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

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 (N3LO) splitting functions Pik(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.