  With nonrelativistic QCD factorization, we present a full next-to-leading
order computation of the polarization observable for $J/psi$ production at
hadron colliders including all important Fock states, i.e. $^3S_1[1,8]$,
$^1S_0[8]$, and $^3P_J[8]$. We find the $^3P_J[8]$ channel contributes a
positive longitudinal component and a negative transverse component. So the
$J/psi$ polarization puzzle may be understood as the transverse components
canceling between $^3S_1[8]$ and $^3P_J[8]$ channels, which results in mainly
the unpolarized (even slightly longitudinally polarized) $J/psi$. This may give
a possible solution to the long-standing $J/psi$ polarization puzzle.
Predictions for $J/psi$ polarization at the LHC are also presented.
