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

TL;DR

This paper uses three-flavor chiral perturbation theory at next-to-leading order to analyze pion and kaon condensation under nonzero isospin and strange chemical potentials, comparing results with lattice QCD data.

Contribution

It provides the NLO effective potential calculations for various condensed phases and compares three-flavor $ ext{ChiPT}$ predictions with lattice QCD results, highlighting differences with two-flavor models.

Findings

Transitions are second order at specific chemical potential values.

Effective potential in pion-condensed phase is independent of strange chemical potential.

Good agreement with lattice QCD for $al<200$ MeV, deviations occur at higher values.

Abstract

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.