Calculation of Feynman integrals by difference equations

TL;DR

The paper tackles the challenging calculation of multi-loop Feynman integrals by reducing diagrams to a small set of master integrals using integration by parts and then solving systems of difference equations in one variable. It develops a factorial-series approach to obtain convergent solutions for these difference equations and demonstrates the method on one-loop and on-shell self-mass cases, generalizing to arbitrary topologies. A comprehensive program, SYS, automates master integral identification, system construction, and high-precision numerical evaluation with epsilon expansions, enabling high-precision results for vacuum, self-mass, vertex, and box diagrams, as well as electron g-2 computations. The approach yields series in a single variable with linear scaling in digits, providing a robust and scalable tool for precise, cross-checkable Feynman-integral calculations across dimensions and topologies.

Abstract

In this paper we describe a method of calculation of master integrals based on the solution of systems of difference equations in one variable. Various explicit examples are given, as well as the generalization to arbitrary diagrams.