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

Abstract

We present $\text{Fuchsia}$ $-$ an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ${\partial }_x\mathbf{f}(x,\epsilon )=\mathbb{A}(x,\epsilon )\mathbf{f}(x,\epsilon )$ finds a basis transformation $\mathbb{T}(x,\epsilon )$, i.e., $\mathbf{f}(x,\epsilon )=\mathbb{T}(x,\epsilon )\mathbf{g}(x,\epsilon )$, such that the system turns into the epsilon form: ${\partial }_x\mathbf{g}(x,\epsilon )=\epsilon \mathbb{S}(x)\mathbf{g}(x,\epsilon )$, 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 $\epsilon$. That makes the construction of the transformation $\mathbb{T}(x,\epsilon )$ 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.

20 pages

References in corpus (8)

• Algorithm FIRE -- Feynman Integral REduction
• FIRE5: a C++ implementation of Feynman Integral REduction
• Lectures on differential equations for Feynman integrals
• Presenting LiteRed: a tool for the Loop InTEgrals REDuction
• Reducing differential equations for multiloop master integrals
• epsilon: A tool to find a canonical basis of master integrals
• Azurite: An algebraic geometry based package for finding bases of loop integrals
• Fuchsia and master integrals for splitting functions from differential equations in QCD

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