J. Bijnens, P.A. Boyle, J. Harrison, N. Hermasson-Truedsson et. al.
Finite-size effects on the leading electromagnetic corrections to the hadronic vaccuum polarisation


We derive an analytic expression for the finite volume corrections to the hadronic vaccuum polarisation at next-to-leading order in the electromagnetic coupling. This is done in $QED_L$ and the leading term is found to be of order $c_{0}/L^3$. The physical argument why the a priori possible $1/L^2$ term vanishes is that the current in neutral and a photon far away thus sees no charge. To further motivate our result, we redo the calculation for charged currents and find that the $1/L^2$ term indeed does not vanish. We show that the result is universal, which implies that in other lattice formulations of QED where $c_{0} = 0$ the finite volume corrections are even more suppressed. We compare to lattice perturbation theory and lattice scalar U(1) gauge theory with stochastically generated photon fields, and there is good agreement between the calculations up to exponentially suppressed finite volume effect. For completeness we also calculate the hadronic vaccuum polarisation in infinite volume using a basis of 2-loop master integrals.

LU TP 18-30