Large-nf Contributions to the Four-Loop Splitting Functions in QCD

TL;DR

This paper calculates the fourth-order nf^2 contributions to three non-singlet and their four nf^3 flavor-singlet splitting functions in QCD, providing analytic forms and discussing their limits, advancing precision in parton distribution evolution.

Contribution

It presents the first complete analytic expressions for four-loop nf^2 contributions to splitting functions in QCD, including the cusp anomalous dimension, with results consistent with previous predictions.

Findings

Analytic forms of four-loop nf^2 splitting functions in Mellin and x-space.

Agreement with all previous lower-order predictions.

Complete nf^2 part of the four-loop cusp anomalous dimension.

Abstract

We have computed the fourth-order nf^2 contributions to all three non-singlet quark-quark splitting functions and their four nf^3 flavour-singlet counterparts for the evolution of the parton distributions of hadrons in perturbative QCD with nf effectively massless quark flavours. The analytic form of these functions is presented in both Mellin N-space and momentum-fraction x-space; the large-x and small-x limits are discussed. Our results agree with all available predictions derived from lower-order information. The large-x limit of the quark-quark cases provides the complete nf^2 part of the four-loop cusp anomalous dimension which agrees with two recent partial computations.