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

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 $\beta$-equilibrium. Our results show that increasing the strength of exchange interactions, characterized by the parameter $\alpha$, 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), finding that the EOS in the crossover regime is significantly stiffer than in the first-order case. In certain regions of parameter space within the crossover scenario, strange quark matter can support the existence of twin star configurations.