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.我们在此介绍Fuchsia——Lee算法的一种实现。针对给定的带有理系数的常微分方程组∂ₓJ(x,ε) = A(x,ε)J(x,ε)，该实现可求解基变换T(x,ε)（即满足J(x,ε)=T(x,ε)J′(x,ε)），使得该方程组转化为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，以维数调节参数ε的洛朗级数展开式实现平凡求解。因此，基变换T(x,ε)的构造对于获取初始方程组的解至关重要。原则上，Fuchsia可处理任意正则微分方程组，但其核心任务是化简费恩曼主积分对应的微分方程。依托费恩曼积分的固有性质，该工具可确保所得解仅包含正则奇点。