Background field method and nonrelativistic QED matching

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.