Four-loop splitting functions in QCD – the gluon-gluon case –

We have computed the even- N moments N ≤ 20 of the gluon-gluon splitting function P gg at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general compact simple gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via structure functions in deep-inelastic scattering. Using our new moments and all available endpoint constraints, we construct improved approximations for the four-loop P gg ( x ) that should be sufficient for a wide range of collider-physics applications. The N 3 LO contributions to the scale derivative of the gluon distribution, resulting from these and the corresponding quark-to-gluon splitting functions, amount to 1% or less at at a standard reference scale with α s = 0.2 .