Four-loop splitting functions at small $x$

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.