Production of ${\chi }_c$ pairs in $k_T$-factorization

We calculate the production of pairs of ${\chi }_c(J)$ mesons with all possible combinations of $J=0,1,2$. The leading order production mechanism is the crossed-channel gluon exchange in the gluon-gluon fusion reaction. The building blocks are the vertices $g^{\ast }{g}^{\ast }\to {\chi }_c(J)$ for off shell gluons. We stick to the color-singlet model and calculate the gluon fusion vertices in the limit of heavy quarks with nonrelativistic motion in the bound state. These vertices are used to construct the $g^* g^* \to \chi_c(J_1) \chi_c(J_2)$ amplitudes. We then calculate hadron-level cross sections using the $k_T$-factorization approach. In our numerical predictions, we use the KMR-type unintegrated gluon distributions. Several differential distributions at the $pp$ center of mass energy $\sqrt{s}=8\mathrm{T}\mathrm{e}\mathrm{V}$ are shown. The salient feature of the $t$ and $u$-channel gluon exchange are the broad distributions in rapidity difference $\mathrm{\Delta }y$ between ${\chi }_c$ mesons.