Fuchsia reduces differential equations for Feynman master integrals to the epsilon form.
To illustrate, let us say we have a system of differential equations of this form:
∂f(x,ϵ)/∂x = 𝕄(x,ϵ) f(x,ϵ),
𝕄(x,ϵ) is a given matrix of rational functions in
ϵ, i.e, a free variable and an infinitesimal parameter.
Our ultimate goal is to find a column vector of unknown functions
f(x,ϵ) as a Laurent series in
ϵ, which satisfies the equations above.
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)
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
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 .
Fuchsia has been announced in proceedings of The Loops and Legs Conference 2016, PoS LL2016 (2016) 030, arXiv:1607.00759.
Since then it is available on-line.
A complete description of the program has been published in Comput. Phys. Commun., arXiv:1701.04269.
If you find Fuchsia useful for your research, please, consider to cite both articles.