Fuchsia and master integrals for splitting functions from differential equations in QCD

TL;DR

The paper advances the automated reduction of differential equations for Feynman master integrals to canonical form using Roman Lee's method and introduces Fuchsia, an open-source Python/SageMath tool. It applies the approach to NLO time-like QCD splitting-function calculations, formulating real-virtual systems as $\partial f/\partial x = \mathbb{M}(x,\epsilon) f$ and deriving the matrix from LiteRed. The authors detail a three-step pipeline—fuchsification, normalization, and factorization—to obtain a canonical, ε-multiplicative form via residue analysis, $\mathbb{P}$-balances, and a constant transformation, illustrating the method with explicit example systems. They discuss performance on sizable sparse matrices, acknowledge current limitations in complex-coefficient polynomial factorization within SageMath, and outline future directions toward multivariate and multi-scale Feynman integrals.

Abstract

We report on the recent progress in reducing differential equations for Feynman master integrals to canonical form with the help of a method proposed by Roman Lee. For the first time, we present Fuchsia --- our open-source implementation of the Lee algorithm written in Python using mathematical routines of a free computer algebra system SageMath. We demonstrate Fuchsia by reducing differential equations for NLO contributions to splitting functions in QCD, which contain both loops and legs integrals.