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 U(1). The formula is quite similar to that in QED, 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 be parity conserving.