  We discuss the resolution of an inconsistency between lattice background
field methods and nonrelativistic QED matching conditions. In particular, we
show that lack of on-shell conditions in lattice QCD with time-dependent
background fields generally requires that certain operators related by
equations of motion should be retained in an effective field theory to
correctly describe the behavior of Green's functions. The coefficients of such
operators in a nonrelativistic hadronic theory are determined by performing a
robust nonrelativistic expansion of QED for relativistic scalar and spin-half
hadrons including nonminimal electromagnetic couplings. Provided that
nonrelativistic QED is augmented with equation-of-motion operators, we find
that the background field method can be reconciled with the nonrelativistic QED
matching conditions without any inconsistency. We further investigate whether
nonrelativistic QED can be employed in the analysis of lattice QCD correlation
function in background fields, but we are confronted with difficulties.
Instead, we argue that the most desirable approach is a hybrid one which relies
on a relativistic hadronic theory with operators chosen based on their
relevance in the nonrelativistic limit. Using this hybrid framework, we obtain
practically useful forms of correlation functions for scalar and spin-half
hadrons in uniform electric and magnetic fields.
