On effects of multiple gluons in $J/\psi$ hadroproduction

The three-gluon contribution to $J/\psi$ hadroproduction is calculated within perturbative QCD in the $k_T$ factorization framework. This mechanism involves double gluon density and enters at a non-leading twist, but it is enhanced at large energies due to large double gluon density at small $x$. We obtain results for differential $p_T$-dependent cross-sections for all $J/\psi$ polarisations. The rescattering contribution is found to provide a significant correction to the standard leading twist cross-section at the energies of the Tevatron or the LHC at moderate $p_T$. We also discuss a possible contribution of the rescattering correction to the anti-shadowing effect for $J/\psi$ production in proton - nucleus collisions.