analytic continuation, E → − iE$_{M}$ , is needed to recover Minkowski space results. As our interests lie only with quantities perturbative in the strength of the field, this analytic con- tinuation can be performed trivially, see [20]. A Euclidean formulation is natural from the point of view of lattice gauge theory simulations; moreover, the Euclidean formulation re- moves instabilities due to non-perturbative effects, i.e. the Schwinger mechanism [21]. With the vector potential specified, we can determine the baryon two-point functions. As neutral and charged baryons propagate differently in electric fields, we handle each separately. We consider first the case of a neutral spin-half particle of mass M described by the field ψ ( x ). The Euclidean space correlation function in the hadronic theory we denote by G$_{αβ}$ ( x$_{4}$, E ), which is given by A. Neutral Spin-Half Baryons G$_{αβ}$ ( x$_{4}$, E ) = ∫ d x 〈 0 | ψ$_{α}$ ( x ) ψ$_{β}$ (0) | 0 〉$_{E}$ , (2) where the subscript denotes that the correlation function is calculated in the background elec- tric field. Integrating over all space projects the correlator onto vanishing three-momentum, which is a good quantum number. For lattice QCD with spatially periodic boundary condi- tions, a sum over lattice sites accomplishes the same thing. The energy of the neutral particle, E ( E ), depends on the strength of the electric field. For weak fields, the energy has the expansion E ( E ) = M + 1 2 4 πα$_{E}$ E 2 + . . . , (3) where α$_{E}$ is the electric polarizability, and the ellipsis denotes higher-order terms in even powers of the field. The quadratic Stark shift is positive due to our Euclidean space treat- ment. The magnetic moment, µ , is also important, and this coupling is entirely anomalous. The single-hadron effective action for a neutral spin-half particle takes the form S$_{E}$ = ∫ d $^{4}$x ψ ( x ) [ / ∂ + E ( E ) − µ ( E ) 4 M σ$_{µν}$F$_{µν}$ ] ψ ( x ) . (4) This action describes the dynamics of the neutron in the ultra-low energy limit of QCD. The electromagnetic field strength tensor is F$_{µν}$ = ∂$_{µ}$A$_{ν}$ − ∂$_{ν}$A$_{µ}$ . For a background electric field, σ$_{µν}$F$_{µν}$ = 2 ⃗ K · ⃗ E , where ⃗ K = i⃗ γγ$_{4}$ is the generator of boosts in the spin-half representation of the Lorentz group. The magnetic moment coupling has been written as E -dependent. In small fields, µ ( E ) has a perturbative expansion in even powers of the field, and satisfies the zero-field limit, µ (0) = µ . Using the effective action in Eq. (4) to determine the unpolarized two-point function, we arrive at Tr[ G ( x$_{4}$, E ) ] = Z ( E ) exp [ − x$_{4}$ E$_{eff}$ ( E )] , (5) 4 volume artifacts [17–19]. As the current study is restricted to one lattice volume, we postpone the investigation of finite volume effects to future work.