Electromagnetic corrections to the hadronic vacuum polarization of the photon within QED$_{\rm L}$ and QED$_{\rm M}$

TL;DR

This paper calculates the leading QED corrections to the hadronic vacuum polarization of the photon, crucial for precise leptonic magnetic moment predictions, using lattice QCD with different finite-volume QED formulations.

Contribution

It provides a detailed comparison of QED$_{ m L}$ and QED$_{ m M}$ methods for computing electromagnetic corrections in lattice QCD, including finite-volume effects and dispersion relations.

Findings

QED corrections are at the percent level for vacuum polarization.

QED$_{ m L}$ and QED$_{ m M}$ yield consistent results within uncertainties.

Finite-volume effects are carefully analyzed and controlled.

Abstract

We compute the leading QED corrections to the hadronic vacuum polarization (HVP) of the photon, relevant for the determination of leptonic anomalous magnetic moments, $a_\ell$. We work in the electroquenched approximation and use dynamical QCD configurations generated by the CLS initiative with two degenerate flavors of non-perturbatively O($a$)-improved Wilson fermions. We consider QED$_{\rm L}$ and QED$_{\rm M}$ to deal with the finite-volume zero modes. We compare results for the Wilson loops with exact analytical determinations. In addition we make sure that the volumes and photon masses used in QED$_{\rm M}$ are such that the correct dispersion relation is reproduced by the energy levels extracted from the charged pions two-point functions. Finally we compare results for pion masses and the HVP between QED$_{\rm L}$ and QED$_{\rm M}$. For the vacuum polarization, corrections with respect to the pure QCD case, at fixed pion masses, turn out to be at the percent level.