Inclusive production of vector quarkonia at the LHC

We discuss prompt production of $J/\psi$ mesons in proton-proton collisions at the LHC within NRQCD k$t$-factorization approach using Kimber-Martin-Ryskin (KMR) and Kutak-Stasto (KS) unintegrated gluon distribution (UGDF). We include both direct color-singlet production $gg \to J/\psi g$ as well as a feed-down from $\chi_{c} \to J/\psi \gamma$ and $\psi' \to J/\psi X$. The corresponding matrix elements for $gg \to J/\psi$, $gg \to \psi'$ and $gg \to \chi_{c}$ include parameters of the nonrelativistic space wave functions of the quarkonia at $r=0$, which are taken from potential models from the literature. We calculate the ratio of the corresponding cross sections for $\chi_{c2}/\chi_{c1}$. We compare our results with ATLAS experimental data. Differential distributions in rapidity and transverse momentum of $J/\psi$ and $\psi'$ are calculated and compared to experimental data of the ALICE and LHCb collaborations. We present results for three different values of energy 2.76 TeV, 7 TeV and 13 TeV.