  Dominant spin-flip effects for the direct and prompt $J/\psi$ polarizations
at TEVATRON run II with collision energy 1.96 TeV and rapidity cut
$|y^{J/\psi}|<0.6$, have been systematically studied, especially, the spin-flip
effect for the transition of $(c\bar{c})_8[^3S_1]$ into $J/\psi$ has been
carefully discussed. It is found that the spin-flip effect shall always dilute
the $J/\psi$ polarization, and with a suitable choice of the parameters
$a_{0,1}$ and $c_{0,1,2}$, the $J/\psi$ polarization puzzle can be solved to a
certain degree. At large transverse momentum $p_t$, $\alpha$ for the prompt
$J/\psi$ is reduced by $\sim50%$ for $f_0 = v^2$ and by $\sim80%$ for $f_0=1$.
We also study the indirect $J/\psi$ polarization from the $b$-decays, which
however is slightly affected by the same spin-flip effect and then shall
provide a better platform to determine the color-octet matrix elements.
