Prompt $\eta_c$ meson production at the LHC in the NRQCD with $k_T$-factorization

In the framework of the $k_T$-factorization, the prompt production of $\eta_c$ mesons at the LHC conditions is studied. Our consideration is based on the off-shell amplitudes for hard partonic subprocesses and on the nonrelativistic QCD (NRQCD) formalism for the formation of bound states. We try two latest parametrizations for noncollinear, or transverse momentum dependent (TMD) gluon densities derived from the Catani-Ciafaloni-Fiorani-Marchesini (CCFM) equation. We use the values of the nonperturbative matrix elements obtained from a combined fit of the $\eta_c$ and $J/\psi$ differential cross sections. Finally, we show an universal set of parameters that provides a reasonable simultaneous description for all of the available data on the prompt $J/\psi$ and $\eta_c$ production at the LHC.