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

Abstract

We consider three-flavor chiral perturbation theory (χPT) at zero temperature and nonzero isospin (µI ) and strange (µS) chemical potentials. The effective potential is calculated to next-to-leading order (NLO) in the π ±-condensed phase, the K±-condensed phase, and the K0/K¯ 0 -condensed phase. It is shown that the transitions from the vacuum phase to these phases are second order and take place when, |µI | = mπ, | 1 2 µI + µS| = mK, and | − 1 2 µI + µS| = mK, 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 µS and in the kaon-condensed phases, it only depends on the combinations ± 1 2 µI + µS and not separately on µI and µS. 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 threeflavor χPT results are in good agreement with lattice QCD for µI < 200 MeV, however for larger values χPT produces values for observables that are consistently above lattice results. For µI > 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-quark, where they reduce to their two-flavor counterparts with renormalized couplings. The disagreement between the predictions of two and three flavor χPT can largely be explained by the differences in the experimental values of the low-energy constants.