The $\theta$-term effects on isospin asymmetric hot and dense quark matter

We investigate the impact of the CP-violating $\theta$ term on isospin symmetry breaking in quark matter and compact star properties using a two-flavor Nambu-Jona-Lasinio (NJL) model. By incorporating the $\theta$ parameter through the Kobayashi-Maskawa-'t Hooft (KMT) determinant interaction, we derive the thermodynamic potential and gap equations under finite temperature, baryon chemical potential, and isospin chemical potential. At zero temperature and baryon density, $\theta$ suppresses conventional chiral ($\sigma$) and pion ($\pi$) condensates while promoting pseudo-scalar ($\eta$) and scalar-isovector ($\delta$) condensates, thereby reducing the critical isospin chemical potential $\mu_I^{\text{crit}}$ for spontaneous symmetry breaking. For $\theta=\pi$, a first-order phase transition emerges at $\mu_I^{\text{crit}} = 0.021$ GeV, accompanied by CP symmetry restoration. Extending the investigation to finite temperature and baryon chemical potential reveals that these $\theta$-term-induced effects persist. Axion effects (modeled via $\theta\equiv a/f_a$) stiffen the equation of state (EOS) of non-strange quark stars, increasing their maximum mass and radii, in agreement with multimessenger constraints from pulsar observations and gravitational wave events. These results establish $\theta$ as a critical parameter modulating both the Quantum Chromodynamics (QCD) phase structure and compact star observables.