The R*-operation for Feynman graphs with generic numerators

TL;DR

This paper extends the R*-operation to Feynman graphs with arbitrary numerators, including tensors, providing a new way to compute divergences and applying it to high-loop calculations in QCD and phi^3 theory.

Contribution

The paper introduces a generalized R*-operation for Feynman graphs with arbitrary numerators and develops a novel infrared counterterm definition, enabling complex high-loop calculations.

Findings

Successfully computed the five-loop beta function in QCD.

Calculated poles of five-loop propagator graphs in phi^3 theory.

Reduced complexity of counterterms using symmetries and relations.

Abstract

The R*-operation by Chetyrkin, Tkachov, and Smirnov is a generalisation of the BPHZ R-operation, which subtracts both ultraviolet and infrared divergences of euclidean Feynman graphs with non-exceptional external momenta. It can be used to compute the divergent parts of such Feynman graphs from products of simpler Feynman graphs of lower loops. In this paper we extend the R*-operation to Feynman graphs with arbitrary numerators, including tensors. We also provide a novel way of defining infrared counterterms which closely resembles the definition of its ultraviolet counterpart. We further express both infrared and ultraviolet counterterms in terms of scaleless vacuum graphs with a logarithmic degree of divergence. By exploiting symmetries, integrand and integral relations, which the counterterms of scaleless vacuum graphs satisfy, we can vastly reduce their number and complexity. A FORM implementation of this method was used to compute the five loop beta function in QCD for a general gauge group. To illustrate the procedure, we compute the poles in the dimensional regulator of all top-level propagator graphs at five loops in four dimensional phi^3 theory.