The elliptic three-loop integrals of hadronic vacuum polarization in chiral perturbation theory

Abstract

This work presents a detailed account of the Feynman integrals required for the three-loop hadronic vacuum polarization calculation performed in arXiv:2510.12885. We explain how to compute each of the three-loop integrals, and outline the mathematical framework underlying their evaluation. This culminates in a practical numerical implementation that enables fast and accurate evaluation of these integrals for arbitrary complex values of the photon virtuality.

Comments: 40 pages; 16 figures; 1 table; implementation code available from https://github.com/mssjo/HVP-numerics