On the status of the 4-loop splitting functions in QCD

Abstract

We briefly discuss recent results on the four-loop splitting functions for the scale dependence of the parton distributions of hadrons. The three non-singlet cases are now completely known. The results agree with almost all expectations, but exhibit some small-$x$ double logarithms not encountered at lower orders. The flavour-singlet cases include, as already obvious from the present partial results, analytical structures not seen in the splitting functions up to now. The numerical effects (and their uncertainties, in the singlet cases) of the four-loop contributions to the scale derivatives amount to about $1\%$ or less down to momentum fractions $x$ as low as about $10^{-4}$ at a standard benchmark point with $α_{\rm s} = 0.2$, with the exception of the total valence quark distribution.