Complete $N_f^2$ contributions to four-loop pure-singlet splitting functions

Abstract

The scale evolution of parton distributions is determined by universal splitting functions. As a milestone towards the computation of these functions to four-loop order in QCD, we compute all contributions to the pure-singlet quark-quark splitting functions that involve two closed fermion loops. The splitting functions are extracted from the pole terms of off-shell operator matrix elements, and the workflow for their calculation is outlined. We reproduce known results for the non-singlet four-loop splitting functions and validate our new pure-singlet results against fixed Mellin moments.

26 pages, 2 figures, journal version

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• Constraints for twist-two alien operators in QCD
• Combination of aN${}^3$LO PDFs and implications for Higgs production cross-sections at the LHC
• Anomalous dimensions at small spins
• Four-loop anomalous dimension of flavor non-singlet quark operator of twist two and Lorentz spin N for general gauge group: transcendental part