  Recently, several surprising experimental observations in the production of
heavy quarkonium have been reported. In $e^+e^-$ annihilation at
$\sqrt{s}=10.6$ GeV, Belle Collaboration finds that $J/\psi$ mesons are
predominantly produced in association with an extra $\bar{c}c$ pair, with
$\sigma(e^+e^- \to J/\psi \bar{c}c) / \sigma(e^+e^- \to J/\psi X) =
0.59^{+0.15}_{-0.13}\pm 0.12$, and the BaBar collaboration reports that the
produced $J/\psi$'s have mostly longitudinal polarization. In $\bar{p}p$
collisions at the Tevatron, the CDF Collaboration reported an excess of
$J/\psi$ and $\psi^{\prime}$ mesons at high $p_{\perp}$ over the perturbative
QCD predictions; non--perturbative approach of NRQCD can accomodate the
magnitude of the production cross section but not the observed experimentally
polarization of quarkonia. In this note we propose possible solutions to these
puzzles, and devise further experimental tests.
