We re-analyze Tavatron data on charmonium hadroproduction in the framework of the color-octet model implemented in the event generator PYTHIA taking into account initial-state radiation of gluons and Altarelli-Parisi evolution of final-state gluons fragmenting into $cbar{c}$ pairs. We obtain new values for the color-octet matrix elements relevant to this production process. We discuss the sensitivity of our results to the transverse momentum lower cut-off employed in the generation to avoid the problematic $p_t{to}0$ region, arguing about the reliability of our previous extraction of the NRQCD matrix elements for the $^3S_1^{(8)}$ and $^1S_0^{(8)}+^3P_J^{(8)}$ contributions. Finally we extrapolate to LHC energies to get predictions on the $J/psi$ direct production rate.
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