Intrinsic charm in the nucleon and charm production at large rapidities in collinear, hybrid and kT-factorization approaches

We discuss the role of intrinsic charm (IC) in the nucleon for forward production of c-quark (or $$ \overline{c} $$ c ¯ -antiquark) in proton-proton collisions for low and high energies. The calculations are performed in collinear-factorization approach with on-shell partons, kT-factorization approach with off-shell partons as well as in a hybrid approach using collinear charm distributions and unintegrated (transverse momentum dependent) gluon distributions. For the collinear-factorization approach we use matrix elements for both massless and massive charm quarks/antiquarks. The distributions in rapidity and transverse momentum of charm quark/antiquark are shown for a few different models of IC. Forward charm production is dominated by gc-fusion processes. The IC contribution dominates over the standard pQCD (extrinsic) gg-fusion mechanism of $$ c\overline{c} $$ c c ¯ -pair production at large rapidities or Feynman-xF. We perform similar calculations within leading-order and next-to-leading order kT-factorization approach. The kT-factorization approach leads to much larger cross sections than the LO collinear approach. At high energies and large rapidities of c-quark or $$ \overline{c} $$ c ¯ -antiquark one tests gluon distributions at extremely small x. The IC contribution has important consequences for high-energy neutrino production in the Ice-Cube experiment and can be, to some extent, tested at the LHC by the SHIP and FASER experiments by studies of the ντ neutrino production.