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$\chi_{cJ}$ (J=0,1,2) production associated with a charm quark pair in hadroncollisions is calculated in the framework of nonrelativistic QCD at theTevatron and LHC. It is found that the color-singlet contribution is small andthe color-octet contribution may be dominant in the large $p_T$ region. Thedifferential cross section of $p\bar p\to\chi_c+c\bar{c}$ is at least one orderof 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 atthe Tevatron. The feeddown contribution from $\chi_{cJ}+c+\bar{c}$ to$J/\psi+c+\bar{c}$ is found to be large compared with $J/\psi$ directproduction and may have important influence on the measurement of$J/\psi+c+\bar{c}$. The validity of fragmentation approximation for the processis also discussed.

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