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

We present Fuchsia — an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients ∂xJ(x,ϵ) = A(x,ϵ)J(x,ϵ) finds a basis transformation T(x,ϵ), i.e., J(x,ϵ) = T(x,ϵ)J′(x,ϵ), such that the system turns into the epsilon form: ∂xJ′(x,ϵ) = ϵS(x)J′(x,ϵ), where 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 T(x,ϵ) crucial for obtaining solutions of the initial system.

In principle, 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.

我们提出了Fuchsia——一种李算法（Lee algorithm）的实现。针对给定的有理系数常微分方程组∂ₓJ(x,ε) = A(x,ε)J(x,ε)，该算法可求解基变换T(x,ε)（即满足J(x,ε) = T(x,ε)J′(x,ε)），使得原方程组转化为ε形式：∂ₓJ′(x,ε) = εS(x)J′(x,ε)，其中S(x)为富克斯矩阵（Fuchsian matrix）。此类形式的方程组可借助多重对数函数（polylogarithms），以维度规整子ε的洛朗级数实现平凡求解，因此构造变换T(x,ε)是获取原方程组解的核心环节。

原则上，Fuchsia可处理任意正则方程组，但其首要任务是化简费恩曼主积分（Feynman master integrals）对应的微分方程。依托费恩曼积分的性质，该工具可保证解仅包含正则奇点（regular singularities）。