  The relativistic corrections to the color-octet $J/\psi$ hadroproduction at
the Tevatron and LHC are calculated up to $\mathcal{O}(v^2)$ in nonrelativistic
QCD factorization frame. The short distance coefficients are obtained by
matching full QCD with NRQCD results for the partonic subprocess $g+g\to J/\psi
({}^1S_0^{[8]},{}^3S_1^{[8]}, {}^3P_J^{[8]})+g$, $q+\bar{q}\to J/\psi
({}^1S_0^{[8]},{}^3S_1^{[8]}, {}^3P_J^{[8]})+g$ and $g+q({\bar{q}})\to J/\psi
({}^1S_0^{[8]},{}^3S_1^{[8]}, {}^3P_J^{[8]})+q({\bar{q}})$. The short distance
coefficient ratios of relativistic correction to leading order for color-octet
states ${^1}S_{0}^{[8]}$, ${^3}S_{1}^{[8]}$, and ${^3}P_{J}^{[8]}$ at large
$p_T$ are approximately -5/6, -11/6, and -31/30, respectively, for each
subprocess, and it is 1/6 for color-singlet state ${^3}S_{1}^{[1]}$. If the
higher order long distance matrix elements are estimated through velocity
scaling rule with adopting $v^2=0.23$ and the lower order long distance matrix
elements are fixed, the leading order cross sections of color-octet states are
reduced by about a factor of $20\sim40%$ at large $p_T$ at both the Tevatron
and the LHC. Comparing with QCD radiative corrections to color-octet states,
relativistic correction is ignored along with $p_T$ increasing. Using long
distance matrix elements extracted from the fit to $J/\psi$ production at the
Tevatron, we can find the unpolarization cross sections of $J/\psi$ production
at the LHC taking into account both QCD and relativistic corrections are
changed by about $20\sim50%$ of that considering only QCD corrections. These
results indicate that relativistic corrections may play an important role in
$J/\psi$ production at the Tevatron and the LHC.
