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

TL;DR

This paper computes all two-fermion-loop contributions to four-loop pure-singlet splitting functions in QCD, advancing the precision of parton distribution evolution calculations.

Contribution

It provides the first complete calculation of these specific four-loop pure-singlet contributions, filling a key gap in high-order QCD splitting functions.

Findings

Reproduces known non-singlet four-loop splitting functions

Validates new pure-singlet results with fixed Mellin moments

Advances the understanding of four-loop QCD evolution equations

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.