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

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, $a_μ^{\text{HVP,LO}} \equiv (g-2)_μ^{\text{HVP,LO}}/2$. For calculations in a finite spatial volume with periodicity $L$, $a_μ^{\text{HVP,LO}}(L)$ admits a transseries expansion with exponentially suppressed $L$ scaling. Using a Hamiltonian approach, we show that the leading finite-volume correction scales as $\exp[- M_πL]$ 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, $F_π(Q^2)$, 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.

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