Phase diagrams in the three-flavor Nambu–Jona-Lasinio model with the Polyakov loop

We present extensive studies on hot and dense quark matter with two light and one heavy flavors in the Nambu–Jona-Lasinio model with the Polyakov loop (so-called PNJL model). First we discuss prescription dependence in choosing the Polyakov loop effective potential and propose a simple and rather sensible ansatz. We look over quantitative comparison to the lattice measurement to confirm that the model captures thermodynamic properties correctly. We then analyze the phase structure by changing the temperature, quark chemical potential, quark masses, and coupling constants. We particularly investigate how the effective ${\mathrm{U}}_{\mathrm{A}}(1)$ restoration and the induced vector-channel interaction at finite density would affect the QCD critical point.

Figure 1

Comparison of the Polyakov loop pressure excess as a function of the temperature. The vertical axis signifies the effective degrees of freedom. RTW05 and RRW06 represent ${\Omega }_{\mathrm{RTW}05}$ and ${\Omega }_{\mathrm{RRW}06}$ with $\ell$ and $\overline{\ell }$ given as a solution of the full gap equation with two quark flavors. For comparison we used the same NJL model parameters, $\Lambda$, $g_{\mathrm{S}}$, and $m_u=m_d$, as in Ref. 15 and quenched the $s$-quark sector to draw the solid curve which represents our ${\Omega }_{\mathrm{Polyakov}}$.

Figure 2

Order parameters at zero density as a function of the temperature. The chiral condensates are normalized by $⟨\overline{u}u{⟩}_0=(246\mathrm{MeV}{)}^3$ and $⟨\overline{s}s{⟩}_0=(267\mathrm{MeV}{)}^3$. The solid and dotted curves represent the $u$-quark and $s$-quark chiral condensates, respectively, and the dashed curve represents the traced Polyakov loop $\ell$.

Figure 3

Effective degrees of freedom associated with $2+1$ flavor quarks with increasing temperature in the NJL model and in the PNJL model. The critical temperature is $T_c=171.6\mathrm{MeV}$ in the NJL model case and $T_c=204.8\mathrm{MeV}$ in the PNJL model case.

Figure 4

Chiral and Polyakov loop susceptibility at zero density as a function of the temperature in the $2+1$ flavor PNJL model.

Figure 5

Quark number susceptibility calculated in the PNJL model for $2+1$ flavor quark matter at zero density.

Figure 6

Ratio of the fourth derivative to the quark number susceptibility calculated in the PNJL model for $2+1$ flavor quark matter at zero density.

Figure 7

Plot for the “interaction measure,” i.e. $(ϵ-3p)/T^4$, and the fermion contribution, ${\Theta }_{\mathrm{F}}^{\mu \mu }=[2m_{ud}(⟨\overline{u}u⟩-⟨\overline{u}u{⟩}_0)+m_s(⟨\overline{s}s⟩-⟨\overline{s}s{⟩}_0)]$.

Figure 8

Plot for $p/ϵ$ as a function of ${ϵ}^{1/4}$.

Figure 9

Normalized light-quark chiral condensate $⟨\overline{u}u⟩/⟨\overline{u}u{⟩}_0$ in the $\mu \mathrm{\text{−}}T$ plane, where $⟨\overline{u}u{⟩}_0$ is the chiral condensate at $T=\mu =0$.

Figure 10

Polyakov loop $\ell$ in the $\mu \mathrm{\text{−}}T$ plane.

Figure 11

Quark number density normalized by the free massless quark density, $N_c{N}_f({\mu }^3/3{\pi }^2+T^2\mu /3)$, in the $\mu \mathrm{\text{−}}T$ plane.

Figure 12

Phase diagram characterized by three quantities, namely, the $u$-quark chiral condensate, the $s$-quark chiral condensate, and the Polyakov loop.

Figure 13

3D plot for the light-quark chiral susceptibility ${\chi }_{ud}$ multiplied by $(T/\Lambda {)}^2$ in the $\mu \mathrm{\text{−}}T$ plane.

Figure 14

3D plot for the quark number susceptibility ${\chi }_q$ multiplied by $(T/\Lambda {)}^2$ in the $\mu \mathrm{\text{−}}T$ plane.

Figure 15

Boundaries of the first-order phase transition region as a function of the quark masses at $\mu =0$, 100, 200, 250, 300, and 350 MeV from the bottom to the top.

Figure 16

First-order transition boundary depending on the strength of the ’t Hooft coupling $g_{\mathrm{D}}$, where $g_{\mathrm{D}0}$ is a value fixed in the vacuum in the Hatsuda-Kunihiro parameter set.

Figure 17

Dependence of the critical point location on the strength of the ’t Hooft coupling constant.

Figure 18

Dependence of the critical point location on the strength of the vector-channel interaction.