  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.
