On threshold resummation of the longitudinal structure function

TL;DR

This paper evaluates a momentum space ansatz for threshold resummation of the non-singlet longitudinal structure function F_L, finding it fails beyond leading logs at three-loop order, indicating limitations of the ansatz.

Contribution

It provides a detailed analysis of the ansatz's validity at higher orders and clarifies its limitations in threshold resummation for F_L.

Findings

The ansatz does not hold beyond leading logarithmic contributions at three loops.

Corrections at order (1-x)^0 also show failure of the ansatz at leading logs.

Failure is linked to specific terms responsible for the breakdown at different orders.

Abstract

The validity of a previously proposed momentum space ansatz for threshold resummation of the non-singlet longitudinal structure function F_L is checked against existing finite order three-loop results. It is found that the ansatz, which is an assumption for the large-x behavior of the physical evolution kernel, does not work beyond the leading logarithmic contributions to the kernel even at {\cal O}(1/(1-x)) order (except at large-\beta_0), which is consistent with a recent observation of Moch and Vogt. Corrections down by one power of 1-x are also studied. At {\cal O}((1-x)^0) order, the corresponding ansatz fails already at the leading logarithmic level, where the situation appears similar to that encountered in the case of the F_i (i=1,2,3) structure functions. At the next-to-leading logarithmic level, the same term (with opposite sign) responsible for the failure of the ansatz at {\cal O}(1/(1-x)) order is found to occur.