Forward production of prompt neutrinos from charm in the atmosphere and at high energy colliders

The high-energy atmospheric neutrino flux is dominated by neutrinos from the decays of charmed hadrons produced in the forward direction by cosmic ray interactions with air nuclei. We evaluate the charm contributions to the prompt atmospheric neutrino flux as a function of the center-of-mass energy $\sqrt{s}$ of the hadronic collision and of the center-of-mass rapidity $y$ of the produced charm hadron. Uncertainties associated with parton distribution functions are also evaluated as a function of $y$. We find that the $y$ coverage of LHCb for forward heavy-flavour production, complemented by the angular coverage of present and future forward neutrino experiments at the LHC, bracket the most interesting $y$ regions for the prompt atmospheric neutrino flux. At $\sqrt{s}=14$ TeV foreseen for the HL-LHC phase, nucleon collisions in air contribute to the prompt neutrino flux prominently below $E_\nu\sim 10^7$~GeV. Measurements of forward charm and/or forward neutrinos produced in hadron collisions up to $\sqrt{s}=100$ TeV, which might become possible at the FCC, are relevant for the prompt atmospheric neutrino flux up to $E_\nu=10^8$ GeV and beyond.