Associated production of $\chi_c$ pairs with a gluon in the collinear-factorization approach

We calculate cross section for production of $\chi_c$ pairs in proton-proton collisions. The cross section for the $g g \to \chi_{c J_1} \chi_{c J_2}$ is considerably smaller (especially for $\chi_{c 1} \chi_{c 1}$ final state) than that obtained recently in the $k_T$-factorization approach. We calculate therefore next-to-leading order contributions with $\chi_c$ pair and one extra associated (mini-)jet. We find these contributions to be much larger than those for the $2 \to 2$ contribution. Especially the emission of a leading gluon (carrying a large momentum fraction of one of the incoming gluons) are important. These emissions in the $k_T$-factorization approach are absorbed into the initial state unintegrated gluon distributions. A smaller contribution to the cross section comes from the production of central gluons emitted with rapidities between the $\chi_c$-mesons. They do lead, however, to an enhancement of the $\chi_c$-pair production at large rapidity distance between the mesons. Our present study explains the size of the cross section for the $\chi_c$ pair production obtained previously in the $k_T$-factorization approach. Several differential distributions are presented.