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

• Published in: The journal of high energy physics Vol. 2017; no. 4; pp. 1 - 43
• Main Author: Meyer, Christoph
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2017; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Canonical forms; Classical and Quantum Gravitation; Differential equations; Elementary Particles; Field theory; High energy physics; Integrals; Mathematical analysis; Mathematical models; Physics; Physics and Astronomy; QCD Phenomenology; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quantum theory; Regular Article - Theoretical Physics; Regulators; Relativity Theory; String Theory; Transformations; Transformations (mathematics); QCD Phenomenology
• ISSN: 1029-8479; 1126-6708; 1127-2236; 1029-8479
• DOI: 10.1007/JHEP04(2017)006

A bstract 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.