On threshold resummation of singlet structure and fragmentation functions

The large-x behavior of the physical evolution kernels appearing in the second order evolution equations of the singlet F_2 structure function and of the F_{phi} structure function in phi-exchange DIS is investigated. The validity of a leading logarithmic threshold resummation, analogous to the one prevailing for the non-singlet physical kernels, is established, allowing to recover the predictions of Soar et al. for the double-logarithmic contributions (ln^i(1-x), i=4,5,6) to the four loop splitting function P^{(3)}_{qg}(x) and P^{(3)}_{gq}(x). Threshold resummation at the next-to-leading logarithmic level is found however to break down in the three loop kernels, except in the "supersymmetric" case C_A=C_F. Assuming a full threshold resummation does hold in this case also beyond three loop gives some information on the leading and next-to-leading single-logarithmic contributions (ln^i(1-x), i=2,3) to P^{(3)}_{qg}(x) and P^{(3)}_{gq}(x). Similar results are obtained for singlet fragmentation functions in e^+e^- annihilation up to two loop, where a large-x Gribov-Lipatov relation in the physical kernels is pointed out. Assuming this relation also holds at three loop, one gets predictions for all large-x logarithmic contributions to the three loop timelike splitting function P^{(2)T}_{gq}(x), which are related to similar terms in P^{(2)}_{qg}(x).