Charmonium pair production at the LHCb in the ICEM using the $k_T$-factorization: DPS versus SPS

In the article, we study pair production of $\psi[1S]$ and $\psi[2S]$ charmonium states in the forward rapidity region of the LHCb experiment at the $\sqrt{s} = 13$ TeV using the improved color evaporation model and the parton Reggeization approach, which is a gauge invariant version of the $k_T$-factorization approach. Taking into account single and double parton scattering mechanisms and using the set of model parameters which have been fixed early, we compare our theoretical predictions with the recent LHCb experimental data as well as with another theoretical predictions. The numerical calculations are performed using the parton--level Monte-Carlo event generator KaTie and modified KMRW model for unintegrated parton distribution functions. We have found absolutely dominant role of the double parton scattering mechanism in the charmonium pair production at the LHCb.