The chiral phase transition and equation of state in the chiral imbalance

The chiral phase transition and equation of state are studied within a new self-consistent mean-field approximation of the two-flavor Nambu$-$Jona-Lasinio model. In this newly developed model, modifications to the chemical potential $\mu$ and chiral chemical potential $\mu_5$ is naturally included by adding vector and axial-vector channels from Fierz-transformed Lagrangian to the standard Lagrangian. In proper-time scheme, the chiral phase transition is a crossover in the $T-\mu$ plane. But when $\mu_5$ is increased, our study shows that there may exist first order phase transition. Furthermore, the chiral imbalance will soften the equation of state of quark matter. The mass-radius relations and tidal deformability of quark stars are calculated. As $\mu_5$ increases, the maximum mass and radius decrease. The vector channel and axial-vector channel have opposite influence on the equation of state. However, when EOS is constrained by astronomical observations, the shape of the mass-radius curve can be used to determine whether there is chiral imbalance in the dense object, and thus indirectly proving the CP violation in the dense matter. Our study shows a different influence of the chiral imbalance on the chiral phase transition in contrary to tree-momentum-cutoff scheme.