Flavour Non-Singlet Splitting Functions at Four Loops in QCD -- The Fermionic Contributions

TL;DR

This paper calculates the four-loop fermionic contributions to flavor non-singlet splitting functions in QCD, providing detailed analytic forms and implications for small and large momentum fractions, with applications to Higgs production.

Contribution

It presents the first complete analytic expressions for four-loop $n_f$ contributions to flavor non-singlet splitting functions in QCD, including small- and large-$x$ limits.

Findings

Analytic forms of splitting functions in Mellin and $x$-space.

Small-$x$ rise confined to $x \,\lesssim\, 10^{-6}$.

Includes $n_f$-part of four-loop quark anomalous dimension.

Abstract

We have determined the fourth-order $n_f$ contributions to the two splitting functions governing the evolution of all flavor differences of quark distributions of hadrons in perturbative quantum chromodynamics with $n_f$ light flavors. The analytic forms of these functions are presented in both Mellin $N$-space and momentum-fraction $x$-space for a general gauge group. In the small-$x$ limit double logarithms occur, but the small-$x$ rise of both splitting functions is confined to extremely small $x$-values, $x \lesssim 10^{-6}$. The large-$x$ limit includes the $n_f$-part of the four-loop quark virtual anomalous dimension. Using this result we obtain also the $n_f$ contributions to the corresponding gluonic quantity and the complete threshold-enhanced logarithms from soft-gluon emission for a large class of inclusive observables, including Higgs boson production in gluon-gluon fusion.