Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond

TL;DR

This paper computes four-loop non-singlet splitting functions in QCD, providing exact results in the large number of colours limit and approximations for other contributions, enhancing precision in collider physics predictions.

Contribution

It presents the first four-loop non-singlet splitting functions in QCD, including exact large-Nc expressions and approximations, advancing high-order perturbative calculations.

Findings

Derived four-loop cusp anomalous dimension for quarks.

Demonstrated numerical stability of N^3LO results.

Provided approximations suitable for collider applications.

Abstract

We present the next-to-next-to-next-to-leading order (N^3LO) 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 N^3LO 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 N=4 super Yang-Mills theory.