Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals

TL;DR

The paper addresses the challenge of obtaining all-$n$ Laurent expansions in $\varepsilon$ for massive Feynman integrals with operator insertions, by representing the integrals as multi-integrals or multi-sums and deriving recurrences in $n$ via an enhanced Almkvist-Zeilberger algorithm and a holonomic/difference-field framework. It then solves these recurrences for the Laurent coefficients, enabling compact representations in terms of indefinite nested sums and products. Two fully-automatic computation pipelines are developed: one for integrals (MultiIntegrate) and one for sums (RhoSum), built on Sigma, HarmonicSums, and EvaluateMultiSums, including explicit handling of harmonic sums and zeta-values. These methods advance high-loop calculations by providing all-$n$ expansion terms in closed forms, suitable for applications in operator matrix elements and related QFT problems.

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.