Large-n contributions to the four-loop splitting functions in QCD

We have computed the fourth-order nf2 contributions to all three non-singlet quark–quark splitting functions and their four nf3 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 nf2 part of the four-loop cusp anomalous dimension which agrees with two recent partial computations.