$\chi_{cJ}$ production associated with a $c\bar c$ pair at hadron colliders

$\chi_{cJ}$ (J=0,1,2) production associated with a charm quark pair in hadron collisions is calculated in the framework of nonrelativistic QCD at the Tevatron and LHC. It is found that the color-singlet contribution is small and the color-octet contribution may be dominant in the large $p_T$ region. The differential cross section of $p\bar p\to\chi_c+c\bar{c}$ is at least one order of magnitude smaller than the next-to-leading order result of $\chi_{cJ}$ inclusive production $p\bar p\to\chi_c+X$, therefore $\chi_{cJ}$ production in $p\bar p\to\chi_c+c\bar{c}$ may have negligible influence on the ratio $R_{\chi_c}=\frac{\sigma_{\chi_{c2}}}{\sigma_{\chi_{c1}}}$ measured by CDF at the Tevatron. The feeddown contribution from $\chi_{cJ}+c+\bar{c}$ to $J/\psi+c+\bar{c}$ is found to be large compared with $J/\psi$ direct production and may have important influence on the measurement of $J/\psi+c+\bar{c}$. The validity of fragmentation approximation for the process is also discussed.