  We attempt to evaluate the effective Lagrangian for a classical background
field interacting with the vacuum of two quantum fields. The integration of one
of the quantum fields in general leads to a non-local term in the effective
Lagrangian and thus becomes intractable during the integration of the other
quantum field. We show that $\phi F \tilde{K}$ interaction is an exception. We
present the complete calculation for the evaluation of the effective Lagrangian
for a pseudoscalar field interacting with photons in the presence of a
background electromagnetic field. Expression for the probability of the vacuum
breaking down into a pseudoscalar-photon pair is evaluated. We conclude that
the gradient of an electric field beyond a certain threshold can give rise to
pseudoscalar-photon pair production.
