Chiral phase transition and equation of state in chiral imbalance *

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