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

We investigate the impact of the CP-violating $θ$ term on isospin symmetry breaking in quark matter and compact star properties using a two-flavor Nambu-Jona-Lasinio (NJL) model. By incorporating the $θ$ 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, $θ$ suppresses conventional chiral ($σ$) and pion ($π$) condensates while promoting pseudo-scalar ($η$) and scalar-isovector ($δ$) condensates, thereby reducing the critical isospin chemical potential $μ_I^{\text{crit}}$ for spontaneous symmetry breaking. For $θ=π$, a first-order phase transition emerges at $μ_I^{\text{crit}} = 0.021$ GeV, accompanied by CP symmetry restoration. Extending the investigation to finite temperature and baryon chemical potential reveals that these $θ$-term-induced effects persist. Axion effects (modeled via $θ\equiv a/f_a$) 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 $θ$ as a critical parameter modulating both the Quantum Chromodynamics (QCD) phase structure and compact star observables.