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.
Categories: hep-ph hep-ex