their two-point functions are no longer time-translationally invariant. The correlator given in Eq. (11) is generally a function of the sink time-slice and not simply a function of the source-sink separation. The full dependence on source time is given in [20]. To implement the background field on the lattice, we modify the SU (3) color gauge links, U$_{µ}$ ( x ), for each quark flavor by multiplying by the color-singlet Abelian links, U ( E ) µ ( x ), for the external field, namely where U ( E ) µ ( x ) = exp[ iqA$_{µ}$ ( x )], with q as the quark electric charge in units of e > 0, and the vector potential is chosen to be: A$_{µ}$ ( x ) = (0 , 0 , −E $^{latt}$x$_{4}$ , 0), as in Eq. (1). The additional transverse links are given by U$_{µ}$ ( x ) −→ U$_{µ}$ ( x ) U ( E ) µ ( x ) U ( E )$_{⊥}$ µ ( x ) , (13) U ( E )$_{⊥}$ ( x ) = exp[ iq E $^{latt}$βx$_{3}$ δ$_{µ}$$_{4}$ δ$_{x}$$_{4}$$_{,β}$$_{−}$$_{1}$ ] . (14) These additional links together with the quantization condition for a torus [28] E latt = 2 πn | q$_{d}$ | βL , (15) E = ( e a$_{t}$a$_{s}$ ) − $^{1}$E $^{latt}$. (16) IV. LATTICE RESULTS Nucleon two-point functions were obtained for each source location on a given con- figuration. Results for multiple source locations on each configuration were then source averaged, yielding lattice correlation functions we denote by: g$_{±}$ ( x$_{4}$, n )$_{i}$. Here i labels the 7 As the gauge field multiplication in Eq. (13) is carried out on pre-existing gauge configura- tions, the sea quarks remain electrically neutral. This approximation is imposed because of computational restrictions. Using Eq. (13), we computed propagators for nine values of the field strength, n , corresponding to the integer appearing in the quantization condition, Eq. (15). We use n = 0, which corresponds to a vanishing external field, as well as n = ± 1, . . . , ± 4. On these lattices, the expansion parameter governing the deformation of a hadron’s pion cloud is given by [20]: ( e E /m 2 $_{π}$) 2 = 0 . 18 n $^{2}$. From the size of this parameter, we anticipate the need to include terms beyond quadratic order in the electric field expansion of hadron energies. In our analysis, we include terms up to quartic order. Larger lattices with their smaller allowed field strengths will be required for better control over systematic uncertainty relating to the electric field expansion of observables. ensure that the flux through every elementary plaquette is q$_{d}$ E latt for down quarks, and q$_{u}$ E latt for up quarks [29, 30]. In the quantization condition, q$_{d}$ = − 1 / 3 is the electric charge of the down quark, and n must take on integer values. The use of non-quantized fields have been investigated in [31], where it was found that boundary gradients in the field strength can lead to energy shifts in the bulk of the lattice as large as the sought after shifts due to the polarizabilities. Notice that E latt is given in lattice units, and the conversion to the physical electric field strength E is given by