Generalized Heisenberg-Euler formula in Abelian gauge theory with parity violation

A generalized Heisenberg-Euler formula is given for an Abelian gauge theory having vector as well as axial vector couplings to a massive fermion. So, the formula is applicable to a parity-violating theory. The gauge group is chosen to be $U(1)$. The formula is quite similar to that in quantum electrodynamics, but there is a complexity in which one factor (related to spin) is expressed in terms of the expectation value. The expectation value is evaluated by the contraction with the one-dimensional propagator in a given background field. The formula affords a basis to the vacuum magnetic birefringence experiment, which aims to probe the dark sector, where the interactions of the light fermions with the gauge fields are not necessarily parity conserving.