On the photon self-energy to three loops in QED

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.

💡 Research Summary

In this paper the authors present a complete analytic calculation of the photon self‑energy Π(p²) in quantum electrodynamics (QED) up to three loops. The work combines modern techniques for handling multi‑loop Feynman integrals—namely the method of differential equations, ε‑factorized canonical bases, and Chen iterated integrals—with a detailed study of the underlying algebraic geometry, which in this case is a one‑parameter K3 surface associated with the equal‑mass three‑loop banana diagram.

The calculation begins by generating all one‑, two‑, and three‑loop 1PI photon‑self‑energy diagrams using QGRAF. After applying the transverse projector and performing Dirac algebra with FORM, the authors map the resulting scalar integrals onto three integral families (labelled A, B, and C). Integration‑by‑parts (IBP) reduction, carried out with Reduze and Kira, reduces the whole set to 36 master integrals (20 from family A and 16 from family B; family C does not introduce new masters).

A central achievement of the paper is the construction of an ε‑factorized differential‑equation system for these masters. Building on the algorithm of Ref.