On quartic colour factors in splitting functions and the gluon cusp anomalous dimension

TL;DR

This paper computes quartic Casimir contributions to four-loop anomalous dimensions and splitting functions, providing new insights into the structure of higher-order QCD corrections and an approximate gluon cusp anomalous dimension for phenomenology.

Contribution

It introduces the first calculation of quartic Casimir invariants in four-loop anomalous dimensions and relates these to splitting functions and the gluon cusp anomalous dimension.

Findings

Quartic Casimir contributions to four-loop anomalous dimensions computed.

Approximate expressions for N^3LO splitting functions including quartic-Casimir effects.

An approximate four-loop gluon cusp anomalous dimension suitable for phenomenology.

Abstract

We have computed the contributions of the quartic Casimir invariants to the four-loop anomalous dimensions of twist-2 spin-N operators at N =< 16. The results provide new information on the structure of the next-to-next-to-next-to-leading order (N^3LO) splitting functions P_{ik}^(3)(x) for the evolution of parton distributions, and facilitate approximate expressions which include the quartic-Casimir contributions to the (light-like) gluon cusp anomalous dimension. These quantities turn out to be closely related, by a generalization of the lower-order `Casimir scaling', to the corresponding quark results. Using these findings, we present an approximate result for the four-loop gluon cusp anomalous dimension in QCD which is sufficient for phenomenological applications.