Sound Attenuation in Quark Matter Due to Pairing Fluctuations

The sound wave in dense quark matter is subject to strong absorption due to diquark field fluctuations above T critical. The result is another facet of Mandelshtam-Leontovich slow relaxation time theory.

The fact that the BCS hierarchy is badly broken in 2SC reflects itself in the value of a dimensionless parameter n 1/3 ξ ∼ k F ξ, where n is the number density and ξ is the characteristic length of pair correlation. In the BCS scenario k F ξ 10 3 v.s. k F ξ 2 in 2S C [1][2][3][4][5]. In BCS pairs are large compared to their separarion, in 2SC they are small compared to their separation (such molecular-like objects are sometimes called Shafroth pairs). The continuous evolution of this parameter as a function of density and of the interaction strength reflects the transition from BCS regime to Bose-Einstein condensation (BEC). The idea of such transformation is due to F. Dyson, the problem was theoretically investigated in [6] for electron pairs and following [6] in [2][3][4] for quarks. Starting from the early 2000th dramatic theoretical and experimental progress has been done in the study of the BCS-BEC crossover in ultracold fermionic atoms [7]. Explaining the BCS-BEC crossover we partly elucidated the problem (ii) from the list above. A remark on terminology is needed. The terms strong-and weak-coupling in superconductivity (and in many-body physics) and in QCD have a different meaning. One can find, of course, some anology. Chiral symmetry breaking in QCD is a kind of a counterpart of the formation of fermion bilinear condensate in BCS. Roughly speaking, weak coupling in superconductivity means that the interaction between particles (electrons) is concentrated within a thin layer of momentum space around the Fermi surface and ω D E F (see (1)). Integration in the vicinity of the Fermi surface is performed (in relativistic case) using the variable ξ defined as where ρ(μ) = p 0 μ 2π 2 , p 0 is the Fermi momentum. The first term in (4) gives rise to Cooper logarithm. The second term takes into account the energy dependence of the density of states near the Fermi surface. Only this term in (4) gives nonzero contribution to the sound absorption [8,9]. In a great number of works the NJL (Nambu and Jona-Lasinio, Vaks and Larkin) model has been used to study color superconductivity. Strong-coupling in this approach is tantamaunt to a large value of the diquark coupling constant. In absence of a fundamental approach and impossibility to perform lattice calculations at nonzero density the NJL model may serve for the orientation purposes with a hope to confront its predictions with future FAIR and NICA data. As a candidate for a strong-coupling model the NJL suffers from a lack of renormalizability and confinement (the latter drawback is proposed to cure adding the Polyakov loop).
On the other hand, the strong-coupling Migdal-Eliashberg theory of superconductivity was developed already more than half-century ago [10,11] and has been successfully applied in many problems including HTSC [12].
Migdal-Eliashberg strong-coupling theory leads to the enhancement of pairing fluctuations and to the broadening of the transition region [13]. Implementation of Migdal theorem [11] to color super-

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conductors will be discussed elsewhere. The fluctuation contribution is characterized by Ginzburg-Levanyuk number Gi which for color superconductor becomes very large This is a huge number compared to the BCS Gi 10 −12 − 10 −14 . An alternative estimate Gi ∼ (k F ξ) −4 ∼ 10 −2 − 10 −3 [4] leads to even larger value.
Precursor pair fluctuations above T c give the dominant contribution to the quark matter transport coefficients. The leading diagram is the Aslamazov-Larkin (AL) one [8], which includes two propagators of the slow pair collective mode singular at T c . Within the ordinary theory of super conductivity this so-called fluctuation propagator (FP) was derived in [8]. For color superconductor it was evaluated using Dyson equation and Matsubara formalism in [14] and from the time-dependent Landau-Ginzburg equation with stohastic Langevin forces in [9]. The FP reads Here ν = ρ(μ) = p 0 μ 2π 2 (see (4)), ε = T − T c T c , D is the diffusion coefficient. At small ω and q the quantity L(q, ω) can be arbitrary large close to T c . The AL diagram for the sound absorption is shown in Fig.1 The two way lines correspond to the FP-s, the solid lines are the quarks Matsubara propagators, the in-and out-vertices are equal to the constants g of phonon-quark interaction. The sound attenuation coefficient is equal to the imaginary part of the retarded polarization operator corresponding to the diagram in Fig.1. In this short presentation we leave out the detailed calculation which may be found in [8] for BCS and in [9] for the quark matter. In the fluctuation region above T c the dominant contribution and the sharp temperature dependence come from the two FP-s. The final result for the imaginary part of the polarization operator reads [9] Im Π = − ω g 2 m 2 2 5 p 4 0 κ 3 (v 2 Here m is the quark mass, p 0 and v 0 are the Fermi momentum and velocity Λ is the ultra-violet cutoff (2), κ 2 = π 8T c D, where D is the diffusion coefficient. As we remarked after Eq.(4), the nonzero contribution to Im Π comes not from crust of the Fermi surface (the first term in (4)), but from the energy dependent density of states given by the second term in (4). Important to note that the evaluation of the AL diagram in [8] and [9] making use of (4) are in complete agreement with the result for this polarization operator obtained in the Eliashberg strong-coupling theory [13]. The physics behind the strong energy dissipation of the sound wave in the precritical region is simple and very general. This is another manifestation of Mandelshtam-Leontovich slow relaxation time theory [15][16][17]. Propagation of the sound wave locally changes the critical temperature and the slow fluctuation pairing can not keep up with this process.