Precision determination of the small-$x$ gluon from charm production at LHCb

The small-$x$ gluon in global fits of parton distributions is affected by large uncertainties from the lack of direct experimental constraints. In this work we provide a precision determination of the small-$x$ gluon from the exploitation of forward charm production data provided by LHCb for three different centre-of-mass (CoM) energies: 5~TeV, 7~TeV and 13~TeV. The LHCb measurements are included in the PDF fit by means of normalized distributions and cross-section ratios between data taken at different CoM values, $R_{13/7}$ and $R_{13/5}$. We demonstrate that forward charm production leads to a reduction of the PDF uncertainties of the gluon down to $x\simeq 10^{-6}$ by up to an order of magnitude, with implications for high-energy colliders, cosmic ray physics and neutrino astronomy.