We compute one of the bulk viscosity coefficients of cold CFL quark matter in the temperature regime where the contribution of mesons, quarks and gluons to transport phenomena is Boltzmann suppressed. In that regime dissipation occurs due to collisions of superfluid phonons, the Goldstone modes associated to the spontaneous breaking of baryon symmetry. We first review the hydrodynamics of relativistic superfluids, and remind that there are at least three bulk viscosity coefficients in these systems. We then compute the bulk viscosity coefficient associated to the normal fluid component of the superfluid. In our analysis we use Son's effective field theory for the superfluid phonon, amended to include scale breaking effects proportional to the square of the strange quark mass m_s. We compute the bulk viscosity at leading order in the scale breaking parameter, and find that it is dominated by collinear splitting and joining processes. The resulting transport coefficient is zeta=0.011 m_s^4/T, growing at low temperature T until the phonon fluid description stops making sense. Our results are relevant to study the rotational properties of a compact star formed by CFL quark matter.
At asymptotic high density and low temperature quarks form Cooper pairs in a color-flavor locked (CFL) configuration. The diquark condensates break spontaneously the baryon symmetry, and this fact makes the CFL phase also superfluid. At low temperatures the transport properties are dominated by the contribution of the superfluid phonon, the Goldstone boson associated to baryon symmetry breaking, in full analogy to what happens in superfluid He4. We discuss how to derive transport properties in the ultracold regime making use of an analogue model of gravity. We also review how this model can be…
We find that for large enough strange quark mass, $m_s^2/4 μΔ>2/3(1-0.024),$ kaon condensed CFL phase of asymptotically dense 3 flavor quark matter is unstable with respect to spontaneous generation of currents of Nambu Goldstone bosons due to spontaneous breaking of baryon number symmetry and hypercharge symmetry in CFL$K^0$ ground state. The total baryon and hypercharge currents still vanish in the ground state. We find that CFL$K^0$ and the new state are separated by the first order phase transition.
We compute the mean free path and shear viscosity in the color-flavor locked (CFL) phase of dense quark matter at low temperature T, when the contributions of mesons, quarks and gluons to the transport coefficients are Boltzmann suppressed. CFL quark matter displays superfluid properties, and transport phenomena in such cold regime are dominated by phonon-phonon scattering. We study superfluid phonons within thermal field theory and compute the mean free path associated to their most relevant collision processes. Small-angle processes turn out to be more efficient in slowing transport phenomen…
The Ginzburg-Landau equations for magnetic and gluomagnetic gauge fields of the semi-superfluid strings in the color superconducting core of the neutron star with diquark condensate in the color-flavor-locking (CFL) phase are derived. The simultaneous coupling of the CFL diquark condensate to the magnetic and gluomagnetic gauge fields is taken into account. The asymptotic values of these string energies are defined from quantization conditions. It is shown that a lattice of semi-superfluid strings with minimal quantum of circulation emerges in the superconducting quark core during rotation of …
The bulk viscosities of two color-superconducting phases, the color-flavor locked (CFL) phase and the 2SC phase, are computed and compared to the result for unpaired quark matter. In the case of the CFL phase, processes involving kaons and the superfluid mode give the largest contribution to the bulk viscosity since all fermionic modes are gapped. In the case of the 2SC phase, ungapped fermionic modes are present and the process u+d <-> u+s provides the dominant contribution. In both cases, the bulk viscosity can become larger than that of the unpaired phase for sufficiently large temperatures…
Transport coefficients of dense quark matter are needed to study properties of compact stars. They can tell us about the cooling, vibrational and rotational properties of the star. We report below a computation of the shear viscosity in the CFL phase at low temperature T. CFL quark matter behaves as a superfluid, and at low T its transport properties are dominated by the collisions of superfluid phonons.
Color-flavor locked (CFL) quark matter at high densities is a color superconductor, which spontaneously breaks baryon number and chiral symmetry. Its low-energy thermodynamic and transport properties are therefore dominated by the H (superfluid) boson, and the octet of pseudoscalar pseudo-Goldstone bosons of which the neutral kaon is the lightest. We study the CFL-K^0 phase, in which the stress induced by the strange quark mass causes the kaons to condense, and there is an additional ultra-light "K^0" Goldstone boson arising from the spontaneous breaking of isospin. We compute the bulk viscosi…
We show that topological superfluid strings/vortices with flux tubes exist in the color-flavor locked (CFL) phase of color superconductors. Using a Ginzburg-Landau free energy we find the configurations of these strings. These strings can form during the transition from the normal phase to the CFL phase at the core of very dense stars. We discuss an interesting scenario for a network of strings and its evolution at the core of dense stars.
We calculate the bulk viscosity of color-superconducting quark matter in the color-flavor-locked (CFL) phase. We assume that the lightest bosons are the superfluid mode H and the kaons K0 and K+, and that there is no kaon condensate. We calculate the rate of strangeness-equilibrating processes that convert kaons into superfluid modes, and the resultant bulk viscosity. We find that for oscillations with a timescale of milliseconds, at temperatures T << 1 MeV, the CFL bulk viscosity is much less than that of unpaired quark matter, but at higher temperatures the bulk viscosity of CFL matter can b…
The radiative viscosity of superfluid $npe$ matter is studied, and it is found that to the lowest order of $δμ/T$ the ratio of radiative viscosity to bulk viscosity is the same as that of the normal matter.
Superfluid vortices in the color-flavor-locked (CFL) phase of dense quark matter are known to be energetically disfavored relative to well-separated triplets of "semi-superfluid" color flux tubes. However, the short-range interaction (metastable versus unstable) has not been established. In this paper we perform numerical calculations using the effective theory of the condensate field, mapping the regions in the parameter space of coupling constants where the vortices are metastable versus unstable. For the case of zero gauge coupling we analytically identify a candidate for the unstable mode,…
The r-mode instability in rotating compact stars is used to constrain the phase of matter at high density. The color-flavor-locked phase with kaon condensation (CFL-K0) and without (CFL) is considered in the temperature range 10^8K < T <10^{11} K. While the bulk viscosity in either phase is only effective at damping the r-mode at temperatures T > 10^{11} K, the shear viscosity in the CFL-K0 phase is the only effective damping agent all the way down to temperatures T > 10^8 K characteristic of cooling neutron stars. However, it cannot keep the star from becoming unstable to gravitational wave e…
We present some results about dissipative processes in fermionic superfluids that are relevant for compact stars. At sufficiently low temperatures the transport properties of a superfluid are dominated by phonons. We report the values of the bulk viscosity, shear viscosity and thermal conductivity of phonons in quark matter at extremely high density and low temperature. Then, we present a new dissipative mechanism that can operate in compact stars and that is named "rocket term". The effect of this dissipative mechanism on superfluid r-mode oscillations is sketched.
At very high densities, QCD is in the color-flavor-locked phase, which is a color-superconducting phase. The diquark condensates break chiral symmetry in the same way as it is broken in vacuum QCD and gives rise to an octet of pseudo-Goldstone bosons and a superfluid mode. The lightest of these are the charged and neutral kaons. For energies below the superconducting gap, the kaons are described by an $O(2)\times O(2)$-symmetric effective scalar field theory with chemical potentials. We use this effective theory to study Bose-condensation of kaons and their properties as functions of the tempe…
We propose a wide universality class of gapless superfluids, and analyze a limit that might be realized in quark matter at intermediate densities. In the breached pairing color superconducting phase heavy $s$-quarks, with a small Fermi surface, pair with light $u$ or $d$ quarks. The groundstate has a superfluid and a normal Fermi component simultaneously. We expect a second order phase transition, as a function of increasing density, from the breached pairing phase to the conventional color-flavor locked (CFL) phase.
We compute the shear viscosity of a superfluid atomic Fermi gas in the unitarity limit. The unitarity limit is characterized by a divergent scattering length between the atoms, and it has been argued that this will result in a very small viscosity. We show that in the low temperature T limit the shear viscosity scales as xi^5/T^5, where the universal parameter 'xi' relates the chemical potential and the Fermi energy, mu=xi E_F. Combined with the high temperature expansions of the viscosity our results suggest that the viscosity has a minimum near the critical temperature T_c. A naive extrapola…
We analyze the contributions of Nambu-Goldstone modes to the shear viscosity of a superfluid atomic Fermi gas close to unitarity. We show that the low temperature experimental values of the shear viscosity to entropy ratio, $η/s$, are reproduced considering an effective shear viscosity which takes into account the collisions among Nambu-Goldstone bosons and the finite size of the optical trap. We predict that, for $T \lesssim 0.1 T_F$, $η/s$ should correlate with the size of the trap and linearly decrease with decreasing temperature. For $T \gtrsim 0.1 T_F$, we find agreement with the experime…
We evaluate the three bulk viscosity coefficients $ζ_1, ζ_2$ and $ζ_3$ in the color-flavor locked (CFL) superfluid phase due to phonons and kaons, which are the lightest modes in that system. We first show that the computation is rather analogous to the computation of the same coefficients in superfluid $^4$He, as due to phonons and rotons. For astrophysical applications, we also find the value of the viscosities when there is a periodic disturbance, and the viscosities also depend on the frequency of the disturbance. In a temperature regime that might be of astrophysical relevance, we find th…
We apply Ginzburg-Landau theory to determine BCS pairing in a strongly-coupled uniform superfluid of three-flavor massless quarks in flavor equilibrium. We elucidate the phase diagram near the critical temperature in the space of the parameters characterizing the thermodynamic potential terms of fourth order in the pairing gap. Within the color and flavor antisymmetric channel with zero total angular momentum, the phase diagram contains an isoscalar (IS) color-antitriplet phase and a color-flavor-locked (CFL) phase, reached by a second order transition from the normal state, as well as states …
It was argued by Brigante et.al that the lower bound on the ratio of the shear viscosity to the entropy density in strongly coupled plasma is translated into microcausality violation in the dual gravitational description. Since transport properties of the system characterize its infrared dynamics, while the causality of the theory is determined by its ultraviolet behavior, the viscosity bound/microcausality link should not be applicable to theories that undergo low temperature phase transitions. We present an explicit model of AdS/CFT correspondence that confirms this fact.
We calculate the correlation functions needed to describe the linear response of superfluid matter, and go on to calculate the differential cross section for neutral-current neutrino scattering in superfluid neutron matter and in color-flavor locked quark matter (CFL). We report the first calculation of scattering rates that includes neutrino interactions with both pair-breaking excitations and low-lying collective excitations (Goldstone modes). Our results apply both above and below the critical temperature, allowing use in simulations of neutrino transport in supernovae and neutron stars.
We compute the contribution of phonons to the shear viscosity $η$ in superfluid neutron stars, assuming neutron pairing in a $^1S_0$ channel. We use a Boltzmann equation amended by a collision term that takes into account the binary collisions of phonons. We use effective field theory techniques to extract the phonon scattering rates, written as a function of the equation of state (EoS) of the system. Our formulation is rather general, and can be used to extract the shear viscosity due to binary collisions of phonons for other superfluids, such as the cold Fermi gas in the unitarity limit. We …
The phonon contribution to the shear viscosity $η$ in superfluid neutron stars is calculated by assuming neutron pairing in a $^1S_0$ channel. The shear viscosity is obtained by means of variational methods for the solution of the Boltzmann equation amended by a collision term which takes into account the binary collisions of phonons. Effective field theory techniques are used to extract the phonon scattering rates in terms of the equation of state (EoS) of the system. We find that $η\propto 1/T^5$, the proportionality factor depending on the EoS of the system. Our results indicate that the ph…
We investigate the color-singlet order parameters and the quark description of the Nambu-Goldstone (NG) bosons in the color-flavor locked (CFL) phase. We put emphasis on the NG boson (phason) called ``H'' associated with the $\mathrm{U_B(1)}$ symmetry breaking. We qualitatively argue the nature of H as the second sound in the hydrodynamic regime. We articulate, based on a diquark picture, how the structural change of the condensates and the associated NG bosons occurs continuously from hadronic to CFL quark matter if the quark-hadron continuity is realized. We sharpen the qualitative differenc…
We calculate the three bulk viscosity coefficients as arising from the collisions among phonons in superfluid neutron stars. We use effective field theory techniques to extract the allowed phonon collisional processes, written as a function of the equation of state of the system. The solution of the dynamical evolution of the phonon number density allows us to calculate the bulk viscosity coefficients as function of the phonon collisional rate and the phonon dispersion law, which depends on the neutron pairing gap. Our method of computation is rather general, and could be used for different su…
We present a detailed analysis of the contribution of small-angle Nambu-Goldstone boson (phonon) collisions to the shear viscosity, $η$, in a superfluid atomic Fermi gas close to the unitarity limit. We show that the experimental values of the shear viscosity coefficient to entropy ratio, $η/s$, obtained at the lowest reached temperature can be reproduced assuming that phonons give the leading contribution to $η$. The phonon contribution is evaluated considering $1 \leftrightarrow 2$ processes and taking into account the finite size of the experimental system. In particular, for very low tempe…
Some results about dissipative processes in superfluids are presented. We focus on fermionic superfluidity and restrict our analysis to the contribution of phonons to bulk viscosity, shear viscosity and thermal conductivity. At sufficiently low temperatures phonons give the dominant contribution to the transport coefficients if all the other low energy excitation of the system are gapped. We first consider a system of cold fermionic atoms close to the unitarity limit. Then we turn to the superfluid phase of quark matter that may be realized at high baryonic density.
We investigate collective excitations coupled with baryon density in a system of massless three-flavor quarks in the collisionless regime. By using the Nambu-Jona-Lasinio (NJL) model in the mean-field approximation, we derive the spectra both for the normal and color-flavor locked (CFL) superfluid phases at zero temperature. In the normal phase, we obtain zero sound as a low-lying collective mode in the particle-hole (vector) channel. In the CFL phase, when the excitation energy, $ω$, is smaller than the threshold given by twice the pairing gap $Δ$ ($ω<2Δ$), a phonon corresponding to fluctuati…
Quark-hadron continuity was proposed as crossover between hadronic matter and quark matter without a phase transition, based on the matching of the symmetry and excitations in both phases. In the limit of a light strange-quark mass, it connects hyperon matter and the color-flavor-locked (CFL) phase exhibiting color superconductivity. Recently, it was proposed that this conjecture could be generalized in the presence of superfluid vortices penetrating both phases (arXiv:1803.05115 [hep-ph]), and it was suggested that one hadronic superfluid vortex in hyperon matter could be connected to one non…
At very high densities, as for example in the core of a neutron star, matter may appear in the color-flavor locked (CFL) phase, which is a superfluid. This phase features topologically stable vortex solutions, which arise in a spinning superfluid as localized configurations carrying quanta of angular momentum. Despite the topological stability of these vortices they are not the lowest energy state of the system at neutron star densities and decay into triplets of semi-superfluid fluxtubes. In these proceedings we report on the progress of our numerical study in the Ginzburg-Landau approach, wh…
We study how vortices in dense superfluid hadronic matter can connect to vortices in superfluid quark matter, as in rotating neutron stars, focusing on the extent to which quark-hadron continuity can be maintained. As we show, a singly quantized vortex in three-flavor symmetric hadronic matter can connect smoothly to a singly quantized non-Abelian vortex in three-flavor symmetric quark matter in the color-flavor locked (CFL) phase, without the necessity for boojums appearing at the transition.
Matter at high density and low temperature is expected to be a color superconductor, which is a degenerate Fermi gas of quarks with a condensate of Cooper pairs near the Fermi surface that induces color Meissner effects. At the highest densities, where the QCD coupling is weak, rigorous calculations are possible, and the ground state is a particularly symmetric state, the color-flavor locked (CFL) phase. The CFL phase is a superfluid, an electromagnetic insulator, and breaks chiral symmetry. The effective theory of the low-energy excitations in the CFL phase is known and can be used, even at m…
We study the shear and bulk viscosity coefficients as well as the thermal conductivity as arising from the collisions among phonons in superfluid neutron stars. We use effective field theory techniques to extract the allowed phonon collisional processes, written as a function of the equation of state and the gap of the system. The shear viscosity due to phonon scattering is compared to calculations of that coming from electron collisions. We also comment on the possible consequences for r-mode damping in superfluid neutron stars. Moreover, we find that phonon collisions give the leading contri…
We study the behavior of Goldstone bosons in color-flavor-locked (CFL) quark matter at nonzero temperature. Chiral symmetry breaking in this phase of cold and dense matter gives rise to pseudo-Goldstone bosons, the lightest of these being the charged and neutral kaons K^+ and K^0. At zero temperature, Bose-Einstein condensation of the kaons occurs. Since all fermions are gapped, this kaon condensed CFL phase can, for energies below the fermionic energy gap, be described by an effective theory for the bosonic modes. We use this effective theory to investigate the melting of the condensate: we d…
We compute the coefficients of bulk viscosity for a non-relativistic superfluid corresponding to a fermionic system close to the unitarity limit. We consider the low temperature regime assuming that the transport properties of the system are dominated by phonons. To compute the coefficients of bulk viscosity we use kinetic theory in the relaxation time approximation and the low energy effective field theory of the corresponding system. We show that the three independent bulk viscosity coefficients, $ζ_1, ζ_2, ζ_3$, associated with irreversible flows vanish for phonons with a linear dispersion …
We consider the phonon contribution to the bulk viscosities $ζ_1, ζ_2$ and $ζ_3$ of a cold relativistic superfluid. We assume the low temperature $T$ regime and that the transport properties of the system are dominated by the phonons. We use kinetic theory in the relaxation time approximation and the low energy effective field theory of the corresponding system. The parametric dependence of the bulk viscosity coefficients is fixed once the equation of state is specified, and the phonon dispersion law to cubic order in momentum is known. We first present a general discussion, valid for any supe…
We compute the thermal conductivity of color-flavor locked (CFL) quark matter. At temperatures below the scale set by the gap in the quark spectrum, transport properties are determined by collective modes. In this work we focus on the contribution from the lightest modes, the superfluid phonon and the massive neutral kaon. The calculation is done in the framework of kinetic theory, using variational solutions of the linearized Boltzmann equation. We find that the thermal conductivity due to phonons is κ^P =1.04 10^{26} mu_{500}^8 Δ_{50}^{-6} erg/(cm s K), where μ_{500} is the chemical potentia…
We analyze the effect of restricted geometries on the contribution of Nambu-Goldstone bosons (phonons) to the shear viscosity, $η$, of a superfluid. For illustrative purpose we examine a simplified system consisting of a circular boundary of radius $R$, confining a two-dimensional rarefied gas of phonons. Considering the Maxwell-type conditions, we show that phonons that are not in equilibrium with the boundary and that are not specularly reflected exert a shear stress on the boundary. In this case it is possible to define an effective (ballistic) shear viscosity coefficient $η\propto ρ_{\rm p…
