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 (N3LO). From this, we extract the exact LL and NLL small-x contributions to the yet unknown N3LO splitting functions, both in the standard \( \overline{\mathrm{MS}} \) scheme and in the Q0\( \overline{\mathrm{MS}} \) scheme usually considered in small-x literature. We show that the impact of unknown subleading logarithmic contributions (NNLL and beyond) at N3LO is significant, thus motivating future work towards their computation. Our results will be also needed in future to match NLL resummation to N3LO 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.