Feynman integrals and difference equations

TL;DR

This paper presents an algorithmic method using difference equations in Mellin space to compute multi-loop Feynman integrals, with implementation tested via a Mathematica package in QCD calculations.

Contribution

It introduces a novel approach employing Pi-Sigma-fields to solve difference equations for Feynman integrals, enabling systematic calculations in perturbative QCD.

Findings

Successful implementation of the Sigma package for multi-loop integrals

Efficient calculation of single-scale Feynman integrals in Mellin space

Validation against recent higher order QCD results

Abstract

We report on the calculation of multi-loop Feynman integrals for single-scale problems by means of difference equations in Mellin space. The solution to these difference equations in terms of harmonic sums can be constructed algorithmically over difference fields, the so-called Pi-Sigma-fields. We test the implementaion of the Mathematica package Sigma on examples from recent higher order perturbative calculations in Quantum Chromodynamics.