A Symbolic Summation Approach to Feynman Integral Calculus
Johannes Bluemlein, Sebastian Klein, Carsten Schneider, Flavia Stan
TL;DR
The paper addresses the problem of obtaining the first coefficients in the Laurent expansion in $\varepsilon$ for Feynman parameter integrals. It proposes a fully algebraic pipeline that converts integrals into hypergeometric multi-sums and then uses symbolic summation tools, including $WZ$-theory and $\Pi\Sigma$-fields, to derive recurrences and solve for $ε$-expansion coefficients in terms of indefinite nested sums/products (often harmonic or $S$-sums). A key contribution is the recurrence solver (Algorithm FLSR) for formal Laurent series and the extension of MultiSum with FSums to handle nonstandard ranges and inhomogeneous recurrences; a divide-and-conquer strategy enables practical extraction of low-order coefficients. The framework is demonstrated on two- and simple three-loop Wilson coefficients, highlighting both its power and its current computational limitations, and it promises broader applicability in symbolic evaluation of high-order quantum-field-theory integrals.
Abstract
Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $ε$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $ε$. 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.