Speed of sound and polytropic index in the Polyakov-loop Nambu-Jona-Lasinio model

We investigate the speed of sound and polytropic index of quantum chromodynamics (QCD) matter in the full phase diagram based on a three-flavor Polyakov-looped Nambu-Jona-Lasinio (PNJL) model. The speed of sound and polytropic index in isothermal and adiabatic cases all have a dip structure at the low chemical potential side of the chiral phase transition boundary, and these quantities reach their global minimum values at the critical end point (CEP) but are not completely zero, where the values in adiabatic are lightly greater than those in isothermal. Different from the speed of sound, the polytropic index rapidly form a peak after crossing the chiral phase transition boundary, which is observed for the first time. Along the hypothetical chemical freeze-out lines, the speed of sound rapidly decreases near the CEP, followed by a small spinodal behavior, while the polytropic index, especially in isothermal, exhibits a more pronounced and nearly close to zero dip structure as it approaches the CEP.

Figure 1

The 3D plot (a) and contour map (b) of squared speed of sound $c_T^2$ in the ${\mu }_B-T$ plane based on a three-flavor PNJL model, as well as the squared speed of sound $c_T^2$ as functions of ${\mu }_B$ (c) at $T=50$, 100, 150, 200 MeV. In (b), the black dash-dotted and solid lines are, respectively, for the chiral crossover and first-order phase transition, the black dot connecting the chiral crossover and first-order phase transition represents the critical end point (CEP), and the gray dashed lines correspond to the contour of the squared speed of sound. In (c), the red star marks the baryon chemical potential of the CEP, and the black dashed line $c_s^2=1/3$ indicating the conformal matter limit is also shown for comparison.

Figure 2

The same as in Fig. 1 but for the squared speed of sound $c_{s/{\rho }_B}^2$.

Figure 3

The 3D plot (a) and contour map (b) of polytropic index ${\gamma }_T$ in the ${\mu }_B-T$ plane based on a three-flavor PNJL model, as well as the polytropic index ${\gamma }_T$ as functions of ${\mu }_B$ (c) at $T=50$, 100, 150, 200 MeV. In (b), the black dash-dotted and solid lines are, respectively, for the chiral crossover and first-order phase transition, the black dot connecting the chiral crossover and first-order phase transition represents the critical end point (CEP), and the gray dashed lines correspond to the contour of the squared speed of sound. The red star in (c) marks the baryon chemical potential of the CEP.

Figure 4

The same as in Fig. 3 but for the polytropic index ${\gamma }_{s/{\rho }_B}$.

Figure 5

The squared speed of sound $c_T^2$ and the factor $P/ϵ$ as functions of ${\mu }_B$ at $T=50$, 100, 150, 200 MeV. The solid and dash-dotted lines represent the results of $c_T^2$ and $P/ϵ$, respectively.

Figure 6

The hypothetical chemical freeze-out lines (a) by rescaling ${\mu }_B$ of the chiral phase transition boundary with factors of 0.95, 0.98, 0.995, 1.01, and 1.02, as well as the squared speed of sound (b) and polytropic index (c) along the hypothetical chemical freeze-out lines. The black dash-dotted and solid lines in (a) are, respectively, for the crossover and first-order phase transition, but the solid and dashed lines in (b) and (c) represent the results of the speed of sound and polytropic index in isothermal and adiabatic cases, respectively. The black dot in (a) connecting the crossover and first-order phase transition represents the CEP, and the red stars in (b) and (c) mark the baryon chemical potential of the CEP.