Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations

TL;DR

The paper tackles the problem of computing the first coefficients of the Laurent expansion in $\varepsilon$ for solutions of coupled differential/difference systems that arise in high-loop Feynman integral calculations. It introduces a solver based on difference field/ring symbolic summation (Sigma) with two main tactics: expand under summation to obtain coefficient sums and extract from recurrences, plus uncoupling of coupled systems to reduce to solvable recurrences, aided by tools like EvaluateMultiSums and HarmonicSums. The authors demonstrate automatic transformations of multi-sums into expressions in terms of indefinite nested sums and products, provide recurrence-based methods to obtain expansions, and apply the framework to a challenging ladder diagram with six massive fermion lines within IBP/Laporta analyses. This work offers a practical, rigorous pipeline for automated ε-expansions of complex master integrals, enhancing the efficiency and reliability of QCD calculations at high perturbative orders.

Abstract

We outline a new algorithm to solve coupled systems of differential equations in one continuous variable $x$ (resp. coupled difference equations in one discrete variable $N$) depending on a small parameter $ε$: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in $ε$ if they are expressible in terms of indefinite nested sums and products. This systematic approach is based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms. The proposed method gives rise to new interesting applications in connection with integration by parts (IBP) methods. As an illustrative example, we will demonstrate how one can calculate the $ε$-expansion of a ladder graph with 6 massive fermion lines.