Electromagnetic finite-size effects to the hadronic vacuum polarisation

TL;DR

This paper investigates finite-size effects in lattice QED calculations of the hadronic vacuum polarisation, showing that certain expected leading effects cancel out, leading to smaller finite-size corrections.

Contribution

It demonstrates that the leading finite-size effects of order 1/L^2 vanish for the hadronic vacuum polarisation in QED, due to current neutrality, and confirms this with numerical and lattice calculations.

Findings

Finite-size effects start at order 1/L^3 instead of 1/L^2.

The cancellation of 1/L^2 effects is universal, including form factors.

Numerical results agree with lattice perturbation theory and scalar QED simulations.

Abstract

In order to reach (sub-)per cent level precision in lattice calculations of the hadronic vacuum polarisation, isospin breaking corrections must be included. This requires introducing QED on the lattice, and the associated finite-size effects are potentially large due to the absence of a mass gap. This means that the finite-size effects scale as an inverse polynomial in $L$ rather than being exponentially suppressed. Considering the $\mathcal{O}(\alpha)$ corrected hadronic vacuum polarisation in QED$_{\mathrm{L}}$ with scalar QED as an effective theory, we show that the first possible term, which is of order $1/L^{2}$, vanishes identically so that the finite-size effects start at order $1/L^{3}$. This cancellation is understood from the neutrality of the currents involved, and we show that this cancellation is universal by also including form factors for the pions. We find good numerical agreement with lattice perturbation theory calculations, as well as, up to exponentially suppressed terms, scalar QED lattice simulations.