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  • Open Access

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

Wen-Li Yuan1,*, Nobutoshi Yasutake2,†, and Renxin Xu1,‡

  • *Contact author: wlyuan@pku.edu.cn
  • †Contact author: nobutoshi.yasutake@it-chiba.ac.jp
  • ‡Contact author: r.x.xu@pku.edu.cn
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Phys. Rev. D 113, 043033 – Published 17 February, 2026

DOI: https://doi.org/10.1103/27k9-r7nc

Abstract

We extend the modified Nambu-Jona-Lasinio 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 f/ρB occurs at zero pressure, further constrains the minimum value of vacuum pressure.

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Phys. Rev. D 113, 043033– Published 17 February, 2026
  • Received 26 August 2025
  • Accepted 22 January 2026
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