Four-loop non-singlet splitting functions in the planar limit and beyond

TLDR: In this article, the authors presented the N3LO contributions to the non-singlet splitting functions for both parton distribution and fragmentation functions in perturbative QCD and derived exact expressions for the terms contributing in the limit of a large number of colours.

Abstract: We present the next-to-next-to-next-to-leading order (N3LO) contributions to the non-singlet splitting functions for both parton distribution and fragmentation functions in perturbative QCD. The exact expressions are derived for the terms contributing in the limit of a large number of colours. For the remaining contributions, approximations are provided that are sufficient for all collider-physics applications. From their threshold limits we derive analytical and high-accuracy numerical results, respectively, for all contributions to the four-loop cusp anomalous dimension for quarks, including the terms proportional to quartic Casimir operators. We briefly illustrate the numerical size of the four-loop corrections, and the remarkable renormalization-scale stability of the N3LO results, for the evolution of the non-singlet parton distribution and the fragmentation functions. Our results appear to provide a first point of contact of four-loop QCD calculations and the so-called wrapping corrections to anomalous dimensions in $$ \mathcal{N}=4 $$ super Yang-Mills theory.