  We analyze the relation between the short-distance behavior of quantum field
theory and the strong-field limit of the background field formalism, for QED
effective Lagrangians in self-dual backgrounds, at both one and two loop. The
self-duality of the background leads to zero modes in the case of spinor QED,
and these zero modes must be taken into account before comparing the
perturbative beta function coefficients and the coefficients of the
strong-field limit of the effective Lagrangian. At one-loop this is familiar
from instanton physics, but we find that at two-loop the role of the zero
modes, and the interplay between IR and UV effects in the renormalization, is
quite different. Our analysis is motivated in part by the remarkable simplicity
of the two-loop QED effective Lagrangians for a self-dual constant background,
and we also present here a new independent derivation of these two-loop
results.
