Low moments of the four-loop splitting functions in QCD

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.