# Finite-volume effects in $(g-2)^{\text{HVP,LO}}_\mu$

## TL;DR

This paper derives an analytic expression for the leading finite-volume effects in lattice QCD calculations of the muon's anomalous magnetic moment contribution from hadronic vacuum polarization, showing exponential suppression with volume size.

## Contribution

It provides a new quantitative formula for finite-volume corrections in lattice QCD for $(g-2)_rac{2}$, incorporating the pion form factor and chiral perturbation theory.

## Key findings

- Leading finite-volume correction scales as exp(-M_pi L)
- The correction depends on the pion Compton amplitude and decay constant
- Multi-particle contributions are negligible at NLO in chiral perturbation theory

## Abstract

An analytic expression is derived for the leading finite-volume effects arising in lattice QCD calculations of the hadronic-vacuum-polarization contribution to the muon's magnetic moment, $a_\mu^{\text{HVP,LO}} \equiv (g-2)_\mu^{\text{HVP,LO}}/2$. For calculations in a finite spatial volume with periodicity $L$, $a_\mu^{\text{HVP,LO}}(L)$ admits a transseries expansion with exponentially suppressed $L$ scaling. Using a Hamiltonian approach, we show that the leading finite-volume correction scales as $\exp[- M_\pi L]$ with a prefactor given by the (infinite-volume) Compton amplitude of the pion, integrated with the muon-mass-dependent kernel. To give a complete quantitative expression, we decompose the Compton amplitude into the space-like pion form factor, $F_\pi(Q^2)$, and a multi-particle piece. We determine the latter through NLO in chiral perturbation theory and find that it contributes negligibly and through a universal term that depends only on the pion decay constant, with all additional low-energy constants dropping out of the integral.