The chiral phase transition and equation of state are studied within a novel self-consistent mean-field ap-proximation of the two-flavor Nambu-Jona-Lasinio model.In this newly developed model,modifications to the chemical μ and chiral chemical μ5 potentials are naturally included by introducing vector and axial-vector channels from Fierz-transformed Lagrangian to the standard Lagrangian.In the proper-time scheme,the chiral phase trans-ition is a crossover in the T-μ plane.However,when μ5 is incorporated,our study demonstrates that a first order phase transition may emerge.Furthermore,the chiral imbalance will soften the equation of state of quark matter.The mass-radius relationship and tidal deformability of quark stars are calculated.The maximum mass and radius de-crease as μ5 increases.Our study also indicates that the vector and axial-vector channels exhibit an opposite influ-ence on the equation of state.