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 F2 structure function and of the Fϕ structure function in ϕ-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 (lni(1−x), i=4,5,6) to the four loop splitting function Pqg(3)(x) and Pgq(3)(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 CA=CF. 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 (lni(1−x), i=2,3) to Pqg(3)(x) and Pgq(3)(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 Pgq(2)T(x), which are related to similar terms in Pqg(2)(x).