where the effective energy, E$_{eff}$ ( E ), depends on the magnetic moment, and is given by E$_{eff}$ ( E ) = E ( E ) − µ ( E ) $^{2}$E 2 8 M 3 = M + 1 2 E 2 ( 4 πα$_{E}$ − µ 2 4 M 3 ) + . . . . (6) In the second line, we have retained terms in the effective energy only up to second order in the electric field. There are two distinct contributions at that order. The first is the expected shift in energy due to the polarizability, while the second is the analogue of the nucleon pole term arising from Born-level couplings to the magnetic moment. Appendix A details the physics underlying this Born-like contribution. While the magnetic moment contribution arises from a relativistic effect, the resulting shift in baryon energy occurs at the same order as that due to the polarizability. 4 Appendix B is devoted to the extraction of E$_{eff}$ ( E ), using unpolarized neutron correlation functions. The correlator in Eq. (5), however, does not allow access to the electric polarizability without knowledge of the magnetic moment. A background field analogue of the Born subtraction is needed. To extract both the magnetic moment and electric polarizability, we use the boost projection operators P$_{±}$ = 1 2 (1 ± K$_{3}$ ) , (7) where K$_{3}$ is the boost operator is the ˆ z -direction. The boost-projected correlation functions are given by G$_{±}$ ( x$_{4}$, E ) ≡ Tr[ P$_{±}$ G ( x$_{4}$, E ) ] (8) = Z ( E ) ( 1 ± E µ 2 M 2 ) exp [ − x$_{4}$ E$_{eff}$ ( E )] . (9) With the additional electric field dependence present in the amplitude, one can separate the electric polarizability from the magnetic moment by simultaneously analyzing both boost- projected correlators. The method we employ to accomplish this will be detailed below. B. Charged Spin-Half Baryons 5 4 In the chiral limit, the contribution from the magnetic moment term is suppressed relative to the electric polarizability by a factor of m π /M . This suppression, however, owes to the singular behavior of the polarizability in that limit, namely α E ∼ 1 /m π . Because our lattice pion mass is larger than physical, we will make no assumption about the dominance of the polarizability near the chiral limit. S$_{E}$ = ∫ d $^{4}$x ψ ( x ) [ D / + E ( E ) − ˜ µ ( E ) 4 M σ$_{µν}$F$_{µν}$ ] ψ ( x ) , (10) Consider now a spin-half baryon with charge Q . The magnetic moment, µ , is a sum of two terms, µ = Q + ˜ µ . The piece proportional to the charge is the Dirac magnetic moment, while that denoted by ˜ µ is the anomalous magnetic moment. Including terms relevant for a uniform external field, the relativistic single-particle action for a charged baryon has the form