Implementation of general background electromagnetic fields on a periodic hypercubic lattice

Abstract

Nonuniform background electromagnetic fields, once implemented in lattice quantum chromodynamics calculations of hadronic systems, provide a means to constrain a large class of electromagnetic properties of hadrons and nuclei, from their higher electromagnetic moments and charge radii to their electromagnetic form factors. We show how nonuniform fields can be constructed on a periodic hypercubic lattice under certain conditions and determine the precise form of the background U (1) gauge links that must be imposed on the quantum chromodynamics gauge-field configurations to maintain periodicity. Once supplemented by a set of quantization conditions on the background-field parameters, this construction guarantees that no nonuniformity occurs in the hadronic correlation functions across the boundary of the lattice. The special cases of uniform electric and magnetic fields, a nonuniform electric field that varies linearly in one spatial coordinate (relevant to the determination of quadruple moment and charge radii), nonuniform electric and magnetic fields with given temporal and spatial dependences (relevant to the determination of nucleon spin polarizabilities) and plane-wave electromagnetic fields (relevant to the determination of electromagnetic form factors) are discussed explicitly.

Nucleon in a periodic magnetic field

“…We expect that the alternative setting for the periodic magnetic field on the lattice (quantized B, ω not quantized), see Ref. [38], which has not been considered in the present paper, will be more advantageous for studying the limit ω → 0. Also, this different approach will allow one to continuously scan the interval of interest in the variable q 2 .…”

Nucleon in a periodic magnetic field: Finite-volume aspects

“…As discussed in Ref. [11], this quantization condition can be implemented on the lattice in two different ways. In the first scenario, the frequency ω is constrained and no constraint is imposed on the magnetic field strength B.…”

“…This is equivalent to setting F 1 = 1, F 2 = κ and β M = 0 in Eqs. (10,11). The subtraction function takes the value…”

“…Different scenarios for implementing periodic background fields in lattice QCD calculations are considered in Ref. [11].…”

The path from finite to infinite volume: Hadronic observables from lattice QCD

“…fields with sufficiently general space and time dependence is introduced in Ref. [66]. As an example, consider the case of an electric field of the form E = E 0 x 3 x 3 , produced by the gauge potential A µ = (−E 0 x 2 3 /2, 0, 0, 0).…”