Fuchsia reduces differential equations for Feynman master integrals to canonical form

## Project description

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].

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