Pion and kaon condensation at zero temperature in three-flavor $\chi$PT at nonzero isospin and strange chemical potentials at next-to-leading order

We consider three-flavor chiral perturbation theory ($\chi$PT) at zero temperature and nonzero isospin ($\mu_{I}$) and strange ($\mu_{S}$) chemical potentials. The effective potential is calculated to next-to-leading order (NLO) in the $\pi^{\pm}$-condensed phase, the $K^{\pm}$-condensed phase, and the $K^0/\bar{K}^0$-condensed phase. It is shown that the transitions from the vacuum phase to these phases are second order and take place when, $|\mu_I|=m_{\pi}$, $|{1\over2}\mu_I+\mu_S|=m_K$, and $|-{1\over2}\mu_I+\mu_S|=m_K$, respectively at tree level and remains unchanged at NLO. The transition between the two condensed phases is first order. The effective potential in the pion-condensed phase is independent of $\mu_S$ and in the kaon-condensed phases, it only depends on the combinations $\pm{1\over2}\mu_I+\mu_S$ and not separately on $\mu_I$ and $\mu_S$. We calculate the pressure, isospin density and the equation of state in the pion-condensed phase and compare our results with recent $(2+1)$-flavor lattice QCD data. We find that the three-flavor $\chi$PT results are in good agreement with lattice QCD for $\mu_I<200$ MeV, however for larger values $\chi$PT produces values for observables that are consistently above lattice results. For $\mu_I>200$ MeV, the two-flavor results are in better agreement with lattice data. Finally, we consider the observables in the limit of very heavy $s$-quarks, where they reduce to their two-flavor counterparts with renormalized couplings. The disagreement between the predictions of two and three flavor $\chi$PT can largely be explained by the differences in the experimental values of the low-energy constants.