Pion and kaon condensation in a 3-flavor Nambu–Jona-Lasinio model

We analyze the phase diagram of a 3-flavor Nambu–Jona-Lasinio model at finite temperature $T$ and chemical potentials ${\mu }_u,{\mu }_d,{\mu }_s$. We study the competition of pion and kaon condensation and we propose a physical situation in which kaon condensation could be led only by light quark finite densities.

Figure 1

Phase diagram for chiral symmetry restoration and meson condensation in the plane (${\mu }_I,{\mu }_Y$) at ${\mu }_q=0$ and $T=0$. Different regions are specified by the nonvanishing of a given condensate, whereas the others are vanishing (${\rho }_{ud},{\rho }_{us}$) or order $\sim m_f/\Lambda$ (${\chi }_u,{\chi }_d,{\chi }_s$). Dashed lines are for the continuous vanishing of pseudoscalar fields, whereas solid lines are for discontinuous behaviors. The solid lines $a$ and $b$ refer to specific paths at fixed values of ${\mu }_Y=230\mathrm{M}\mathrm{e}\mathrm{V}$ (line $a$) and ${\mu }_I=200\mathrm{M}\mathrm{e}\mathrm{V}$ (line $b$).

Figure 2

Scalar and pseudoscalar condensates vs ${\mu }_I$, for $T=0$, ${\mu }_q=0$, and ${\mu }_Y=230\mathrm{M}\mathrm{e}\mathrm{V}$. The path followed in the phase diagram of Fig. 1 is that of the solid line $a$.

Figure 3

Scalar and pseudoscalar condensates vs ${\mu }_Y$, for $T=0$, ${\mu }_q=0$, and ${\mu }_I=200\mathrm{M}\mathrm{e}\mathrm{V}$. The path followed in the phase diagram of Fig. 1 is that of the solid line $b$

Figure 4

Phase diagram for chiral symmetry restoration and meson condensation in the plane (${\mu }_I,{\mu }_Y$) at ${\mu }_q=0$ and $T=0$. Different regions are specified by the nonvanishing of a given condensate, whereas the others are vanishing (${\rho }_{ud},{\rho }_{us},{\rho }_{ds}$) or order $\sim m_f/\Lambda$ (${\chi }_u,{\chi }_d,{\chi }_s$). Dashed lines are for the continuous vanishing of pseudoscalar fields, whereas solid lines are for discontinuous behaviors.

Figure 5

Phase diagram for chiral symmetry restoration and meson condensation in the plane (${\mu }_I,{\mu }_Y$) at ${\mu }_q=0$ and $T=100\mathrm{M}\mathrm{e}\mathrm{V}$, which is above the temperature of the tricritical point. Different regions are specified by the nonvanishing of a given condensate, whereas the others are vanishing (${\rho }_{ud},{\rho }_{us}$) or order $\sim m_f/\Lambda$ (${\chi }_u,{\chi }_d,{\chi }_s$). Dashed lines are for the continuous vanishing of pseudoscalar fields or for crossover transitions for scalar fields, whereas solid lines are for discontinuous behaviors.

Figure 6

Phase diagram for chiral symmetry restoration and pion condensation in the plane (${\mu }_I,{\mu }_Y$) at ${\mu }_q=0$ and $T=140\mathrm{M}\mathrm{e}\mathrm{V}$, which is above the highest temperature $\sim 110\mathrm{M}\mathrm{e}\mathrm{V}$ to have kaon condensation. Different regions are specified by the nonvanishing of a given condensate, whereas the others are vanishing (${\rho }_{ud}$) or order $\sim m_f/\Lambda$ (${\chi }_u,{\chi }_d,{\chi }_s$). Dashed lines are for the continuous vanishing of pseudoscalar fields or for crossover transitions for scalar fields.

Figure 7

Phase diagram for chiral symmetry restoration and kaon condensation in the plane (${\mu }_q,{\mu }_s$) at ${\mu }_I=0^{-}$ and $T=0$. Different regions are specified by the nonvanishing of a given condensate, whereas the others are vanishing (${\rho }_{us},{\rho }_{ds}$) or order $\sim m_f/\Lambda$ (${\chi }_u,{\chi }_d,{\chi }_s$). Dashed lines are for the continuous vanishing of pseudoscalar fields, whereas solid lines are for discontinuous behaviors.

Figure 8

Phase diagram for chiral symmetry restoration and kaon condensation in the plane (${\mu }_q,{\mu }_s$) at ${\mu }_I=0^{-}$ and $T=0$, and for $m_K=300\mathrm{M}\mathrm{e}\mathrm{V}$. Different regions are specified by the nonvanishing of a given condensate, whereas the others are vanishing (${\rho }_{us},{\rho }_{ds}$) or order $\sim m_f/\Lambda$ (${\chi }_u,{\chi }_d,{\chi }_s$). Dashed lines are for the continuous vanishing of pseudoscalar fields, whereas solid lines are for discontinuous behaviors.