Towards parton distribution functions with small-$x$ resummation: HELL 2.0

In global fits of parton distribution functions (PDFs) a large fraction of data points, mostly from the HERA collider, lies in a region of $x$ and $Q^2$ that is sensitive to small-$x$ logarithmic enhancements. Thus, the proper theoretical description of these data requires the inclusion of small-$x$ resummation. In this work we provide all the necessary ingredients to perform a PDF fit to deep-inelastic scattering (DIS) data which includes small-$x$ resummation in the evolution of PDFs and in the computation of DIS structure functions. To this purpose, not only we include the resummation of DIS massless structure functions, but we also consider the production of a massive final state (e.g. a charm quark), and the consistent resummation of mass collinear logarithms through the implementation of a variable flavour number scheme at small $x$. As a result, we perform the small-$x$ resummation of the matching conditions in PDF evolution at heavy flavour thresholds. The resummed results are accurate at next-to-leading logarithmic (NLL) accuracy and matched, for the first time, to next-to-next-to-leading order (NNLO). Furthermore, we improve on our previous work by considering a novel all-order treatment of running coupling contributions. These results, which are implemented in a new release of HELL, version 2.0, will allow to fit PDFs from DIS data at the highest possible theoretical accuracy, NNLO+NLL, thus providing an important step forward towards precision determination of PDFs and consequently precision phenomenology at the LHC and beyond.