Multiloop integrals in dimensional regularization made simple

TL;DR

This paper introduces a method to simplify multi-loop Feynman integrals in dimensional regularization by choosing an optimal basis, transforming their differential equations into a canonical form for easier solutions.

Contribution

It proposes criteria for selecting an optimal basis of integrals, enabling straightforward solutions of differential equations in multi-loop calculations, extending techniques from supersymmetric theories to general quantum field theories.

Findings

Differential equations become elementary in canonical form.

Solutions are expressed in simple, compact functions.

Method applied successfully to a two-loop example.

Abstract

Scattering amplitudes at loop level can be expressed in terms of Feynman integrals. The latter satisfy partial differential equations in the kinematical variables. We argue that a good choice of basis for (multi-)loop integrals can lead to significant simplifications of the differential equations, and propose criteria for finding an optimal basis. This builds on experience obtained in supersymmetric field theories that can be applied successfully to generic quantum field theory integrals. It involves studying leading singularities and explicit integral representations. When the differential equations are cast into canonical form, their solution becomes elementary. The class of functions involved is easily identified, and the solution can be written down to any desired order in epsilon within dimensional regularization. Results obtained in this way are particularly simple and compact. In this letter, we outline the general ideas of the method and apply them to a two-loop example.