Forward light-by-light scattering and electromagnetic correction to hadronic vacuum polarization

TL;DR

This paper develops a method to accurately compute electromagnetic corrections to the hadronic vacuum polarization in lattice QCD by separating short- and long-distance effects using a scale and relating the long-distance part to light-by-light scattering amplitudes.

Contribution

It introduces a novel separation scale approach to handle UV and IR divergences in electromagnetic corrections to HVP and relates the long-distance contribution to dispersive sum rules involving gamma-gamma fusion.

Findings

Precisely determined the UV divergence using operator-product expansion.

Validated the dispersive sum rule by reproducing two-loop QED vacuum polarization.

Predicted the lattice-QCD integrand for the gamma* gamma* to pi0 process.

Abstract

Lattice QCD calculations of the hadronic vacuum polarization (HVP) have reached a precision where the electromagnetic (e.m.) correction can no longer be neglected. This correction is both computationally challenging and hard to validate, as it leads to ultraviolet (UV) divergences and to sizeable infrared (IR) effects associated with the massless photon. While we precisely determine the UV divergence using the operator-product expansion, we propose to introduce a separation scale $\Lambda\sim400\;$MeV into the internal photon propagator, whereby the calculation splits into a short-distance part, regulated in the UV by the lattice and in the IR by the scale $\Lambda$, and a UV-finite long-distance part to be treated with coordinate-space methods, thereby avoiding power-law finite-size effects altogether. In order to predict the long-distance part, we express the UV-regulated e.m. correction to the HVP via the forward hadronic light-by-light (HLbL) scattering amplitude and relate the latter via a dispersive sum rule to $\gamma^*\gamma^*$ fusion cross-sections. Having tested the relation by reproducing the two-loop QED vacuum polarization (VP) from the tree-level $\gamma^*\gamma^*\to e^+e^-$ cross-section, we predict the expected lattice-QCD integrand resulting from the $\gamma^*\gamma^*\to\pi^0$ process.