Algorithmic transformation of multi-loop Feynman integrals to a canonical basis

TL;DR

This paper presents an algorithm and a Mathematica package for transforming differential equations of multi-loop Feynman integrals into a canonical form, simplifying their computation in high-energy physics predictions.

Contribution

It introduces the first algorithm capable of computing rational transformations to canonical form for multi-variable differential equations in Feynman integrals, with an implementation in CANONICA.

Findings

Successfully transforms complex multi-loop integrals into canonical form.

Applies to problems with multiple kinematic variables and dimensional regulators.

Identifies new integral topologies relevant for LHC top-quark production.

Abstract

The evaluation of multi-loop Feynman integrals is one of the main challenges in the computation of precise theoretical predictions for the cross sections measured at the LHC. In recent years, the method of differential equations has proven to be a powerful tool for the computation of Feynman integrals. It has been observed that the differential equation of Feynman integrals can in many instances be transformed into a so-called canonical form, which significantly simplifies its integration in terms of iterated integrals. The main result of this thesis is an algorithm to compute rational transformations of differential equations of Feynman integrals into a canonical form. Apart from requiring the existence of such a rational transformation, the algorithm needs no further assumptions about the differential equation. In particular, it is applicable to problems depending on multiple kinematic variables and also allows for a rational dependence on the dimensional regulator. First, the transformation law is expanded in the dimensional regulator to derive differential equations for the coefficients of the transformation. Using an ansatz in terms of rational functions, these differential equations are then solved to determine the transformation. This thesis also presents an implementation of the algorithm in the Mathematica package CANONICA, which is the first publicly available program to compute transformations to a canonical form for differential equations depending on multiple variables. The main functionality and its usage are illustrated with some simple examples. Furthermore, the package is applied to state-of-the-art integral topologies appearing in recent multi-loop calculations. These topologies depend on up to three variables and include previously unknown topologies contributing to higher-order corrections to the cross section of single top-quark production at the LHC.