Pion and kaon condensation at zero temperature in three-flavor \u03c7PT 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 {K}^0/{overline{K}}^0 -condensed phase. It is shown that the transitions from the vacuum phase to these phases are second order and take place when, left|{mu}_Iright|={m}_{pi },left|frac{1}{2}{mu}_I+{mu}_Sright|={m}_K , and left|-frac{1}{2}{mu}_I+{mu}_Sright|={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 μS and in the kaon-condensed phases, it only depends on the combinations pm frac{1}{2}{mu}_I+{mu}_S 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 three-flavor 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.