Polarization of $J/\psi$ and $psi(2S)$ Mesons Produced in $p\bar{p}$ Collisions at 1.96 TeV

We have measured the polarizations of $\jpsi$ and $\psiprime$ mesons as functions of their transverse momentum $\pt$ when they are produced promptly in the rapidity range $|y|<0.6$ with $\pt \geq 5 \pgev$. The analysis is performed using a data sample with an integrated luminosity of about $800 \ipb$ collected by the CDF II detector. For both vector mesons, we find that the polarizations become increasingly longitudinal as $\pt$ increases from 5 to $30 \pgev$. These results are compared to the predictions of non-relativistic quantum chromo-dynamics and other contemporary models. The polarizations of $\jpsi$ and $\psiprime$ mesons from $B$-hadron decays are also reported.