  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.
