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

TL;DR

This paper details the computation and numerical evaluation of complex three-loop Feynman integrals for hadronic vacuum polarization within chiral perturbation theory, enabling precise calculations in quantum field theory.

Contribution

It provides a comprehensive method for calculating three-loop integrals and implements a numerical approach for their efficient evaluation in relevant physical scenarios.

Findings

Developed a practical numerical implementation for three-loop integrals.

Enabled fast and accurate evaluation for arbitrary complex photon virtuality.

Facilitated advanced calculations in hadronic vacuum polarization studies.

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.