We consider $J/psi$ photoproduction in ion--ion ultraperipheral collisions (UPCs) at the LHC and RHIC in the coherent and incoherent quasielastic channels with and without accompanying forward neutron emission and analyze the role of nuclear gluon shadowing at small $x$, $x=10^{-4}-10^{-2}$, in these processes. We find that despite the good agreement between large nuclear gluon shadowing and the ALICE data in the coherent channel, in the incoherent channel, the leading twist approximation predicts the amount of nuclear suppression which is by approximately a factor of $1.5$ exceeds that seen in the data. We hypothesize that part of the discrepancy can be accounted for by the incoherent inelastic process of $J/psi$ photoproduction with nucleon dissociation. To separate the high-photon-energy and low-photon-energy contributions to the $d sigma_{AAto AAJ/psi}(y)/dy$ cross section, we consider ion--ion UPCs accompanied by neutron emission due to electromagnetic excitation of one or both colliding nuclei. We describe the corresponding PHENIX data and make predictions for the LHC kinematics. In addition, in the incoherent quasielastic case, we show that the separation between the low-photon-energy and high-photon-energy contributions can be efficiently performed by measuring the correlation between the directions of $J/psi$ and the emitted neutrons.
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