Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals

TL;DR

This paper develops advanced algorithms combining multivariate Almkvist--Zeilberger and holonomic approaches to efficiently derive recurrence relations for Feynman integrals, enabling their expansion in terms of nested sums and products.

Contribution

It introduces an enhanced algorithmic framework for transforming Feynman integrals into recurrence relations and solving them to express expansions in nested sums and products.

Findings

Efficient recurrence relations for specific Feynman integrals are derived.

The method determines when Laurent series coefficients can be expressed in nested sums.

Compact solutions for the Laurent series coefficients are obtained without algebraic sum relations.

Abstract

Two-point Feynman parameter integrals, with at most one mass and containing local operator insertions in $4+\ep$-dimensional Minkowski space, can be transformed to multi-integrals or multi-sums over hyperexponential and/or hypergeometric functions depending on a discrete parameter $n$. Given such a specific representation, we utilize an enhanced version of the multivariate Almkvist--Zeilberger algorithm (for multi-integrals) and a common summation framework of the holonomic and difference field approach (for multi-sums) to calculate recurrence relations in $n$. Finally, solving the recurrence we can decide efficiently if the first coefficients of the Laurent series expansion of a given Feynman integral can be expressed in terms of indefinite nested sums and products; if yes, the all $n$ solution is returned in compact representations, i.e., no algebraic relations exist among the occurring sums and products.