Kaon superfluidity in the early Universe

Previously, it was found that pion superfluidity could be realized in the quantum chromodynamics (QCD) epoch of the early Universe, when lepton flavor asymmetry $|l_e+l_{\mu }|$ is large enough to generate a charge chemical potential $\left|{\mu }_Q\right|$ larger than vacuum pion mass. By following the same logic, kaon superfluidity might also be possible when $|l_e+l_{\mu }|$ is so large that $\left|{\mu }_Q\right|$ 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.