  Models with essential non-perturbative QCD dynamics for the production of
charmonium are extrapolated to give predictions of prompt J/psi production at
the LHC. Differences of up to an order of magnitude occurs, although the
different models all describe the Tevatron data on high-pT charmonium. An
important point is here the treatment of higher order perturbative QCD effects.
We also discuss the large rate of prompt J/psi from these models as a
background to CP violation studies.
