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

TL;DR

The authors investigate how the CP-violating QCD $\theta$ term, implemented via the two-flavor NJL model with a Kobayashi–Maskawa–’t Hooft determinant, influences isospin-asymmetric quark matter and the structure of nonstrange quark stars. They derive the mean-field thermodynamic potential $\Omega$ and gap equations at finite temperature, baryon chemical potential, and isospin chemical potential, including scalar and pseudoscalar condensates $\sigma,\pi,\eta,\delta$, with $\theta$ driving mixing among channels. At $T=0$, $\mu_B=0$ the results show $\theta$ suppresses $\sigma$ and $\pi$ while enhancing $\eta$ and $\delta$, lowering $\mu_I^{\text{crit}}$ for isospin breaking, and producing a first-order transition at $\theta=\pi$ with CP restoration; these effects persist at finite $T$ and $\mu_B$. The study also demonstrates that axion-like coupling (via $\theta$) stiffens the quark-star EOS, increasing $M_{\max}$ and radii in agreement with multimessenger constraints. Overall, $\theta$ emerges as a crucial parameter modulating QCD phase structure and compact-star observables, with implications for axion phenomenology in dense matter.

Abstract

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.

Figures (13)

• Figure 1: The four condensates $\sigma$, $\pi$, $\eta$ and $\delta$ scaled by the chiral condensate $\sigma_0$, as function of isospin chemical potential $\mu_I$ for several values of $\theta \equiv a/f_a$. Top left plot corresponds to $\theta=0$, top right to $\theta=\pi/3$, bottom left to $\theta=2\pi/3$, and finally bottom right to $\theta=\pi$.
• Figure 2: The critical isospin chemical potential for spontaneous isospin symmetry breaking, $\mu_{I}^{\text{crit}}$ as function of CP-violating parameter $\theta$.
• Figure 3: The variations of the pseudoscalar condensate $\eta$ with respect to $\theta$ for several isospin chemical potentials. The black dots are the positions of the critical $\theta_c$ when spontaneous isospin symmetry breaking occurs.
• Figure 4: The condensates $\pi$ and $\delta$ as functions of $\theta$ for several isospin chemical potentials. The black dots are the positions of the critical $\theta_c$ when spontaneous isospin symmetry breaking occurs.
• Figure 5: The normalized thermodynamic potential as a function of $\theta$ for different isospin chemical potentials. The black dots are the positions of the critical $\theta_c$ when spontaneous isospin symmetry breaking occurs.