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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mrow><a:mo stretchy="false">|</a:mo><a:msub><a:mrow><a:mi>l</a:mi></a:mrow><a:mrow><a:mi mathvariant="normal">e</a:mi></a:mrow></a:msub><a:mo>+</a:mo><a:msub><a:mrow><a:mi>l</a:mi></a:mrow><a:mrow><a:mi>μ</a:mi></a:mrow></a:msub><a:mo stretchy="false">|</a:mo></a:mrow></a:math> is large enough to generate a charge chemical potential <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:mo stretchy="false">|</f:mo><f:msub><f:mi>μ</f:mi><f:mi mathvariant="normal">Q</f:mi></f:msub><f:mo stretchy="false">|</f:mo></f:math> larger than vacuum pion mass. By following the same logic, kaon superfluidity might also be possible when <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo stretchy="false">|</k:mo><k:msub><k:mi>l</k:mi><k:mi mathvariant="normal">e</k:mi></k:msub><k:mo>+</k:mo><k:msub><k:mi>l</k:mi><k:mi>μ</k:mi></k:msub><k:mo stretchy="false">|</k:mo></k:math> is so large that <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:mo stretchy="false">|</p:mo><p:msub><p:mi>μ</p:mi><p:mi mathvariant="normal">Q</p:mi></p:msub><p:mo stretchy="false">|</p:mo></p:math> 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 <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mi>σ</u:mi></u:math> condensations, it would get killed by the more favored homogeneous pion superfluidity. If we introduce mismatch between <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mi>s</w:mi></w:math> and <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"><y:mi>d</y:mi></y:math> quarks, kaon superfluidity would require so large <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:mi>s</ab:mi></ab:math> quark density that such a state is impossible in the early Universe.