# Electromagnetic finite-size effects to the hadronic vacuum polarization

## TL;DR

This paper derives an analytic expression for electromagnetic finite-size effects on the hadronic vacuum polarization in lattice QED, crucial for improving muon g-2 predictions by reducing uncertainties.

## Contribution

It provides the first next-to-leading order analytical formula for finite-volume electromagnetic corrections to the hadronic vacuum polarization in scalar QED, confirming universality and agreement with numerical simulations.

## Key findings

- Leading finite-size correction is proportional to 1/L^3.
- No 1/L^2 term due to charge neutrality.
- Analytical results match numerical loop integrals and lattice simulations.

## Abstract

In order to reduce the current hadronic uncertainties in the theory prediction for the anomalous magnetic moment of the muon, lattice calculations need to reach sub-percent accuracy on the hadronic-vacuum-polarization contribution. This requires the inclusion of $\mathcal{O}(\alpha)$ electromagnetic corrections. The inclusion of electromagnetic interactions in lattice simulations is known to generate potentially large finite-size effects suppressed only by powers of the inverse spatial extent. In this paper we derive an analytic expression for the $\mathrm{QED}_{\mathrm{L}}$ finite-volume corrections to the two-pion contribution to the hadronic vacuum polarization at next-to-leading order in the electromagnetic coupling in scalar QED. The leading term is found to be of order $1/L^{3}$ where $L$ is the spatial extent. A $1/L^{2}$ term is absent since the current is neutral and a photon far away thus sees no charge and we show that this result is universal. Our analytical results agree with results from the numerical evaluation of loop integrals as well as simulations of lattice scalar $U(1)$ gauge theory with stochastically generated photon fields. In the latter case the agreement is up to exponentially suppressed finite-volume effects. For completeness we also calculate the hadronic vacuum polarization in infinite volume using a basis of 2-loop master integrals.