Secondary pairing in gapless color-superconducting quark matter

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We calculate secondary pairing in a model of a color 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 color-flavor-locked quark matter. The secondary pairing opens up a gap Deltaₛ in the quark spectrum, and we confirm Hong's prediction that in typical secondary channels Deltaₛ is proportional to Gₛ² for coupling strength Gₛ. This shows that the large density of states of the quadratically gapless mode greatly enhances the secondary pairing over the standard BCS result, in which Delta is proportional to exp(-const/G). In all of the secondary channels that we analyzed 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.