. Reference frames may be characterized by an antisymmetric
acceleration tensor, whose components are identified as the inertial
accelerations of the frame (the translational acceleration and the frequency of
rotation of the frame). This te... | [
"J. W. Maluf",
"F. F. Faria and S. C. Ulhoa"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-07 | 2026-02-26 | We consider the interpretation of tetrad fields as reference frames in spacetime. Reference frames may be characterized by an antisymmetric acceleration tensor, whose components are identified as the inertial accelerations of the frame (the translational acceleration and the frequency of rotation of the frame). This te... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We consider the interpretation of tetrad fields as reference frames in spacetime. Reference frames may be characterized by an antisymmetric acceleration tensor, whose components are identified as the inertial accelerations of the frame ... | |
arxiv:0704.0992 | 0704.0992 | 1 | 10.1103/PhysRevD.76.084013 | 7f187ca7f2eb956f4549d8af85a9974f751ced89043bf16286a9da57eb85932e | Compatibility of radial, Lorenz and harmonic gauges | We observe that the radial gauge can be consistently imposed \emph{together}
with the Lorenz gauge in Maxwell theory, and with the harmonic traceless gauge
in linearized general relativity. This simple observation has relevance for
some recent developments in quantum gravity where the radial gauge is
implicitly utilize... | [
"Elena Magliaro",
"Claudio Perini",
"Carlo Rovelli"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-09 | 2026-02-26 | We observe that the radial gauge can be consistently imposed together with the Lorenz gauge in Maxwell theory, and with the harmonic traceless gauge in linearized general relativity. This simple observation has relevance for some recent developments in quantum gravity where the radial gauge is implicitly utilized.
The... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We observe that the radial gauge can be consistently imposed together with the Lorenz gauge in Maxwell theory, and with the harmonic traceless gauge in linearized general relativity. This simple observation has relevance for some recent... | |
arxiv:0704.0993 | 0704.0993 | 1 | 10.1088/0264-9381/24/13/F01 | 9b6f2611d9ca2d09b89906f2375971781c0c56ff71ea190da30dcdbecc16fe5e | Late-time tails of a Yang-Mills field on Minkowski and Schwarzschild
backgrounds | We study the late-time behavior of spherically symmetric solutions of the
Yang-Mills equations on Minkowski and Schwarzschild backgrounds. Using
nonlinear perturbation theory we show in both cases that solutions having
smooth compactly supported initial data posses tails which decay as $t^{-4}$ at
timelike infinity. Mo... | [
"Piotr Bizo\\'n",
"Tadeusz Chmaj",
"Andrzej Rostworowski"
] | [
"gr-qc",
"hep-th",
"math.AP"
] | gr-qc | [] | 2007-04-07 | 2026-02-26 | In a classical paper [1] Eardley and Moncrief proved that solutions of the Yang-Mills equations on the 3+1 Minkowski spacetime starting from smooth initial data remain smooth for all future times. A different proof allowing for initial data with only finite energy was given later by Klainerman and Machedon [2] . Once g... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In a classical paper [1] Eardley and Moncrief proved that solutions of the Yang-Mills equations on the 3+1 Minkowski spacetime starting from smooth initial data remain smooth for all future times. A different proof allowing for... | |
arxiv:0704.0995 | 0704.0995 | 1 | 10.1007/s10773-008-9715-0 | dd6b448311a375118bbdc39b4d3a39071315dfe386b975d7b86a79460ffad2d3 | Composite Structure and Causality | We study the question of whether a composite structure of elementary
particles, with a length scale $1/\Lambda$, can leave observable effects of
non-locality and causality violation at higher energies (but $\lesssim
\Lambda$). We formulate a model-independent approach based on
Bogoliubov-Shirkov formulation of causalit... | [
"Satish D. Joglekar"
] | [
"hep-ph",
"hep-th"
] | hep-ph | [] | 2007-04-07 | 2026-02-26 | We study the question of whether a composite structure of elementary particles, with a length scale 1/Λ, can leave observable effects of non-locality and causality violation at higher energies (but Λ). We formulate a model-independent approach based on Bogoliubov-Shirkov formulation of causality. We analyze the relatio... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We study the question of whether a composite structure of elementary particles, with a length scale 1/Λ, can leave observable effects of non-locality and causality violation at higher energies (but Λ). We formulate a model-independent a... | |
arxiv:0704.0996 | 0704.0996 | 1 | 10.1088/1126-6708/2007/06/006 | f69ac31255e6b4a923349569454920c1545eacc7b27247ae394de93d9602bdb5 | Brane World Black Rings | Five dimensional neutral rotating black rings are described from a
Randall-Sundrum brane world perspective in the bulk black string framework. To
this end we consider a rotating black string extension of a five dimensional
black ring into the bulk of a six dimensional Randall-Sundrum brane world with
a single four bran... | [
"Anurag Sahay",
"Gautam Sengupta"
] | [
"hep-th"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | 1 1 Introduction.
Higher dimensional spacetimes are now an essential aspect of effective field theories arising from fundamental theories of quantum gravity. The general assumption implicit in such constructions was that the extra spatial dimensions are compactified to ultrashort length scales. Hence quantum gravity e... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "1 1 Introduction.\n\nHigher dimensional spacetimes are now an essential aspect of effective field theories arising from fundamental theories of quantum gravity. The general assumption implicit in such constructions was that the ext... | |
arxiv:0704.1012 | 0704.1012 | 1 | 10.1007/s00220-008-0486-5 | 838f454f53f02d408684cc79be323717bb7fcb9c8f0dc4e365ebb3ce4064e731 | Unstable and Stable Galaxy Models | To determine the stability and instability of a given steady galaxy
configuration is one of the fundamental problems in the Vlasov theory for
galaxy dynamics. In this article, we study the stability of isotropic spherical
symmetric galaxy models $f_{0}(E)$, for which the distribution function $f_{0}$
depends on the par... | [
"Yan Guo and Zhiwu Lin"
] | [
"astro-ph",
"gr-qc",
"math.AP"
] | astro-ph | [] | 2007-04-09 | 2026-02-26 | To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric galaxy models f 0 (E), for which the distribution function f 0 depends on the particle e... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "To determine the stability and instability of a given steady galaxy configuration is one of the fundamental problems in the Vlasov theory for galaxy dynamics. In this article, we study the stability of isotropic spherical symmetric gala... | |
arxiv:0704.1015 | 0704.1015 | 1 | 10.1088/1126-6708/2007/09/009 | eee001158438da16f89e504870899999a2451030055488cc13427cd82c5f0edd | An Alternative Topological Field Theory of Generalized Complex Geometry | We propose a new topological field theory on generalized complex geometry in
two dimension using AKSZ formulation. Zucchini's model is $A$ model in the case
that the generalized complex structuredepends on only a symplectic structure.
Our new model is $B$ model in the case that the generalized complex structure
depends... | [
"Noriaki Ikeda and Tatsuya Tokunaga"
] | [
"hep-th",
"math.DG"
] | hep-th | [] | 2007-04-08 | 2026-02-26 | We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is A model in the case that the generalized complex structure depends on only a symplectic structure. Our new model is B model in the case that the generalized complex structure depends on... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is A model in the case that the generalized complex structure depends on only a symplectic structure. Ou... | |
arxiv:0704.1026 | 0704.1026 | 1 | 10.1088/1475-7516/2007/06/012 | 1c2ebbe6c3e5ef5ffd8ad09c4914f26dadf1ef436d45e1cd1551a12f113eb113 | The Measure for the Multiverse and the Probability for Inflation | We investigate the measure problem in the framework of inflationary
cosmology. The measure of the history space is constructed and applied to
inflation models. Using this measure, it is shown that the probability for the
generalized single field slow roll inflation to last for $N$ e-folds is
suppressed by a factor $\ex... | [
"Miao Li",
"Yi Wang"
] | [
"hep-th",
"astro-ph",
"gr-qc"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | We investigate the measure problem in the framework of inflationary cosmology. The measure of the history space is constructed and applied to inflation models. Using this measure, it is shown that the probability for the generalized single field slow roll inflation to last for N e-folds is suppressed by a factor exp(-3... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "We investigate the measure problem in the framework of inflationary cosmology. The measure of the history space is constructed and applied to inflation models. Using this measure, it is shown that the probability for the generalized sin... | |
arxiv:0704.1035 | 0704.1035 | 1 | 10.1016/j.physletb.2007.04.008 | 72da920b72c04bb6caa70aff13305a9c1a3f19dc4f9f2734855072b760cf35ef | Anisotropic brane gravity with a confining potential | We consider an anisotropic brane world with Bianchi type I and V geometries
where the mechanism of confining the matter on the brane is through the use of
a confining potential. The resulting equations on the anisotropic brane are
modified by an extra term that may be interpreted as the x-matter, providing a
possible p... | [
"M. Heydari-Fard and H. R. Sepangi"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-08 | 2026-02-26 | The type Ia supernovae (SNe Ia) [1] observations provide the first evidence for the accelerating expansion of the present universe. These results, when combined with the observations on the anisotropy spectrum of cosmic microwave background (CMB) [2] and the results on the power spectrum of large scale structure (LSS) ... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The type Ia supernovae (SNe Ia) [1] observations provide the first evidence for the accelerating expansion of the present universe. These results, when combined with the observations on the anisotropy spectrum of cosmic microwave ... | |
arxiv:0704.1037 | 0704.1037 | 1 | 10.1103/PhysRevA.76.033409 | 504ddf332cb486b8d2f68811d4e02a8b3e44da6734d539f94af4feb8366e651a | Controlled collisions of a single atom and ion guided by movable
trapping potentials | We consider a system composed of a trapped atom and a trapped ion. The ion
charge induces in the atom an electric dipole moment, which attracts it with an
r^{-4} dependence at large distances. In the regime considered here, the
characteristic range of the atom-ion interaction is comparable or larger than
the characteri... | [
"Z. Idziaszek",
"T. Calarco",
"P. Zoller"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-08 | 2026-02-26 | Techniques developed in atomic physics during the last two decades allow the preparation of single atoms and ions in the laboratory. Single neutral atoms can be stored and laser cooled in far off-resonance laser traps (FORT) [1, 2] , and arrays of atoms can be prepared in an optical lattice via a Mott insulator phase, ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Techniques developed in atomic physics during the last two decades allow the preparation of single atoms and ions in the laboratory. Single neutral atoms can be stored and laser cooled in far off-resonance laser traps (FORT) [1... | |
arxiv:0704.1038 | 0704.1038 | 1 | 10.1016/j.nuclphysb.2007.07.026 | 766e7a4219705289485bd116dc5634201dacdf8ed85b8060d4f8ba56d6c22ef5 | Counting BPS operators in N=4 SYM | The free field partition function for a generic U(N) gauge theory, where the
fundamental fields transform in the adjoint representation, is analysed in
terms of symmetric polynomial techniques. It is shown by these means how this
is related to the cycle polynomial for the symmetric group and how the large N
result may ... | [
"F.A. Dolan"
] | [
"hep-th"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may ... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques. It is shown by these means how this is rel... | |
arxiv:0704.1040 | 0704.1040 | 1 | d006cfb78bb11b0580d2546a119242f89886581647bc1485d955750afa5f30f4 | Recent Results on Thermal Casimir Force between Dielectrics and Related
Problems | We review recent results obtained in the physics of the thermal Casimir force
acting between two dielectrics, dielectric and metal, and between metal and
semiconductor. The detailed derivation for the low-temperature behavior of the
Casimir free energy, pressure and entropy in the configuration of two real
dielectric p... | [
"B. Geyer",
"G.L. Klimchitskaya",
"V.M. Mostepanenko"
] | [
"quant-ph",
"cond-mat.stat-mech",
"hep-th"
] | quant-ph | [] | 2007-04-08 | 2026-02-26 | The Casimir effect 1 is the force and also the specific polarization of the vacuum arising in restricted quantization volumes and originating from the zero-point oscillations of quantized fields. This force acts between two closely spaced macrobodies, between an atom or a molecule and macrobody or between two atoms or ... | [
{
"section_type": "BACKGROUND",
"section_title": "INTRODUCTION",
"text": "The Casimir effect 1 is the force and also the specific polarization of the vacuum arising in restricted quantization volumes and originating from the zero-point oscillations of quantized fields. This force acts between two closel... | ||
arxiv:0704.1042 | 0704.1042 | 1 | 9f451056a440032725ead173c36ec150fe683509dc3e530d946da326af93a5e0 | Entangling and disentangling capacities of nonlocal maps | Entangling and disentangling capacities are the key manifestation of the
nonlocal content of a quantum operation. A lot of effort has been put recently
into investigating (dis)entangling capacities of unitary operations, but very
little is known about capacities of non-unitary operations. Here we investigate
(dis)entan... | [
"Berry Groisman"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-08 | 2026-02-26 | Entanglement content is one of the fundamental ways to characterize nonlocal quantum resources (nonlocal states and operations). For pure bipartite states the ultimate measure of entanglement, the von Neumann entropy of entanglement, had been recently discovered [1] . A universal measure of entanglement for mixed state... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Entanglement content is one of the fundamental ways to characterize nonlocal quantum resources (nonlocal states and operations). For pure bipartite states the ultimate measure of entanglement, the von Neumann entropy of entangl... | ||
arxiv:0704.1058 | 0704.1058 | 1 | 10.1088/0264-9381/24/11/R01 | 418b473054216467def87b4ea566874d20859da100de8f3f0f5e2c48ba15e9c9 | Dark energy, gravitation and supernovae | The discovery of the acceleration of the rate of expansion of the Universe
fosters new explorations of the behavior of gravitation theories in the
cosmological context. Either the GR framework is valid but a cosmic component
with a negative equation of state is dominating the energy--matter contents or
the Universe is ... | [
"Pilar Ruiz-Lapuente"
] | [
"gr-qc",
"astro-ph",
"hep-ph"
] | gr-qc | [] | 2007-04-09 | 2026-02-26 | 2 Contents 1. Introduction 2. Gravity and extra dimensions 3. The dark energy field and scalar-tensor gravity 4. The cosmological constant? 5. The d L test from high-z SNe Ia 6. Testing the adequacy of the FRW metric 7. Next decade experiments 8. Complementarity 9. Conclusion 3 1. Introduction
The discovery of the acc... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "2 Contents 1. Introduction 2. Gravity and extra dimensions 3. The dark energy field and scalar-tensor gravity 4. The cosmological constant? 5. The d L test from high-z SNe Ia 6. Testing the adequacy of the FRW metric 7. Next decade... | |
arxiv:0704.1060 | 0704.1060 | 1 | 10.1016/j.physletb.2007.05.028 | 5c344ad79790e17129e4b9cd0d618b59413fdc31e40f21d7653b7ad44038512e | Deformation of Dijkgraaf-Vafa Relation via Spontaneously Broken N=2
Supersymmetry | It is known that the fermionic shift symmetry of the N=1, U(N) gauge model
with a superpotential of an adjoint chiral superfield is replaced by the second
(spontaneously broken) supersymmetry in the N=2, U(N) gauge model with a
prepotential and Fayet-Iliopoulos parameters. Based on a diagrammatic analysis,
we demonstra... | [
"Hiroshi Itoyama",
"Kazunobu Maruyoshi"
] | [
"hep-th"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | For more than two decades, effective superpotential has been a central object in the nonperturbative study of N = 1 supersymmetric theories. This object is protected from perturbative corrections in the conventional sense [1] , and yet receives important nonperturbative corrections (see for example [2, 3] ). In recent ... | [
{
"section_type": "BACKGROUND",
"section_title": "I. Introduction",
"text": "For more than two decades, effective superpotential has been a central object in the nonperturbative study of N = 1 supersymmetric theories. This object is protected from perturbative corrections in the conventional sense [1] ,... | |
arxiv:0704.1069 | 0704.1069 | 1 | 10.1088/1367-2630/9/7/224 | fee712c164d1c7eafe992b56c90448f22ae70b6370c0e412bad7fab2dd2807b2 | Optical Zeno Gate: Bounds for Fault Tolerant Operation | In principle the Zeno effect controlled-sign gate of Franson et al's (PRA 70,
062302, 2004) is a deterministic two-qubit optical gate. However, when
realistic values of photon loss are considered its fidelity is significantly
reduced. Here we consider the use of measurement based quantum processing
techniques to enhanc... | [
"Patrick M. Leung and Timothy C. Ralph"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-09 | 2026-02-26 | Photons are a natural choice for making qubits because the quantum information encoded can have a long decoherence time and is easy to manipulate and measure. Also, photonic qubits are the only type of qubits that are feasible for long distance quantum communication. Quantum information processing requires universal tw... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "Photons are a natural choice for making qubits because the quantum information encoded can have a long decoherence time and is easy to manipulate and measure. Also, photonic qubits are the only type of qubits that are feasible ... | |
arxiv:0704.1073 | 0704.1073 | 1 | 10.1103/PhysRevD.76.064020 | 148c809db5725248492c69a93d04e4a35fb7ed85a244bc3f944f0d0c6d4b58f3 | Shell-mediated tunnelling between (anti-)de Sitter vacua | We give an extensive study of the tunnelling between arbitrary (anti-)de
Sitter spacetimes separated by an infinitesimally thin relativistic shell in
arbitrary spacetime dimensions. In particular, we find analytically an exact
expression for the tunnelling amplitude.
The detailed spacetime structures that can arise a... | [
"Stefano Ansoldi and Lorenzo Sindoni"
] | [
"gr-qc",
"hep-th"
] | gr-qc | [] | 2007-04-09 | 2026-02-26 | The study of vacuum decay initiated more than 30 years ago with the work of Callan and Coleman [1, 2] ; in the following years the interest in the subject increased and the possible interplay of true vacuum bubbles with gravitation was also studied [3, 4] , together with bubbles collisions and their importance in the e... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "The study of vacuum decay initiated more than 30 years ago with the work of Callan and Coleman [1, 2] ; in the following years the interest in the subject increased and the possible interplay of true vacuum bubbles with gravita... | |
arxiv:0704.1079 | 0704.1079 | 1 | 10.1088/1126-6708/2007/06/011 | 07092b48df99e03a14bc7cb9c6dea5dc9b655146e584c77ad1809f8431641638 | Instanton Induced Neutrino Majorana Masses in CFT Orientifolds with
MSSM-like spectra | Recently it has been shown that string instanton effects may give rise to
neutrino Majorana masses in certain classes of semi-realistic string
compactifications. In this paper we make a systematic search for supersymmetric
MSSM-like Type II Gepner orientifold constructions admitting boundary states
associated with inst... | [
"L.E. Ibanez",
"A.N. Schellekens",
"A. M. Uranga"
] | [
"hep-th",
"hep-ph"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | String Theory, as the leading candidate for a unified theory of Particle Physics and Gravity, should be able to describe all observed particle phenomena. One the most valuable experimental pieces of information obtained in the last decade concerns neutrino masses. Indeed the evidence from solar, atmospheric, reactor an... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "String Theory, as the leading candidate for a unified theory of Particle Physics and Gravity, should be able to describe all observed particle phenomena. One the most valuable experimental pieces of information obtained in the las... | |
arxiv:0704.1084 | 0704.1084 | 1 | 10.1088/1751-8113/40/25/S21 | a7acfb95a2edadd76e4c81acce00f735d0f24f7fbf0f6b4f279d6055f94998ba | Vacuum Energy and Renormalization on the Edge | The vacuum dependence on boundary conditions in quantum field theories is
analysed from a very general viewpoint. From this perspective the
renormalization prescriptions not only imply the renormalization of the
couplings of the theory in the bulk but also the appearance of a flow in the
space of boundary conditions. F... | [
"M. Asorey",
"D. Garcia-Alvarez and J. M. Munoz-Castaneda"
] | [
"hep-th"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | The emergence of the dark energy as one of the basic ingredients of the current standard cosmological scenario, and the absence of an even vague understanding of its possible origin, opens a window to the analysis of all possible mechanisms that generate background energy (see e.g. [1] for a review of recent proposals)... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The emergence of the dark energy as one of the basic ingredients of the current standard cosmological scenario, and the absence of an even vague understanding of its possible origin, opens a window to the analysis of all possible ... | |
arxiv:0704.1085 | 0704.1085 | 1 | 10.1016/j.physletb.2007.06.056 | 8bd7acc2c1e6a1228e15ea29ec0203416a3e3b53fc075678499862e4db6f75e6 | Gauge Mediation in String Theory | We show that a large class of phenomenologically viable models for gauge
mediation of supersymmetry breaking based on meta-stable vacua can be realized
in local Calabi-Yau compactifications of string theory. | [
"Teruhiko Kawano",
"Hirosi Ooguri",
"Yutaka Ookouchi"
] | [
"hep-th",
"hep-ph"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | The use of meta-stable vacua in supersymmetric model building has attracted much attention lately, especially after the discovery [1] that generic supersymmetric field theories in four dimensions such as the supersymmetric QCD with massive flavors have metastable vacua with broken supersymmetry. In [2] , realistic mode... | [
{
"section_type": "BACKGROUND",
"section_title": "Introduction",
"text": "The use of meta-stable vacua in supersymmetric model building has attracted much attention lately, especially after the discovery [1] that generic supersymmetric field theories in four dimensions such as the supersymmetric QCD wit... | |
arxiv:0704.1086 | 0704.1086 | 1 | 10.1103/PhysRevD.75.124002 | dcf0d7701a7b1489c0a47a9a712c25a75b13e093f228ff30b9d6ef70503ef573 | Higher harmonics increase LISA's mass reach for supermassive black holes | Current expectations on the signal to noise ratios and masses of supermassive
black holes which the Laser Interferometer Space Antenna (LISA) can observe are
based on using in matched filtering only the dominant harmonic of the inspiral
waveform at twice the orbital frequency. Other harmonics will affect the
signal-to-... | [
"K.G. Arun",
"Bala R. Iyer",
"B. S. Sathyaprakash",
"Siddhartha Sinha"
] | [
"gr-qc",
"astro-ph"
] | gr-qc | [] | 2007-04-09 | 2026-02-26 | There is strong observational evidence for the existence of supermassive black holes (SMBHs) with masses in the range of 10 6 M ⊙ -10 9 M ⊙ (see e.g. Ref. [1] and references therein) in most galactic nuclei [2] . Therefore, mergers of galaxies, as evidenced by high-redshift surveys, should give rise to binaries contain... | [
{
"section_type": "OTHER",
"section_title": "A. Supermassive black hole binaries and LISA",
"text": "There is strong observational evidence for the existence of supermassive black holes (SMBHs) with masses in the range of 10 6 M ⊙ -10 9 M ⊙ (see e.g. Ref. [1] and references therein) in most galactic nuc... | |
arxiv:0704.1087 | 0704.1087 | 1 | 57ae57ad802e32ae02271f91071064ea51c6198a0a0e354c32d1360cad7aef45 | EPR, Bell, Schrodinger's cat, and the Monty Hall Paradox | The purpose of this manuscript is to provide a short pedagogical explanation
why "quantum collapse" is not a metaphysical event, by pointing out the analogy
with a "classical collapse" which is associated with the Monty Hall Paradox. | [
"Doron Cohen"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-09 | 2026-02-26 | This manuscript constitutes a short self-contained version of some selected sections taken from "lecture notes in Quantum Mechanics" [quant-ph/0605180](∼250p). In particular section 47.4 regarding the notion of collapse has attracted some attention. In this section I suggest to use the "Monty Hall Paradox" as a pedagog... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "This manuscript constitutes a short self-contained version of some selected sections taken from \"lecture notes in Quantum Mechanics\" [quant-ph/0605180](∼250p). In particular section 47.4 regarding the notion of collapse has attra... | ||
arxiv:0704.1090 | 0704.1090 | 1 | 10.1088/1126-6708/2007/10/071 | 2623c387ee17c729321fff1ce4314279a5e01c6ab9c899752e543243c7e44ed3 | Bouncing Universe with Quintom Matter | The bouncing universe provides a possible solution to the Big Bang
singularity problem. In this paper we study the bouncing solution in the
universe dominated by the Quintom matter with an equation of state (EoS)
crossing the cosmological constant boundary. We will show explicitly the
analytical and numerical bouncing ... | [
"Yi-Fu Cai",
"Taotao Qiu",
"Yun-Song Piao",
"Mingzhe Li",
"Xinmin Zhang"
] | [
"gr-qc",
"astro-ph",
"hep-ph",
"hep-th"
] | gr-qc | [] | 2007-04-09 | 2026-02-26 | The bouncing universe provides a possible solution to the Big Bang singularity problem. In this paper we study the bouncing solution in the universe dominated by the Quintom matter with an equation of state (EoS) crossing the cosmological constant boundary. We will show explicitly the analytical and numerical bouncing ... | [
{
"section_type": "OTHER",
"section_title": "Untitled Section",
"text": "The bouncing universe provides a possible solution to the Big Bang singularity problem. In this paper we study the bouncing solution in the universe dominated by the Quintom matter with an equation of state (EoS) crossing the cosmo... | |
arxiv:0704.1092 | 0704.1092 | 1 | 10.1063/1.2746128 | b7695653f6647e8e619ed5ff510d9714b7a8657922337075d92123288adfc2c0 | Simplifying additivity problems using direct sum constructions | We study the additivity problems for the classical capacity of quantum
channels, the minimal output entropy and its convex closure. We show for each
of them that additivity for arbitrary pairs of channels holds iff it holds for
arbitrary equal pairs, which in turn can be taken to be unital. In a similar
sense, weak add... | [
"Motohisa Fukuda",
"Michael M. Wolf"
] | [
"quant-ph"
] | quant-ph | [] | 2007-04-09 | 2026-02-26 | A central question in classical and quantum information theory is, how much information can be transmitted through a given noisy channel. For classical channels the maximal asymptotically achievable rate-the capacity-was derived in the seminal work of Shannon [1] . For quantum channels, however, the matter is complicat... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "A central question in classical and quantum information theory is, how much information can be transmitted through a given noisy channel. For classical channels the maximal asymptotically achievable rate-the capacity-was derive... | |
arxiv:0704.1096 | 0704.1096 | 1 | 10.1103/PhysRevA.76.053603 | 4fce9d3a936e06bb27860d451986f511743ca489f3216c4b6002564bb2e9de92 | Formation of a sonic horizon in isotropically expanding Bose-Einstein
condensates | We propose a simple experiment to create a sonic horizon in isotropically
trapped cold atoms within currently available experimental techniques.
Numerical simulation of the Gross-Pitaevskii equation shows that the sonic
horizon should appear by making the condensate expand. The expansion is
triggered by changing the in... | [
"Yasunari Kurita",
"Takao Morinari"
] | [
"cond-mat.other",
"gr-qc",
"hep-th"
] | cond-mat.other | [] | 2007-04-09 | 2026-02-26 | For the exploration of cosmology and gravitational physics, it is necessary to have a deep understanding of quantum filed theory in curved spacetime: It is widely believed that everything except for the spacetime itself should originate from quantum fluctuations in the early Universe. Quantum effects on curved spacetim... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "For the exploration of cosmology and gravitational physics, it is necessary to have a deep understanding of quantum filed theory in curved spacetime: It is widely believed that everything except for the spacetime itself should ... | |
arxiv:0704.1104 | 0704.1104 | 1 | 10.1103/PhysRevD.76.047701 | 125b45f03427d978496db2f46aa6048dda50a3d8b80ada4086123ac0491cebaa | On the induction of the four-dimensional Lorentz-breaking non-Abelian
Chern-Simons action | A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is
generated as a one-loop perturbative correction via an appropriate
Lorentz-breaking coupling of the non-Abelian gauge field to the spinor field.
This term is shown to be regularization dependent but nevertheless it can be
found unambiguously in... | [
"M. Gomes",
"J. R. Nascimento",
"E. Passos",
"A. Yu. Petrov",
"A. J. da Silva"
] | [
"hep-th"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is generated as a one-loop perturbative correction via an appropriate Lorentz-breaking coupling of the non-Abelian gauge field to a spinor field. This term is shown to be regularization dependent but nevertheless it can be found unambiguously in d... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "A four-dimensional Lorentz-breaking non-Abelian Chern-Simons like action is generated as a one-loop perturbative correction via an appropriate Lorentz-breaking coupling of the non-Abelian gauge field to a spinor field. This term is show... | |
arxiv:0704.1109 | 0704.1109 | 1 | 10.1007/s10773-009-0204-x | 0150dbf53a2658b7aa805f3b8407fa9fb78488a4b5c888bd44f7c9a11f6c9fcb | Spinning Strings, Black Holes and Stable Closed Timelike Geodesics | The existence and stability under linear perturbation of closed timelike
curves in the spacetime associated to Schwarzschild black hole pierced by a
spinning string are studied. Due to the superposition of the black hole, we
find that the spinning string spacetime is deformed in such a way to allow the
existence of clo... | [
"Valeria M. Rosa and Patricio S. letelier"
] | [
"gr-qc"
] | gr-qc | [] | 2007-04-09 | 2026-02-26 | The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the superposition of the black hole, we find that the spinning string spacetime is deformed in such a way to allow the existence of clo... | [
{
"section_type": "ABSTRACT",
"section_title": "Abstract",
"text": "The existence and stability under linear perturbation of closed timelike curves in the spacetime associated to Schwarzschild black hole pierced by a spinning string are studied. Due to the superposition of the black hole, we find that t... | |
arxiv:0704.1111 | 0704.1111 | 1 | 10.1103/PhysRevD.76.054506 | c1e7189384c09b0d8453508ecbcfe6308ebf06fe3577cf0b8d3327ef3e292ae1 | QCD in One Dimension at Nonzero Chemical Potential | Using an integration formula recently derived by Conrey, Farmer and
Zirnbauer, we calculate the expectation value of the phase factor of the
fermion determinant for the staggered lattice QCD action in one dimension. We
show that the chemical potential can be absorbed into the quark masses; the
theory is in the same chi... | [
"L. Ravagli and J.J.M. Verbaarschot"
] | [
"hep-th",
"hep-lat"
] | hep-th | [] | 2007-04-09 | 2026-02-26 | In spite of significant recent progress, QCD at nonzero chemical potential remains a notoriously hard problem [1, 2, 3] . In particular, first principle nonperturbative results at low temperature are absent because the phase of the fermion determinant invalidates probabilistic methods to evaluate the partition function... | [
{
"section_type": "BACKGROUND",
"section_title": "I. INTRODUCTION",
"text": "In spite of significant recent progress, QCD at nonzero chemical potential remains a notoriously hard problem [1, 2, 3] . In particular, first principle nonperturbative results at low temperature are absent because the phase of... |