New results on small-x resummation for splitting functions

TL;DR

This paper advances small-$x$ resummation techniques for splitting functions by deriving new analytical results and implementing a more robust, numerically stable resummation framework within HELL 4.0.

Contribution

It introduces new all-order analytical results for key anomalous dimensions and Green functions, enabling a properly resummed $qg$ splitting kernel for the first time.

Findings

New analytical all-order results for anomalous dimensions and Green functions.

Implementation of a more stable and accurate small-$x$ resummation in HELL 4.0.

Enhanced numerical behavior of the resummation framework.

Abstract

We revisit the basic steps necessary to obtain next-to-leading-logarithmic accurate small-$x$ results for the DGLAP splitting functions, and their implementations within the HELL framework. We derive new analytical all-order results for the leading-logarithmic $gg$ anomalous dimension, the $qg$ and $gg$ finite Green functions, and most importantly for the $qg$ anomalous dimension, which allows us to arrive for the first time at a properly resummed $qg$ splitting kernel. We use these results as cornerstones of a new implementation of small-$x$ splitting-function resummation which is more solid and numerically better behaved with respect to those available thus far. All of these novelties are included in the upcoming 4.0 version of HELL.