Factorizing the position-space photon propagator in QED corrections to lattice QCD correlators

TL;DR

This paper introduces methods to efficiently compute electromagnetic corrections in lattice QCD by factorizing volume-sums in the photon propagator, improving computational performance for relevant correlators.

Contribution

It proposes integral representations of the photon propagator that enable factorization of volume-sums, including a novel autoconvolution approach, and compares their effectiveness in lattice QCD calculations.

Findings

Factorization reduces computational complexity.

Autoconvolution method shows promising performance.

Applicable to electromagnetic corrections in hadronic vacuum polarization.

Abstract

Electromagnetic corrections to the $n$-point functions of lattice QCD can be evaluated using a position-space photon propagator defined in infinite volume. Here we address the computational challenge arising from the volume-squared sum over the endpoints of the photon propagator. We consider a class of integral representations of the photon propagator that lead to a factorization of the two volume-sums, the Fourier representation being one instance thereof. An alternative choice is based on expressing the free scalar propagator as the autoconvolution of the corresponding five-dimensional propagator. We compare the performance of three different choices in the context of electromagnetic corrections to the hadronic vacuum polarization, on a gauge ensemble of size $48^3\times128$ with a pion mass of 286 MeV. As an outlook, we discuss more generally the factorization of sums over internal vertices, taking as an example the hadronic light-by-light contribution to the muon $(g-2)$.