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We present a complete evaluation for the prompt $\eta_c$ production at theLHC at next-to-leading order in $\alpha_s$ in nonrelativistic QCD. By assumingheavy quark spin symmetry, the recently observed $\eta_c$ production data byLHCb results in a very strong constraint on the upper bound of the color-octetlong distance matrix element $1S0$ of $J/\psi$. We find this upper bound isconsistent with our previous study of the $J/\psi$ yield and polarization andcan give good descriptions for the measurements, but inconsistent with someother theoretical estimates. This may provide important information forunderstanding the nonrelativistic QCD factorization formulism.

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