Re-visiting Direct $J/\psi$ Production at the Fermilab Tevatron

We re-analyze the direct $\jp$ production processes at the Fermilab Tevatron in view of the recent observation at the B-factories, where both $\jp$ inclusive and exclusive production rates are found to be about an order of magnitude larger than the leading order estimates of non-relativistic QCD. The charm quark fragmentation to $\jp$, which is the dominant color-singlet process at high $p_T$, is normalized by the B-factory measurements. The process receives further enhancement due to the charm sea contribution which has so far been ignored in most analyses. After summing up all sub-process contributions, we find that the color-singlet mechanism alone can account for 20% to 90% of the observed direct $\jp$ high $p_T$ production. The polarization rate of the directly produced $\jp$ is sensitive to the fraction of the color-octet contribution, which is employed to fill the gap between the color-singlet prediction and experimental data. With a bigger $K$-factor for the charm quark fragmentation probability, we envisage a smaller matrix element for the color-octet $^3S_1^{(8)}$ state, and this can be examined at the LHC in near future.