Timelike Single-logarithm-resummed Splitting Functions
  We calculate the single logarithmic contributions to the quark singlet and gluon matrix of timelike splitting functions at all orders in the modified minimal-subtraction (MSbar) scheme. We fix two of the degrees of freedom of this matrix from the analogous results in the massive-gluon regularization scheme by using the relation between that scheme and the MSbar scheme. We determine this scheme transformation from the double logarithmic contributions to the timelike splitting functions and the coefficient functions of inclusive particle production in e+ e- annihilation now available in both schemes. The remaining two degrees of freedom are fixed by reasonable physical assumptions. The results agree with the fixed-order results at next-to-next-to-leading order in the literature. 
S. Albino, P. Bolzoni, B. A. Kniehl, A. V. Kotikov
hep-ph