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We discuss prompt production of $J/\psi$ mesons in proton-proton collisionsat the LHC within NRQCD $k_t$-factorization approach using Kimber-Martin-Ryskin(KMR) unintegrated gluon distributions (UGDF). We include both directcolor-singlet production ($g g \to J/\psi g$) as well as a feed-down from$\chi_c \to J/\psi \gamma$ and $\psi' \to J/\psi X$ decays. The production ofthe decaying mesons ($\chi_c$ or $\psi'$) is also calculated within NRQCD$k_t$-factorization approach. The corresponding matrix elements for $g g \toJ/\psi$ g, $g g \to \psi'g$ and $g g \to \chi_c$ include parameters of thenonrelativistic space wave functions of the quarkonia at $r = 0$, which aretaken from potential models in the literature. We get the ratio of thecorresponding cross section ratio for $\chi_c(2)$-to-$\chi_c(1)$ atmidrapidities much closer to experimental data than obtained in a recentanalysis. Differential distributions in rapidity and transverse momentum of$J/\psi$ and $\psi'$ are calculated and compared with experimental data of theALICE and LHCb collaborations. We discuss a possible onset of gluon saturationeffects in the production of $J/\psi$ and $\chi_{c}$ mesons at forward/backwardrapidities. We show that it is neccessary to modify the standard KMR UGDF todescribe ALICE and LHCb data. A mixed UGDF scenario was proposed. Then we candescribe the experimental data for $J/\psi$ production within modeluncertainties with color-singlet component only. Therefore our theoreticalresults leave only a relatively small room for the color-octet contributions.

physical review. d/physical review. d.

Prompt inclusive production of $J/\psi$, $\psi'$ and $\chi_{c}$ mesons  at the LHC in forward directions within the NRQCD $k_t$-factorization  approach - search for the onset of gluon saturation