Soft Gluons in Logarithmic Summations

TL;DR

This paper unifies various logarithmic summation techniques in QCD using the Collins-Soper resummation method, deriving new unified equations for different kinematic regimes and large Bjorken x.

Contribution

It introduces a unified resummation framework that encompasses multiple known logarithmic summations in QCD through soft gluon approximations.

Findings

Derived $k_T$ resummation for two-scale processes.

Obtained unified evolution equations for large and small x.

Connected single- and double-logarithm summations in a single framework.

Abstract

We demonstrate that all the known single- and double-logarithm summations for a parton distribution function can be unified in the Collins-Soper resummation technique by applying soft approximations appropriate in different kinematic regions to real gluon emissions. Neglecting the gluon longitudinal momentum, we obtain the $k_T$ (double-logarithm) resummation for two-scale QCD processes, and the Balitsky-Fadin-Kuraev-Lipatov (single-logarithm) equation for one-scale processes. Neglecting the transverse momentum, we obtain the threshold (double-logarithm) resummation for two-scale processes, and the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (single-logarithm) equation for one-scale processes. If keeping the longitudinal and transverse momenta simultaneously, we derive a unified resummation for large Bjorken variable $x$, and a unified evolution equation appropriate for both intermediate and small $x$.