Are there any challenges in the charmonia production and polarization at the LHC?

We analyze the inclusive prompt production of $\psi^\prime$, $\chi_c$, $J/\psi$ and $\eta_c$ mesons at the LHC using the $k_T$-factorization approach. Our consideration is based on the off-shell production amplitudes for hard partonic subprocesses, nonrelativistic QCD formalism for the formation of bound states and transverse momentum dependent (or unintegrated) gluon densities in a proton derived from Catani-Ciafaloni-Fiorani-Marchesini equation. The nonperturbative color octet transitions are treated in terms of the multipole radiation theory. We extract the corresponding long-distance matrix elements from a combined fit to transverse momentum distributions measured at various LHC experiments. We make predictions for the polarization parameters $\lambda_\theta$, $\lambda_\phi$, $\lambda_{\theta \phi}$ and the frame-independent parameter $\lambda^*$ and compare them to the available $\psi^\prime$ and $J/\psi$ data. Finally, we present a universal set of parameters that provides a reasonable simultaneous description for the whole body of the LHC data (on the $p_T$ distributions, relative production rates and polarization observables) for the whole charmonium family.