BCS theory of hadronic matter at high densities

The equilibrium between the so-called 2SC and CFL phases of strange quark matter at high densities is investigated in the framework of a simple schematic model of the NJL type. Equal densities are assumed for quarks $u,d$ and $s$. The 2SC phase is here described by a color-flavor symmetric state, in which the quark numbers are independent of the color-flavor combination. In the CFL phase the quark numbers depend on the color-flavor combination, that is, the number of quarks associated with the color-flavor combinations $ur,dg,sb$ is different from the number of quarks associated with the color flavor combinations $ug,ub,dr,db,sr,sg$. We find that the 2SC phase is stable for a chemical potential $\mu$ below $\mu_c=0.505$ GeV, while the CFL phase is stable above, the equilibrium pressure being $P_c=0.003$ GeV$^4$. We have used a 3-momentum regularizing cutoff $\Lambda=0.8$ GeV, which is somewhat larger than is usual in NJL type models. This should be adequate if the relevant chemical potential does not exceed 0.6 GeV.