$2+1$ flavor Polyakov–Nambu–Jona-Lasinio model at finite temperature and nonzero chemical potential

We extend the Polyakov-loop improved Nambu–Jona-Lasinio model to $2+1$ flavor case to study the chiral and deconfinement transitions of strongly interacting matter at finite temperature and nonzero chemical potential. The Polyakov loop, the chiral susceptibility of light quarks ($u$ and $d$), and the strange quark number susceptibility as functions of temperature at zero chemical potential are determined and compared with the recent results of lattice QCD simulations. We find that there is always an inflection point in the curve of strange quark number susceptibility accompanying the appearance of the deconfinement phase, which is consistent with the result of lattice QCD simulations. Predictions for the case at nonzero chemical potential and finite temperature are made as well. We give the phase diagram in terms of the chemical potential and temperature and find that the critical end point moves down to low temperature and finally disappears with the decrease of the strength of the ’t Hooft flavor-mixing interaction.

Figure 1

Left panel: calculated Polyakov-loop $\Phi$, scaled chiral condensates for light quarks ${\varphi }_l$ and for strange quarks ${\varphi }_s$ as functions of temperature at vanishing quark chemical potential in the case of considering the polynomial Polyakov-loop effective potential ${\mathcal{U}}_{\mathrm{pol}}$, the improved Polyakov-loop effective potential ${\mathcal{U}}_{\mathrm{imp}}$, respectively. Right panel: calculated temperature dependence of the derivatives $\partial \Phi /\partial T$, $\partial {\varphi }_l/\partial T$, and $\partial {\varphi }_s/\partial T$ with the two kinds of Polyakov-loop potentials.

Figure 2

Calculated results of the renormalized chiral susceptibility $m_l^2\Delta {\chi }_l/T^4$ as a function of temperature at zero quark chemical potential for the two Polyakov-loop effective potentials.

Figure 3

Left panel: calculated results of the scaled strange quark number susceptibility ${\chi }_s/T^2$ as a function of temperature (thinner curves are for the case with $K{\Lambda }^5=12.36$ and thicker ones for $K{\Lambda }^5=0$). Right panel: plots of the calculated $\partial ({\chi }_s/T^2)/\partial T$.

Figure 4

Calculated results of the scaled strange quark chiral condensate as a function of temperature with different values of the current mass of strange quark for the improved effective potential (left panel) and the temperature dependence of the corresponding Polyakov-loop $\Phi$ at several strange quark masses (right panel).

Figure 5

The same as Fig. 2 but with $m_s=m_l=5.5\mathrm{MeV}$.

Figure 6

The same as Fig. 3 but with $m_s=m_l=5.5\mathrm{MeV}$ and $K{\Lambda }^5=12.36$.

Figure 7

Calculated derivatives of the strange quark number susceptibility versus the temperature in the cases: $m_l=5.5\mathrm{MeV}$, $m_s=140.7\mathrm{MeV}$, $m_l=m_s=5.5\mathrm{MeV}$, and $m_l=m_s=3.0\mathrm{MeV}$.

Figure 8

Calculated results of the scaled light quark chiral condensate (left panel) and its derivative with respect to temperature (right panel) as functions of temperature for several values of the quark chemical potential.

Figure 9

The same as Fig. 8, but for the condensate of strange quarks.

Figure 10

Calculated results of the Polyakov-loop $\Phi$ (left panel) and $\partial \Phi /\partial T$ (right panel) as functions of temperature at several values of quark chemical potentials.

Figure 11

Calculated results of the Polyakov-loop $\Phi$ (thin curves) and scaled $\Delta \Phi$ (thick curves) as functions of temperature at several values of the chemical potential.

Figure 12

Left panel: calculated quark chemical potential dependence of the scaled chiral condensates ${\varphi }_l$ and ${\varphi }_s$ at temperature $T=100\mathrm{MeV}$. Right panel: plots of $\partial {\varphi }_l/\partial \mu$ and $\partial {\varphi }_s/\partial \mu$ with respect to the chemical potential.

Figure 13

Calculated strange quark condensate as a function of chemical potential with different values of $K{\Lambda }^5$ at $T=100\mathrm{MeV}$.

Figure 14

Calculated Polyakov-loop $\Phi$ (thin curves) and the scaled $\Delta \Phi$ (thick curves) as functions of quark chemical potential at several values of temperature.

Figure 15

Left panel: calculated phase diagram in terms of temperature and quark chemical potential of the $2+1$ flavor PNJL with physical masses, viz. $m_l=5.5\mathrm{MeV}$, $m_s=140.7\mathrm{MeV}$, and with different flavor-mixing interaction strengths: $K{\Lambda }^5=12.36$ (thicker curves) and $K{\Lambda }^5=8.0$ (thinner curves). Right panel: phase diagrams for $m_l=m_s=5.5\mathrm{MeV}$ (thicker curves) and in the chiral limit $m_l=m_s=0$ (thinner curves), respectively, with $K{\Lambda }^5$ fixed at 12.36.