The superfluid phases of quark matter: Ginzburg-Landau theory and color neutrality

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 reached by a first order transition. We complement the general Ginzburg-Landau approach by deriving the high-density asymptotic form of the Ginzburg-Landau free energy from the weak-coupling gap equation. The dynamically-screened, long-range color magnetic interactions are taken into account in solving the gap equation. We find that in the limit of weak coupling, the IS phase is less favorable near the transition temperature than the CFL phase. In view of the fact that deconfined quark matter must be color charge neutral, we incorporate the constraint of overall color neutrality into the Ginzburg-Landau theory and the gap equation. This constraint yields a disparity in the chemical potential between colors and reduces the size of the gap, in the presence of the anisotropy of the order parameters in color space. In comparison with the case in which there are no chemical potential differences between colors and hence the superfluid generally has nonzero net color charge, we find that while the constraint of color neutrality has only negligible effects on the gap in the weak coupling regime, it appreciably destabilizes the IS phase in the strong coupling regime without affecting the CFL phase.