fuchsia 16.6.21

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

In concrete terms, let’s say you have a system of differential equations of this form:

d/dx J = M(x, eps) * J,

where J is a column vector of unknown functions, M is a matrix of rational functions, x is a free variable, and eps is an infinitesimal parameter

Fuchsia will, if possible, transform this system to an equivalent Fuchsian system of this form:

d/dx J’ = eps * S(x) * J’

where S(x) = Sum_i { S_i / (x - x_i) }

… with the transformation itself defined by matrix T like this:

J = T * J’

Such a transformation is useful, because from it one can easily find J’ (and therefore J) as a series in eps.

You can learn about the algorithm Fuchsia uses to perform this transformation from Roman Lee’s paper at [1].

Fuchsia is available both as a command line utility and as a (Python) library for SageMath [2]. It will run on most Unix-like operating systems. You can learn about it’s installation and usage from [3].

[1] https://arxiv.org/abs/1411.0911v1 [2] http://www.sagemath.org/ [3] http://www.gituliar.net/fuchsia/fuchsia.pdf