Bulk viscosities of a cold relativistic superfluid: color-flavor locked quark matter

We consider the phonon contribution to the bulk viscosities $\zeta_1, \zeta_2$ and $\zeta_3$ 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 superfluid, then we focus on the color-flavor locked superfluid because all the parameters needed in the analysis can be computed in the high density limit of QCD, and also because of the possible astrophysical applications. For the three independent bulk viscosity coefficients we find that they scale with the temperature as $\zeta_i \sim 1/T$, and that in the conformal limit only the third coefficient $\zeta_3$ is non-zero.