$\eta_c$ production at LHC and indications on the understanding of $J/\psi$ production

We present a complete evaluation for the prompt $\eta_c$ production at the LHC at next-to-leading order in $\alpha_s$ in nonrelativistic QCD. By assuming heavy quark spin symmetry, the study of $\eta_c$ production results in a very strong constraint on the upper bound of the color-octet long distance matrix element $\mopa$ of $J/\psi$. We find this upper bound is consistent with our previous study of the $J/\psi$ yield and polarization and can give good descriptions for the measurements, but in conflict with most other theoretical studies. This may provide important information for understanding the mechanism of charmonium production.