χcJ production associated with a c(c)over-bar 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 (p) over bar -> chi(c) + c (c) over bar is at least 1 order of magnitude smaller than the next-to-leading order result of chi(cJ) inclusive production p (p) over bar -> chi(c) + X; therefore chi(cJ) production in p (p) over bar -> chi(c) + c (c) over bar may have negligible influence on the ratio R-chi c = sigma chi c2/sigma chi c1 measured by CDF Collaboration at the Tevatron. The feed-down contribution from chi(cJ) + c + (c) over bar is found to be large compared with J/psi direct production and may have important influence on the measurement of J/psi + c + (c) over bar. The validity of fragmentation approximation for the process is also discussed.