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 ) 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 .
Fuchsia is available both as a command line utility and as a (Python) library for SageMath . It will run on most Unix-like operating systems.
Documentation with more information, installation and usage details is here .
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