fuchsia 16.7.25

Fuchsia reduces differential equations for Feynman master integrals to canonical form

Fuchsia reduces differential equations for Feynman master integrals to canonical form.

In concrete terms, let us say we have a system of differential equations of this form:

∂f(x,ϵ)/∂x = 𝕄(x,ϵ) f(x,ϵ)

where 𝕄(x,ϵ) is a given matrix of rational functions in x and ϵ, i.e, a free variable and an infinitesimal parameter. Our ultimately goal is to find a column vector of unknown functions f(x,ϵ) as a Laurent series in ϵ, which satisfies our equations.

With the help of Fuchsia we can find a transformation matrix 𝕋(x,ϵ) which turns our system to the equivalent Fuchsian system of this form:

∂g(x,ϵ)/∂x = ϵ 𝕊(x) g(x,ϵ)

where 𝕊(x) = ∑ᵢ 𝕊ᵢ/(x-xᵢ) and f(x,ϵ) = 𝕋(x,ϵ) g(x,ϵ).

Such a transformation is useful, because we can easily solve the equivalent system for g(x,ϵ) (see [1]) and then, multiplying it by 𝕋(x,ϵ), find f(x,ϵ).

You can learn about the algorithm used in Fuchsia to find such transformations from Roman Lee’s paper [2].

Fuchsia is available both as a command line utility and as a (Python) library for SageMath [3]. It will run on most Unix-like operating systems.

Documentation with more information, installation and usage details is here [4].

• [1] https://arxiv.org/abs/1304.1806

• [2] https://arxiv.org/abs/1411.0911

• [3] http://www.sagemath.org/

• [4] http://www.gituliar.net/fuchsia/fuchsia.pdf