A Symbolic Summation Approach to Feynman Integral Calculus

TL;DR

This paper introduces a symbolic summation framework for efficiently computing the Laurent series coefficients of Feynman parameter integrals, transforming integrals into hypergeometric sums and developing new summation algorithms.

Contribution

It presents novel algorithms for expressing Feynman integrals as hypergeometric sums and extracting Laurent series coefficients using advanced summation techniques.

Findings

Successfully transforms Feynman integrals into hypergeometric sums.

Develops algorithms to compute Laurent series coefficients.

Enhances summation methods for complex boundary conditions.

Abstract

Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $\epsilon$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $\epsilon$. In a first step, the integrals are expressed by hypergeometric multi-sums by means of symbolic transformations. Given this sum format, we develop new summation tools to extract the first coefficients of its series expansion whenever they are expressible in terms of indefinite nested product-sum expressions. In particular, we enhance the known multi-sum algorithms to derive recurrences for sums with complicated boundary conditions, and we present new algorithms to find formal Laurent series solutions of a given recurrence relation.