Transforming differential equations of multi-loop Feynman integrals into canonical form

TL;DR

This paper introduces an algorithm that transforms differential equations of multi-loop Feynman integrals into a canonical form, simplifying calculations in quantum field theory, and demonstrates its effectiveness on complex examples.

Contribution

The paper presents a new algorithm for finding rational transformations to canonical bases of differential equations in multi-loop Feynman integrals, applicable to multi-scale problems.

Findings

Algorithm successfully transforms complex differential equations into canonical form.

Applicable to multi-scale problems with rational dependence on regulators.

Demonstrated effectiveness on non-trivial multi-loop examples.

Abstract

The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.