On the electron self-energy to three loops in QED

TL;DR

This paper calculates the electron self-energy in QED up to three loops using advanced mathematical techniques, providing compact analytic expressions and numerical evaluation methods, and revisiting three-loop renormalization constants.

Contribution

It introduces a novel approach using elliptic integrals and an epsilon-factorized basis for three-loop calculations in QED, enabling full control over differential forms.

Findings

Derived compact analytic expressions for three-loop electron self-energy.

Developed generalized series expansions for numerical evaluation.

Recomputed three-loop renormalization constants in the on-shell scheme.

Abstract

We compute the electron self-energy in Quantum Electrodynamics to three loops in terms of iterated integrals over kernels of elliptic type. We make use of the differential equations method, augmented by an $\epsilon$-factorized basis, which allows us to gain full control over the differential forms appearing in the iterated integrals to all orders in the dimensional regulator. We obtain compact analytic expressions, for which we provide generalized series expansion representations that allow us to evaluate the result numerically for all values of the electron momentum squared. As a by product, we also obtain $\epsilon$-resummed results for the self-energy in the on-shell limit $p^2 = m^2$, which we use to recompute the known three-loop renormalization constants in the on-shell scheme.