Nonstrange and strange quark matter at finite temperature within modified NJL model and protoquark stars

TL;DR

This work addresses the finite-temperature equation of state for nonstrange and strange quark matter within a modified NJL framework that incorporates exchange interactions via a Fierz transformation, across two- and three-flavor sectors in $β$-equilibrium. By introducing a tunable weight $α$ for exchange channels, the authors derive self-consistent mass gaps $M_f$ and effective chemical potentials $\tilde{μ}_f$, compute the thermodynamic potential $\Omega_M(T,\tilde{μ})$, and enforce thermodynamic consistency through the condition that the minimum of the free energy per baryon $f/ρ_B$ occurs at $P=0$, which constrains the vacuum pressure $B$. They find that increasing $α$ converts the chiral transition from first-order to a crossover and stiffens the EOS due to repulsive exchange interactions, with neutrino trapping and finite temperature having relatively modest effects under isothermal conditions; the vacuum pressure plays a crucial role in determining surface densities and the maximum masses of protoquark stars, with distinct implications for strange versus nonstrange configurations. The results provide a self-consistent framework for hot quark-star modeling and highlight key parameters, such as $α$ and $B$, that govern stellar structure and stability, pointing toward future inclusion of color superconductivity and diquark condensates for a more complete description.

Abstract

We extend the modified Nambu-Jona-Lasinio (NJL) model -- incorporating exchange interactions via the Fierz transformation -- to finite temperatures in both two- and three-flavor scenarios, and investigate the properties of protoquark stars in $β$-equilibrium. Our results show that increasing the strength of exchange interactions, characterized by the parameter $α$, changes the chiral phase transition from first-order to crossover. We examine the effects of finite temperature, lepton fraction, and exchange interactions on the equation of state (EOS). We find that, in the crossover regime, the EOS is significantly stiffer than in the first-order case due to the substantial contribution of repulsive interactions in the exchange channels, while it remains relatively insensitive to variations in temperature and lepton fraction. Imposing the thermodynamic consistency, which requires the minimum of free energy per baryon $f / ρ_B$ occurs at zero pressure, further constrains the minimum value of vacuum pressure.

Figures (5)

• Figure 1: Upper panel: The constituent quark masses of $u,\ d, \ s$ quarks versus quark chemical potential $\mu$ both for strange quark matter and nonstrange quark matter in the absence of neutrinos ($Y_l = 0.0$), under various values of $\alpha$ and temperature $T$. The results for several representative parameter sets are shown: ($\alpha=0.0$, $T=40$ MeV), ($\alpha=0.0$, $T=0$ MeV), ($\alpha=0.5$, $T=0$ MeV), ($\alpha=0.8$, $T=0$ MeV), ($\alpha=0.9$, $T=0$ MeV). Middle panel: The corresponding results for quark number densities of $u,\ d, \ s$ quarks as functions of $\mu$. Lower panel: The electron chemical potential $\mu_e$ as a function of $\mu$.
• Figure 2: The constituent quark masses $M_u$, $M_d$, and $M_s$, the electron chemical potential $\mu_e$, and the $d$ quark chemical potential $\mu_d$ are shown as functions of the $u$ quark chemical potential in the neutrino-trapped scenario, for a representative case with $\alpha = 0.5$, $Y_l = 0.1$, and $T = 40$ MeV. Results are presented for both strange and nonstrange quark matter.
• Figure 3: The constituent quark masses of $u,\ d,\ s$ quarks versus quark chemical potential $\mu$ for various choices of the three- and two-flavor NJL model parameters with neutrino trapped scenario for the study of their effects (see text for details). Lower panel: The pressure as a function of baryon number density, expressed in units of the nuclear saturation density $\rho_0$, for both strange quark matter and nonstrange quark matter.
• Figure 4: Upper panel: EOS for several representative parameter sets calculated within NJL model. Results are shown for the three-flavor case with ($\alpha=0.5$, $Y_l=0.1$, $T=40$ MeV, $B^{1/4}=133\;\text{MeV}$), ($\alpha=0.8$, $Y_l=0.1$, $T=40$ MeV, $B^{1/4}=133\;\text{MeV}$), ($\alpha=0.8$, $Y_l=0.1$, $T=10$ MeV, $B^{1/4}=133\;\text{MeV}$), ($\alpha=0.8$, $Y_l=0.1$, $T=10$ MeV, $B^{1/4}=95\;\text{MeV}$), ($\alpha=0.8$, $Y_l=0.4$, $T=10$ MeV, $B^{1/4}=95\;\text{MeV}$), as well as for the two-flavor case with ($\alpha=0.8$, $Y_l=0.1$, $T=10$ MeV, $B^{1/4}=110\;\text{MeV}$), ($\alpha=0.8$, $Y_l=0.1$, $T=40$ MeV, $B^{1/4}=110\;\text{MeV}$), ($\alpha=0.8$, $Y_l=0.1$, $T=10$ MeV, $B^{1/4}=90\;\text{MeV}$), ($\alpha=0.8$, $Y_l=0.4$, $T=10$ MeV, $B^{1/4}=90\;\text{MeV}$), ($\alpha=0.9$, $Y_l=0.1$, $T=10$ MeV, $B^{1/4}=90\;\text{MeV}$). The panel shows the full EOS, featuring the phase with unrestored chiral symmetry on the left and the restored quark phase corresponding to self-bound strange quark matter with a nonzero vacuum pressure $B$ on the right. Middle panel: Free energy per baryon $f/\rho_B$ as a function of pressure $P$, and the squared speed of sound $c_s^2$ as a function of energy density $\epsilon$, corresponding to the parameter sets shown in the upper panel for self-bound strange and nonstrange quark matter. The gray segments of the $f/\rho_B$ curves represent thermodynamically unstable EOSs 1967RPPh...30..615F. Lower panel: The $f/\rho_B$ as functions of $\mu$ and $\rho_B/\rho_0$ for strange and nonstrange quark matter.
• Figure 5: The mass-radius (M-R) relations for strange and nonstrange protoquark stars corresponding to the same parameter sets shown in Fig. \ref{['fig:3f_EOS_cs']}.