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Single inclusive jet photoproduction at very forward rapidities in $pp$ and $pPb$ collisions at the LHC

The particle production at very forward rapidities is expected to be sensitive to the non-linear effects in the QCD dynamics at high energies. In this paper we present, for the first time, the predictions of the Color Dipole formalism for the single inclusive jet photoproduction in $pp$ and $pPb$ collisions considering the very forward rapidities probed by the CMS-CASTOR calorimeter, which will be characterized by a jet in the rapidity range of $5.2 \le Y \le 6.6$, a rapidity gap in the rapidity range probed by the central CMS detector and one of the incident hadrons remaining intact in the final state. The transverse momentum distributions are estimated considering the more recent phenomenological models for the dipole-proton scattering amplitude, which are based on the Color Glass Condensate formalism and are able to describe the inclusive and exclusive $ep$ HERA data. Our results indicate that a future experimental analysis of this process is, in principle, feasible and useful to constrain the description of the QCD dynamics at high energies.