Zimmermann's Forest Formula, Infrared Divergences and the QCD Beta Function

TL;DR

This paper reviews Zimmermann's forest formula and its extension to subtract both ultraviolet and infrared divergences, demonstrating its application in computing the five-loop QCD beta function and high-order Higgs decay rates.

Contribution

It introduces the R*-operation, a generalization of the R-operation, for efficient calculation of renormalization constants including infrared divergences.

Findings

Computed the five-loop QCD beta function using the R*-operation.

Applied the method to Higgs boson decay rates at N^4LO.

Demonstrated the efficiency of the generalized subtraction technique.

Abstract

We review Zimmermann's forest formula, which solves Bogoliubov's recursive $R$-operation for the subtraction of ultraviolet divergences in perturbative Quantum Field Theory. We further discuss a generalisation of the $R$-operation which subtracts besides ultraviolet also Euclidean infrared divergences. This generalisation, which goes under the name of the $R^*$-operation, can be used efficiently to compute renormalisation constants. We will discuss several results obtained by this method with focus on the QCD beta function at five loops as well as the application to hadronic Higgs boson decay rates at N${}^4$LO. This article summarizes a talk given at the Wolfhart Zimmermann Memorial Symposium.