Secondary pairing in gapless colour-superconducting quark matter

We calculate secondary pairing in a model of a colour superconductor with a quadratic gapless dispersion relation for the quasiquarks of the primary pairing. Our model mimics the physics of the sector of blue-up and red-strange quarks in gapless colour-flavour-locked quark matter. The secondary pairing opens up a gap Δ_s in the quark spectrum, and we confirm Hong's prediction that in typical secondary channels Δ_s ∝ G ²_s for coupling strength G_s. This shows that the large density of states of the quadratically gapless mode greatly enhances the secondary pairing over the standard BCS result Δ ∝ exp(-const/G). In all of the secondary channels that we analysed we find that the secondary gap, even with this enhancement, is from ten to hundreds of times smaller than the primary gap at reasonable values of the secondary coupling, indicating that secondary pairing does not generically resolve the magnetic instability of the gapless phase.