content_image/398360.jpg content_image/398363.jpg content_image/398365.jpg content_image/398367.jpg
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. 

## A. Neutral Spin-Half Baryons 

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 

<img src='content_image/398360.jpg'>

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 

<img src='content_image/398363.jpg'>

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 

<img src='content_image/398365.jpg'>

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 

<img src='content_image/398367.jpg'>

