Finite volume corrections to the electromagnetic mass of composite particles

The long-range electromagnetic interaction presents a challenge for numerical computations in $\mathrm{QCD}+\mathrm{QED}$. In addition to power-law finite volume effects, the standard lattice gauge theory approach introduces nonlocality through removal of photon zero-momentum modes. The resulting finite volume effects must be quantitatively understood; and, to this end, nonrelativistic effective field theories are an efficient tool, especially in the case of composite particles. Recently an oddity related to nonlocality of the standard lattice approach was uncovered by the Budapest-Marseille-Wuppertal collaboration. Explicit contributions from antiparticles appear to be required so that finite volume QED results for a pointlike fermion can be reproduced in the effective field theory description. We provide transparency for this argument by considering pointlike scalars and spinors in finite volume QED using the method of regions. For the more germane case of composite particles, we determine that antiparticle modes contribute to the finite volume electromagnetic mass of composite spinors through terms proportional to the squares of timelike form factors evaluated at threshold. We extend existing finite volume calculations to one order higher, which is particularly relevant for the electromagnetic mass of light nuclei. Additionally, we verify that the analogous finite volume contributions to the nucleon mass in chiral perturbation theory vanish in accordance with locality.

Figure 1

Electromagnetic mass corrections at $\mathcal{O}(\alpha )$. Circles denote the relativistic charge and charge-squared couplings of the photon. The latter coupling is only present in the case of scalar electrodynamics.

Figure 2

The fully relativistic sunset diagram reduces to the antiparticle tadpole topology of the effective theory in the limit of large photon virtuality. The antiparticle tadpoles give the contributions in Eqs. (17) and (18) that are required for the effective field theory to reproduce the QED results.

Figure 3

Particle-antiparticle annihilation diagrams in the full and effective theory. The necessarily large photon virtuality reduces to a contact interaction in the effective theory.

Figure 4

Photon-mediated tadpole diagrams in the effective theory. The circles and squares denote the various photon couplings present in the NRQED Lagrangian density. For each possible square vertex, the sum of particle and antiparticle loops vanishes for a given circle vertex due to charge conjugation.

Figure 5

Strong nucleon-antinucleon annihilation.