Kaon superfluidity in the early universe

Previously, it was found that pion superfluidity could be realized in the QCD epoch of the early universe, when lepton flavor asymmetry $|l_{\rm e}+l_\mu|$ is large enough to generate a charge chemical potential $|\mu_{\rm Q}|$ larger than vacuum pion mass. By following the same logic, kaon superfluidity might also be possible when $|l_{\rm e}+l_\mu|$ is so large that $|\mu_{\rm Q}|$ becomes larger than vacuum kaon mass. Such a possibility is checked by adopting Ginzburg-Landau approximation within the three-flavor Polyakov--Nambu--Jona-Lasinio model. Consider the case with full chemical balance, though kaon superfluidity could be stable compared to the chiral phases with only $\sigma$ condensations, it would get killed by the more favored homogeneous pion superfluidity. If we introduce mismatch between $s$ and $d$ quarks, kaon superfluidity would require so large $s$ quark density that such a state is impossible in the early universe.