Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions

We have investigated the next-to-next-to-leading order (NNLO) corrections to inclusive hadron production in e^+e^- annihilation and the related parton fragmentation distributions, the time-like' counterparts of the space-like’ deep-inelastic structure functions and parton densities. We have re-derived the corresponding second-order coefficient functions in massless perturbative QCD, which so far had been calculated only by one group. Moreover we present, for the first time, the third-order splitting functions governing the NNLO evolution of flavour non-singlet fragmentation distributions. These results have been obtained by two independent methods relating time-like quantities to calculations performed in deep-inelastic scattering. We briefly illustrate the numerical size of the NNLO corrections, and make a prediction for the difference of the yet unknown time-like and space-like splitting functions at the fourth order in the strong coupling constant.