Bulk viscosity in kaon-condensed color-flavor-locked quark matter

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-K⁰ phase, in which the stress induced by the strange quark mass causes the kaons to condense, and there is an additional ultra-light 'K⁰' Goldstone boson arising from the spontaneous breaking of isospin. We compute the bulk viscosity of matter in the CFL-K⁰ phase, which arises from the beta-equilibration processes K⁰ ↔ H + H and K⁰ + H ↔ H. We find that the bulk viscosity varies as T⁷, unlike the CFL phase where it is exponentially Boltzmann suppressed by the kaon's energy gap. However, in the temperature range of relevance for r-mode damping in compact stars, the bulk viscosity in the CFL-K⁰ phase turns out to be even smaller than in the uncondensed CFL phase, which already has a bulk viscosity much smaller than all other known color-superconducting quark phases.