  The next to leading order (NLO) QCD corrections to J/psi polarization of
hadronproduction at Tevatron and LHC are calculated. The results show that the
J/psi polarization is extremely changed from more transversal polarization at
leading order (LO) into more longitudinal polarization at NLO. Although it
gives more longitudinal polarization than the recent experimental result on the
J/psi polarization at Tevatron. It sheds light on the solution to the large
discrepancy of J/psi polarization between theoretical predication and
experimental measurement, and suggests that the next important step is to
calculate the NLO correction for color octet state J/psi^{(8)}
hadronproduction. Our calculations are performed in two ways where the
polarizations are summed analytically or not, and they are checked with each
other. It also gives a K factor for total cross section (ratio of NLO to LO) of
about 2 and shows that the NLO corrections boost the J/psi production for about
2 order of magnitude in high transverse momentum p_t region of J/psi, which
confirms the calculation by Campbell, Maltoni and Tramontano.
