f-feng / fuchsia

A tool for reducing differential equations for Feynman master integrals to an epsilon form.

Description

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),

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 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)

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].

Publications

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.

References

• [1] https://arxiv.org/abs/1304.1806
• [2] https://arxiv.org/abs/1411.0911
• [3] http://www.sagemath.org/
• [4] https://arxiv.org/abs/1701.04269