On the photon self-energy to three loops in QED

TL;DR

This paper calculates the photon self-energy in QED up to three loops using differential equations and iterated integrals, providing analytical results and numerical expansions for all kinematic regions.

Contribution

It introduces a novel method employing differential equations and a complete epsilon-factorization to analytically compute three-loop photon self-energy in QED.

Findings

Analytical three-loop photon self-energy results obtained.

Series expansions valid across the entire kinematic space.

Photon wave function renormalization constant confirmed at three loops.

Abstract

We compute the photon self-energy to three loops in Quantum Electrodynamics. The method of differential equations for Feynman integrals and a complete $\epsilon$-factorization of the former allow us to obtain fully analytical results in terms of iterated integrals involving integration kernels related to a K3 geometry. We argue that our basis has the right properties to be a natural generalization of a canonical basis beyond the polylogarithmic case and we show that many of the kernels appearing in the differential equations, cancel out in the final result to finite order in $\epsilon$. We further provide generalized series expansions that cover the whole kinematic space so that our results for the self-energy may be easily evaluated numerically for all values of the momentum squared. From the local solution at $p^2=0$, we extract the photon wave function renormalization constant in the on-shell scheme to three loops and confirm its agreement with previously obtained results.