Finite-volume and thermal effects in the leading-HVP contribution to muonic $(g-2)$

The leading finite-volume and thermal effects, arising in numerical lattice QCD calculations of $ {a}_{\mu}^{\mathrm{HVP},\mathrm{LO}}\equiv {\left(g-2\right)}_{\mu}^{\mathrm{HVP},\mathrm{LO}}/2 $, are determined to all orders with respect to the interactions of a generic, relativistic effective field theory of pions. In contrast to earlier work [1] based in the finite-volume Hamiltonian, the results presented here are derived by formally summing all Feynman diagrams contributing to the Euclidean electromagnetic-current two-point function, with any number of internal pion loops and interaction vertices. As was already found in ref. [1], the leading finite-volume corrections to $ {a}_{\mu}^{\mathrm{HVP},\mathrm{LO}} $ scale as exp[−mL] where m is the pion mass and L is the length of the three periodic spatial directions. In this work we additionally control the two sub-leading exponentials, scaling as exp[−$ \sqrt{2} $mL] and exp[−$ \sqrt{3} $mL]. As with the leading term, the coefficient of these is given by the forward Compton amplitude of the pion, meaning that all details of the effective theory drop out of the final result. Thermal effects are additionally considered, and found to be sub-percent-level for typical lattice calculations. All finite-volume corrections are presented both for $ {a}_{\mu}^{\mathrm{HVP},\mathrm{LO}} $ and for each time slice of the two-point function, with the latter expected to be particularly useful in correcting small to intermediate current separations, for which the series of exponentials exhibits good convergence.