On threshold resummation of singlet structure and fragmentation functions

Abstract

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 ϕ 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 ( ln i ( 1 − x ) , i = 4 , 5 , 6 ) to the four loop splitting function P q g ( 3 ) ( x ) and P g q ( 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 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 q g ( 3 ) ( x ) and P g q ( 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 P g q ( 2 ) T ( x ) , which are related to similar terms in P q g ( 2 ) ( x ) .