On the photon self-energy to three loops in QED

Abstract

We compute the photon self-energy to three loops in Quantum Electrodynamics. The method of differential equations for Feynman integrals and a complete $ϵ$-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 $ϵ$. 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.

Comments: 25 pages, 3 figures, results in computer-readable form attached to the ancillary files. v2: Updated version including corrected plots and ancillary files, removing the wrong rescaling with p^2. Additionally, we have added the series expansion around p^2=0 of the fully renormalized photon propagator to three loops