  In this work, we present more detail of the calculation on the NLO QCD
corrections to polarization of direct J/psi production via color singlet at
Tevatron and LHC, as well as the results for Upsilon for the first time. Our
results show that the J/psi polarization status drastically changes from
transverse polarization dominant at LO into longitudinal polarization dominant
in the whole range of the transverse momentum $p_t$ of J/psi when the NLO
corrections are counted. For Upsilon production, the p_t distribution of the
polarization status behaves almost the same as that for J/psi except that the
NLO result is transverse polarization at small p_t range. Although the
theoretical evaluation predicts a larger longitudinal polarization than the
measured value at Tevatron, it may provide a solution towards the previous
large discrepancy for J/psi and Upsilon polarization between theoretical
predication and experimental measurement, and suggests that the next important
step is to calculate the NLO corrections to hadronproduction of color octet
state J/psi^(8) and Upsilon^(8). Our calculations are performed in two ways,
namely we do and do not analytically sum over the polarizations, and then check
them with each other.
