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 transition from the vacuum phase to these phases are second order and takes 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 $\mu_K$ and in the kaon-condensed phases, it only depends on the combinations $\pm{1\over2}\mu_I+\mu_K$ and not separately on $\mu_I$ and $\mu_K$. 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 three-flavor $\chi$PT is in excellent agreement with lattice QCD near the second order phase transition at $\mu_{I}=m_{\pi}$ when using the central values of the low energy constants (LECs), which includes the vacuum pion (and kaon) masses, pion (and kaon) decay constants and the LECs ($L_{i}$) of the $\mathcal{O}(p^{4})$ $\chi$PT Lagrangian. For larger values of the isospin chemical potential (and zero strange quark chemical potential), while $\chi$PT and lattice QCD results are consistent for the observables including pressure, isospin density and energy density, the central values of LECs produce observables that are overestimates compared to the lattice results.

中文翻译：

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非零同位旋和奇特化学势下三味 χPT 在零温度下的 Pion 和 kaon 缩合

我们考虑在零温度和非零同位旋 ($\mu_{I}$) 和奇怪的 ($\mu_{S}$) 化学势下的三味手性扰动理论 ($\chi$PT)。在 $\pi^{\pm}$-condensed 阶段、$K^{\pm}$-condensed 阶段和 $K^0/ \bar{K}^0$-凝聚相。结果表明，从真空阶段到这些阶段的转变是二阶的，并且发生在 $|\mu_I|=m_{\pi}$, $|{1\over2}\mu_I+\mu_S|=m_K$,和 $|-{1\over2}\mu_I+\mu_S|=m_K$，分别在树级别并且在 NLO 保持不变。两个凝聚相之间的转变是一阶的。pion 凝聚相中的有效电位与 $\mu_K$ 和 kaon 凝聚相中的有效电位无关，它只取决于组合 $\pm{1\over2}\mu_I+\mu_K$ 而不是单独取决于 $\mu_I$ 和 $\mu_K$。我们计算了 pion 凝聚相中的压力、同位旋密度和状态方程，并将我们的结果与最近的 $(2+1)$-风味晶格 QCD 数据进行比较。我们发现，当使用低能量常数 (LECs) 的中心值时，三味 $\chi$PT 在 $\mu_{I}=m_{\pi}$ 处的二阶相变附近与晶格 QCD 非常一致)，其中包括 $\mathcal{O}(p^{4})$ $\chi 的真空 pion（和 kaon）质量、pion（和 kaon）衰减常数和 LECs $PT 拉格朗日。对于较大的同位旋化学势值（和零奇异夸克化学势），而 $\chi$PT 和晶格 QCD 结果对于包括压力、同位旋密度和能量密度在内的可观测值是一致的，