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Flavor Nonsinglet Splitting Functions at Four Loops in QCD: Fermionic Contributions

B. A. Kniehl ^1,*, S. Moch ^1,†, V. N. Velizhanin ^1,‡, and A. Vogt ^2,§

¹II. Institute for Theoretical Physics, Hamburg University, D-22761 Hamburg, Germany
²Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom

^*Contact author: bernd.kniehl@desy.de
^†Contact author: sven-olaf.moch@desy.de
^‡Contact author: vitaly.velizhanin@desy.de
^§Contact author: andreas.vogt@liverpool.ac.uk

Phys. Rev. Lett. 135, 071902 – Published 12 August, 2025

DOI: https://doi.org/10.1103/hkg5-88hr

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, $x ≲ 10^{- 6}$ . 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.

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Phys. Rev. Lett. 135, 071902– Published 12 August, 2025

Vol. 135, Iss. 7 — 15 August 2025

Received 18 May 2025
Revised 17 June 2025
Accepted 7 July 2025

DOI: https://doi.org/10.1103/hkg5-88hr

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Flavor Nonsinglet Splitting Functions at Four Loops in QCD: Fermionic Contributions

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