Four-loop splitting functions at small $x$

TL;DR

This paper expands small-$x$ resummed DGLAP splitting functions to four-loop order, extracting key contributions and highlighting the importance of subleading terms, while providing an improved resummation code for future research.

Contribution

It provides the first explicit extraction of LL and NLL small-$x$ contributions at N$^3$LO and introduces an improved resummation implementation in HELL 3.0.

Findings

Exact LL and NLL contributions at N$^3$LO are derived.

Subleading logarithmic effects at N$^3$LO are significant.

Updated resummation code HELL 3.0 is released.

Abstract

We consider the expansion of small-$x$ resummed DGLAP splitting functions at next-to-leading logarithmic (NLL) accuracy to four-loop order, namely next-to-next-to-next-to-leading order (N$^3$LO). From this, we extract the exact LL and NLL small-$x$ contributions to the yet unknown N$^3$LO splitting functions, both in the standard $\overline{MS}$ scheme and in the $Q_0 \overline{MS}$ scheme usually considered in small-$x$ literature. We show that the impact of unknown subleading logarithmic contributions (NNLL and beyond) at N$^3$LO is significant, thus motivating future work towards their computation. Our results will be also needed in future to match NLL resummation to N$^3$LO evolution. In turn, we propose an improved implementation of the small-$x$ resummation and therefore release a new version of the resummation code (HELL 3.0) which contains these changes.