Low moments of the four-loop splitting functions in QCD

TL;DR

This paper calculates the first four moments of four splitting functions at four loops in QCD, showing the perturbative series is well-behaved for certain momentum fractions, aiding precision in parton distribution evolution.

Contribution

It provides the first four moments of four-loop splitting functions in QCD, offering analytical results for validation and future comprehensive calculations.

Findings

Perturbative expansion is well-behaved with small alpha_s effects.

Results are valid for momentum fractions x >~ 0.1.

Analytical expressions are provided for a general gauge group.

Abstract

We have computed the four lowest even-N moments of all four splitting functions for the evolution of flavour-singlet parton densities of hadrons at the fourth order in the strong coupling constant alpha_s. The perturbative expansion of these moments, and hence of the splitting functions for momentum fractions x >~ 0.1, is found to be well behaved with relative alpha_s-coefficients of order one and sub-percent effects on the scale derivatives of the quark and gluon distributions at alpha_s ~< 0.2. More intricate computations, including other approaches such as the operator-product expansion, are required to cover the full x-range relevant to LHC analyses. Our results are presented analytically for a general gauge group for detailed checks and validations of such future calculations.