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The particle production at very forward rapidities is expected to besensitive to the non-linear effects in the QCD dynamics at high energies. Inthis paper we present, for the first time, the predictions of the Color Dipoleformalism for the single inclusive jet photoproduction in $pp$ and $pPb$collisions considering the very forward rapidities probed by the CMS-CASTORcalorimeter, 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 CMSdetector and one of the incident hadrons remaining intact in the final state.The transverse momentum distributions are estimated considering the more recentphenomenological models for the dipole-proton scattering amplitude, which arebased on the Color Glass Condensate formalism and are able to describe theinclusive and exclusive $ep$ HERA data. Our results indicate that a futureexperimental analysis of this process is, in principle, feasible and useful toconstrain the description of the QCD dynamics at high energies.

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