Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation

We have derived the coefficients of the highest three1/x-enhanced small-x logarithms of all timelike splitting functions and the coefficient functions for the transverse fragmentation function in one-particle inclusive e+e− annihilation at (in principle) all orders in massless perturbative QCD. For the longitudinal fragmentation function we present the respective two highest contributions. These results have been obtained from KLN-related decompositions of the unfactorized fragmentation functions in dimensional regularization and their structure imposed by the mass-factorization theorem. The resummation is found to completely remove the huge small-x spikes present in the fixed-order results, allowing for stable results down to very small values of the momentum fraction and scaling variable x. Our calculations can be extended to (at least) the corresponding αsnln2n−ℓx contributions to the above quantities and their counterparts in deep-inelastic scattering.