On third-order timelike splitting functions and top-mediated Higgs decay into hadrons

We employ relations between spacelike and timelike deep-inelastic processes in perturbative QCD to calculate the next-to-next-to-leading order (NNLO) contributions to the timelike quark-quark and gluon-gluon splitting functions for the evolution of flavour-singlet fragmentation distributions. We briefly address the end-point behaviour and the numerical size of these third-order corrections, and write down the second moments of all four timelike splitting functions. In the same manner we re-derive the NNLO result for the Higgs-boson decay rate into hadrons in the limit of a heavy top quark and five massless flavours, and confirm the recent N3LO computation of this quantity.