Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form

TL;DR

Fuchsia implements Lee's algorithm to transform differential equations for Feynman master integrals into epsilon form, enabling straightforward ε-expansions. It formalizes fuchsification, normalization, and factorization, with a block-triangular optimization to handle large, sparse systems. The paper details the mathematical framework, algorithmic steps, and practical usage via CLI and SageMath/Python, and discusses performance and limitations (Maple acceleration for heavy cases). This public tool thus provides a concrete, openly available pathway to efficiently compute master integrals by reducing their governing differential equations to a canonical epsilon form. The work facilitates scalable, symbolic solutions in high-energy physics computations.

Abstract

We present $\text{Fuchsia}$ $-$ an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $\partial_x\,\mathbf{f}(x,ε) = \mathbb{A}(x,ε)\,\mathbf{f}(x,ε)$ finds a basis transformation $\mathbb{T}(x,ε)$, i.e., $\mathbf{f}(x,ε) = \mathbb{T}(x,ε)\,\mathbf{g}(x,ε)$, such that the system turns into the epsilon form: $\partial_x\, \mathbf{g}(x,ε) = ε\,\mathbb{S}(x)\,\mathbf{g}(x,ε)$, where $\mathbb{S}(x)$ is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator $ε$. That makes the construction of the transformation $\mathbb{T}(x,ε)$ crucial for obtaining solutions of the initial equations. In principle, $\text{Fuchsia}$ can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals.