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 $gg\to {\chi }_{c{J}_1}{\chi }_{c{J}_2}$ is considerably smaller (especially for ${\chi }_{c1}{\chi }_{c1}$ 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.