Quasi-particle Specific Heats for the Crystalline Color Superconducting Phase of QCD

We calculate the specific heats of quasi-particles of two-flavor QCD in its crystalline phases for low temperature. We show that for the different crystalline structures considered here there are gapless modes contributing linearly in temperature to the specific heat. We evaluate also the phonon contributions which are cubic in temperature. These features might be relevant for compact stars with an inner shell in a color superconducting crystalline phase.


Introduction
A number of theoretical studies have recently been devoted to QCD at low temperatures T and high densities. Besides the theoretical interest for the different QCD phases, the possibility of applications to compact stars, where dense quark matter might exist, has driven much of the recent interest.
At high density and small T , quarks at the Fermi surface are expected to condense, giving rise to color superconductivity, see [1,2] and, for reviews, [3]. At the largest densities QCD with three flavors should be in the so called CFL (color-flavor locked) phase. For decreasing density one expects pairing between quarks of non vanishing total momentum, resulting in crystalline gap behaviors [4,5,6,7] (for reviews see [8,9]). Crystalline phases had already been discussed for electric superconductors [10,11], so that one sometimes also refers to the crystalline phases as LOFF phases. A crystalline color superconductor shell might exist inside compact stars, between the external hadronic crust and the internal CFL core.
In the present letter we discuss dispersion laws and specific heats of the quasi-particles present in the LOFF phase. These properties are relevant for the calculation of the thermal conductivity and neutrino emissivity and would affect the cooling of the compact stars.
Thus far crystalline color superconductivity has been studied only in a two-flavor model. Different crystal structures have been computed, with the conclusion, based on a Ginzburg-Landau discussion, that the most favored structure at zero temperature is a cubic structure very close to a face centered cube (FCC). However at non zero temperature the situation is far from being clear (see the discussion in Ref. [8]), therefore we will discuss other crystalline structures as well.
The results that one can easily predict for the specific heat in the LOFF phase with two flavors are a T 3 behavior for the phonons and a µ 2 T behavior for the unpaired quarks. Our subsequent analysis (Sections 3 and 4) confirms this expectation. For the paired quarks one would naively expect an exponential suppression exp(−∆/T ), which is the behavior of the regular 2SC phase. However our analysis shows (see Sect. 3) that this is not the case, and for quark flavors that undergo LOFF pairing the specific heat has a µ 2 T dependence. The reason for this behavior is in the fermi quasi-particle dispersion law (see Sect. 2) that for the LOFF phase is gapless. As a conclusion, in a neutron star, provided a LOFF phase is present, the specific heat and the thermal conductivity should be dominated by fermions, with paired and unpaired quarks on the same footing. A similar dominance is expected also for the LOFF phase with three flavors. In this case there are no unpaired quarks and therefore a a clear distinction should emerge in the thermal properties (specific heat, thermal conductivity) of the homogeneous case (CFL) and the inhomogeneous one (LOFF) because all the quarks in the former case would have the exponential suppression, while in the latter the T dependence is expected to be linear.
The plan of the paper is as follows. In section 2 we discuss the disper-sion laws of the fermionic quasi-particles for different crystal structures of the LOFF phase of high density QCD. We discuss the one-plane wave structure, the strip, and the face centered cube. In section 3 we calculate the contribution of the fermionic quasi-particles to the specific heat for the three mentioned structures. In section 4 we use the effective phonon Lagrangian to calculate the phonon specific heats. Section 5 is devoted to the conclusions.

Fermi quasi-particle dispersion law
In this section we derive the Fermi quasi-particle dispersion law in the twoflavor QCD LOFF phase for a few different crystalline structures. We will work with an inhomogeneous condensate given bŷ where α, β are color indices and i, j = 1, 2 are flavor indices. Notice that the gap term pairs together the color 1 and 2, whereas the fermions with color 3 are unpaired. We work in presence of a difference in the chemical potential of the u and the d quarks. We define As usual the quasi-particle dispersion law is obtained by looking at the zeros of the inverse propagator. For fermionic quasi-particles one has to solve the eigenvalue equation where S −1 is the inverse propagator in the LOFF phase in the Nambu-Gorkov formalism and χ β j are the Green eigenfunctions. Following the notations in [8] we write S −1 as follows: where E is the quasi-particle energy and v is the Fermi velocity, that in QCD with massless quarks satisfies v = |v| = 1. Let us define Performing the unitary transformation one measures the energy of each flavor from its Fermi energy. The resulting equations for F α i and G α i are independent of color and flavor indices, that therefore will be omitted below: We solve these equations for three different crystalline structures, corresponding to different decompositions of (1) in plane waves: 1. One plane wave; 2. Two antipodal plane waves; 3. The face centered cubic (FCC) structure, formed by eight plane waves with momenta pointing to the vertices of a cube.
The first case is discussed for metals in [11] by Fulde and Ferrel (FF); for QCD is analyzed in [4]. The second case has been examined at T = 0 by [10]; we will refer to it as the strip below. The last case is the preferred solution at T = 0 in Ginzburg-Landau analysis of Ref. [7]. The reason why we examine these different structures is in the fact that the preferred structure at T = 0 can be different from that at T = 0, see e.g [6] and the discussion in [8].

One plane wave
For the FF condensate we take the direction of the Cooper pair total momentum 2q along the z−axis. The gap is therefore a complex number. If we take G =Ĝ(r)e iq·r , F =F (r)e −iq·r , we get from Eqs. (7) [ These are the standard Gorkov equations for a uniform superconductor with energy E − qv z . The eigenfunctions are simple plane waveŝ and the quasi-particle spectrum is given by: where ξ = k · v is the residual longitudinal momentum, i.e. the longitudinal momentum measured from the Fermi surface. Because of the transformation (6), quasi-particle energies are computed from the corresponding Fermi energies µ u,d . Eq. (11) is the dispersion law of quasi-particle (E ± ≥ 0) or hole states (E ± < 0). We notice that in this case, as in [12], [13], [14] and [15] , we are in presence of gapless superconductivity. An anisotropic dispersion law was also obtained in [4] by a different procedure (variational method). Their result was obtained for finite µ and reduces to (11) if one considers only the leading order in the asymptotic µ → ∞ limit, which is the limit we consider in the present paper.

Strip and FCC structures
Both the strip and the cubic structure have real ∆(r). The solutions of Eqs. (7) are Bloch functions with u(r) and w(r) periodic functions and k in the first Brillouin zone. They satisfy The corresponding quasi-particles are gapless [10]. In fact, for E = 0 and ξ = 0, the system (13) has two solutions. We have w ± = ±u ± , with u ± solutions of v · ∇u ± (r) = ±∆(r)u ± (r) , and given by Here the integration is over a path joining the origin to the point r.
To find the dispersion law of quasi-particles for small values of ξ one uses degenerate perturbation theory and gets with where V c is the volume of a unit cell of the lattice.
Let us now specialize to the case of the strip, i.e. the crystalline structure formed by two plane waves with wave vectors ±2q. The inhomogeneous gap is given by ∆(r) = ∆ cos 2qz (18) and one gets where I 0 (z) is the modified Bessel function of the zeroth order.
Let us now turn to the FCC structure. Summing the eight plane waves, the condensate can be put in the form ∆(r) = ∆ cos 2qx cos 2qy cos 2qz .
If we define one obtains for the cube where the integration is over the elementary cell of volume (π/q) 3 . Let us notice that whereas for the strip in Eq. (19) one has a closed formula, for the FCC only numerical or approximate expressions can be given.
Since the parameter x = ∆/qv is expected to be small, expanding A (fcc) ± as a power series one obtains In conclusion for both the strip and the FCC the quasi-fermion spectrum is gapless. This result remains valid also for massive quarks, since the effect of the quark mass can be accounted for by reducing the quark velocity from the value v = 1 valid for the massless case [16].

Specific heat of the Fermi quasi-particles
The contribution of the Fermi quasi-particles to the specific heat per unit volume is where, for the two flavor case (and |v| = 1), while n(E, T ) is the Fermi distribution function and the angular integration is over the directions of v.
We should also remind that, in the case of two flavors, the two quarks of color 3 are unpaired and as such each contributes to the specific heat as follows: Let us stress however that quarks that do not participate in the standard spin 0 condensate can nevertheless couple with spin 1, with much smaller gaps. Examples of these models can be found in [17]. Let us now specialize Eq. (24) to the three crystalline structure under scrutiny. We limit the analysis to the small T range.

One plane wave
The dispersion law of quasi-particles is given by (11). Using (24) one gets in the small temperature limit (T ≪ ∆) and for ∆ < q The specific heat depends linearly on temperature because the quasi-particle dispersion law (11) gives rise to gapless modes. There is also a contribution to the specific heat that comes from gapped modes, but this contribution is exponentially suppressed with the temperature.

Strip and FCC structures
From Eq. (24), using (16) that is valid for the strip and the FCC alike, one obtains, for T ≪ ∆: This expression can be evaluated in closed form for the strip, when A ± = A (s) . One gets where 1 F 2 denotes the generalized hypergeometric function [18]. Differently from the analysis of [10], here v is not small and we can take z = ∆/qv → 0 near the second order phase transition. Since for small z one has 1 F 2 (−1/2; 1/2, 1; z 2 ) ≃ 1 − z 2 , it is easily seen that the normal Fermi liquid result (26) is obtained for z = 0. On the other hand, at finite ∆, the specific heat turns out to be smaller. For the face centered cube, A (fcc) ± can be computed by evaluating the power expansion. One gets for this case Using the expansion (23) one can get an approximate formula giving a power series of ∆/qv.

Specific heat of phonons
Besides Fermi quasi-particles we consider also the massless Nambu-Goldstone bosons (NGB). Even though we expect a parametrically smaller O(T 3 ) contribution in this case, the contribution of the NGB might be relevant for future applications, e.g. for the calculation of thermal conductivity. For two flavor QCD in the crystalline phase the only NGB are phonons [19,20].

One or two plane waves
Let us begin with the phonon Lagrangian for the plane wave where φ is the phonon field, ∇ = n · ∇, ∇ ⊥ = ∇ − n∇ and n along the z−axis, i.e. the direction of q. In [19] we found with cos θ q = δµ 2 /q = 0.833. Here δµ 2 is the δµ value corresponding to the second order phase transition from the LOFF phase to the normal one. The dispersion law, relating the phonon quasi-momentum k and energy ω, therefore is Let N be the total number of oscillatory modes. There is an oscillator for any quark pair and the total number of quark pairs is given by the available phase space, i.e. the pairing region. If V is the available volume, then where g = 4 × 2 and we divide by two due to the fact that there are two quarks in the pair. The integration over the pairing region gives A factor 2δ arises from the integration over the longitudinal residual momentum ξ; δ is the ultraviolet cutoff, of the order of µ while ζ is the fraction of the phase space available for pairing in the LOFF phase. It is estimated of the order of ∆/q. The ratio N/V can be expressed in terms of a cutoff frequency by a procedure analogous to the introduction of the Debye frequency for ordinary crystals. We write where f (ω) is given by Substituting in (34) we see that ω D is of the order of µ(∆/q) 1/3 therefore large (excluding the second order phase transition region). More precisely we get the formula These results also hold in the case of the strip, i.e. two antipodal plane waves. As a matter of fact one can prove, following the same procedure as in [19], that the effective Lagrangian for the phonon field is still given by (31).
The specific heat per unit volume at small temperatures (ω D ≫ T ), is given by (k B =h = 1): and therefore This result holds for the two considered structures (one or two plane waves).

Face-centered-cube
For the FCC crystal structure the phonon Lagrangian is given by [21] (41) In [20] we found Eq. (36) is still valid with Here the velocities v 2 r = v 2 r (n) are the eigenvalues of the matrix with n 1 = sin θ cos ϕ, n 2 = sin θ sin ϕ, n 3 = cos θ.
To get the Debye frequency and the specific heat we use the numerical result corresponding to the values (42) of the parameters. For the specific heat we get, similarly to Eq. (40), the result One can note that numerically c (fcc) v > c (s) v .

Conclusions and Outlook
We have considered three different crystalline structures for high density QCD in the LOFF phase: one plane wave, the strip, and the favorite facecentered-cubic strucure. Gapless fermionic quasi-particles are found to exist even in the presence of quark mass terms. These gapless fermions provide for the dominant contributions to the thermal properties for low temperatures, such as of possible interest for compact stars. Typically they contribute to specific heats with terms linear in T which are dominant at low T . The massless phonons contribute instead as T 3 while gapped particles in the homogeneous case (2SC) have an exponential suppression at low T . The previous calculations could be applied to the evaluation of transport properties of compact stars. At the present a reliable model for the LOFF phase in compact stars is still lacking, since the study of the inhomogeneous superconducting phase of QCD with three flavors has not yet been performed.
Nevertheless one can make some conjecture by imagining the existence of the LOFF phase inside the star, but outside the inner core, where one might suppose the existence of quark matter in the CFL phase. In this analysis the thermal properties of the homogeneous and inhomogeneous phases would be different since for the homogeneous case the fermion specific heats would be exponentially suppressed, while for the inhomogeneous case the vanishing with the temperature would be linear. An application of previous results might be the evaluation of the thermal conductivity in the LOFF phase for the existing model of 2-flavor QCD. We plan to come to this subject in the future.