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

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 <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>β</a:mi> </a:math> equilibrium. Our results show that increasing the strength of exchange interactions, characterized by the parameter <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>α</c:mi> </c:math> , 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 <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>f</e:mi> <e:mo>/</e:mo> <e:msub> <e:mi>ρ</e:mi> <e:mi>B</e:mi> </e:msub> </e:math> occurs at zero pressure, further constrains the minimum value of vacuum pressure.