Advanced Computer Algebra Algorithms for the Expansion of Feynman Integrals

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.

💡 Research Summary

The paper addresses the problem of analytically evaluating two‑point Feynman parameter integrals that appear in perturbative quantum field theory, specifically those defined in 4 + ε dimensional Minkowski space, containing at most one mass scale and local operator insertions. Such integrals are notoriously difficult to handle because they involve multiple continuous parameters and often require expansion in the dimensional regulator ε. The authors propose a systematic, computer‑algebra‑driven pipeline that transforms any integral of this class into either a multi‑integral or a multi‑sum representation built from hyper‑exponential or hyper‑geometric terms that depend on a discrete Mellin‑type variable n.

The first major component of the pipeline is an enhanced multivariate Almkvist–Zeilberger (AZ) algorithm tailored for multi‑integrals. While the classical AZ algorithm generates creative telescoping relations for single integrals, the authors extend it to handle several integration variables simultaneously. They introduce a multivariate difference operator that acts on all integration variables, automatically incorporates boundary contributions, and yields linear recurrence relations in n whose coefficients are rational functions of n and the auxiliary parameters. The resulting recurrences are of minimal order and can be solved symbolically.

The second component deals with multi‑sums. Here the authors combine the holonomic systems approach with the difference‑field (Π‑Σ) theory. By representing each hyper‑geometric summand as an element of a Π‑extension over a base difference field, they apply Σ‑extension techniques to construct a telescoping relation. This produces a linear recurrence in n for the whole sum, together with a certificate that guarantees correctness. The method is implemented using the Sigma package and related tools, allowing automatic handling of nested sums of arbitrary depth.

Once a recurrence has been obtained, the authors match its initial conditions to the coefficients of the Laurent expansion in ε. They compute the first few ε‑coefficients (typically ε⁰, ε¹, …) by direct series expansion of the original integral, then feed these values into the recurrence to obtain a closed‑form solution for each coefficient as a function of n. Crucially, they test whether the solution can be expressed in terms of indefinite nested sums and products (so‑called d‑finite expressions). This is done by checking algebraic independence of the occurring sums and products via Gröbner‑basis techniques in the underlying difference field. If the test succeeds, the algorithm returns a compact representation where no further algebraic relations exist among the building blocks.

The paper demonstrates the effectiveness of the method on several non‑trivial examples from high‑order perturbative calculations, including three‑loop self‑energy diagrams and two‑loop box integrals with massive propagators. In each case the algorithm reproduces known results, and in some instances it uncovers new nested‑sum structures that were not previously identified. The authors also discuss computational complexity, noting that the multivariate AZ step dominates the runtime for integrals with many variables, while the Σ‑Π step scales well with the depth of nesting.

In summary, the authors deliver a unified framework that (i) converts a broad class of two‑point Feynman integrals into hyper‑exponential/hyper‑geometric multi‑integrals or multi‑sums, (ii) derives minimal linear recurrences in the discrete parameter n using an extended Almkvist–Zeilberger algorithm and a Π‑Σ difference‑field approach, (iii) solves these recurrences to decide and construct explicit ε‑expansion coefficients in terms of indefinite nested sums and products, and (iv) provides a proof of algebraic independence for the resulting expressions. This work significantly advances the automation of high‑precision perturbative calculations and offers a powerful tool for both theoretical physicists and computer‑algebra researchers.