Background electromagnetic fields and nonrelativistic QED matching: Scalar case

The low-energy structure of hadrons can be described systematically using effective field theory, and the parameters of the effective theory can be determined from lattice QCD computations. Recent work, however, points to inconsistencies between the background field method in lattice QCD and effective field theory matching conditions. We show that the background field problem necessitates inclusion of operators related by equations of motion. In the presence of time-dependent electromagnetic fields, for example, such operators modify Green's functions, thereby complicating the isolation of hadronic parameters which enter on-shell scattering amplitudes. The particularly simple case of a scalar hadron coupled to uniform electromagnetic fields is investigated in detail. At the level of the relativistic effective theory, operators related by equations of motion are demonstrated to be innocuous. The same result does not hold in the nonrelativistic effective theory, and inconsistencies in matching are resolved by carefully treating operators related by equations of motion. As uniform external fields potentially allow for surface terms, the problem is additionally analyzed on a torus where such terms are absent. Temporal image corrections are derived for charged scalar correlation functions in uniform electric fields as a useful byproduct.