Pion and kaon condensation at zero temperature in three-flavor χPT 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 (+/-)-condensed phase, the K-+/--condensed phase, and the K0/K<overbar></mml:mover>0-condensed phase. It is shown that the transitions from the vacuum phase to these phases are second order and take place when, <mml:mfenced close="|" open="|">mu I</mml:mfenced>=m pi,<mml:mfenced close="|" open="|"><mml:mfrac>12</mml:mfrac>mu I+mu S</mml:mfenced>=mK, and <mml:mfenced close="|" open="|">-<mml:mfrac>12</mml:mfrac>mu I+mu S</mml:mfenced>=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 mu (S) and in the kaon-condensed phases, it only depends on the combinations +/- <mml:mfrac>12</mml:mfrac>mu I<mml:mo>+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-quark, 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.