On threshold resummation of singlet structure and fragmentation functions

TL;DR

This paper investigates the threshold resummation properties of singlet structure and fragmentation functions at large-x, establishing leading logarithmic resummation validity and exploring its breakdown at higher orders, with implications for splitting functions.

Contribution

It demonstrates the validity of leading logarithmic threshold resummation for singlet kernels and explores its limitations at next-to-leading order, providing predictions for higher-order splitting functions.

Findings

Leading logarithmic resummation holds for singlet kernels at large-x.

Next-to-leading logarithmic resummation breaks down at three loops, except in supersymmetric cases.

Predictions for three-loop splitting functions based on resummation assumptions.

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_{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).