Inclusive prompt ${\chi }_{c,b}(1^{++})$ production at the LHC

We study the prompt production of the ${\chi }_c(1^{+})$ and ${\chi }_b(1^{+})$ mesons at high energies. Unlike $\chi (0^{+},2^{+})$ production, $\chi (1^{+})$ mesons cannot be created at LO via the fusion of two on-mass-shell gluons, that is $gg\to {\chi }_{c,b}(1^{+})$ are not allowed. However, the available experimental data show that the cross sections for ${\chi }_c(1^{+})$ and ${\chi }_c(2^{+})$ are comparable. We therefore investigate four other $\chi (1^{+})$ production mechanisms: namely, (i) the standard NLO process $gg\to {\chi }_{c,b}(1^{+})+g$, (ii) via gluon virtuality, (iii) via gluon reggeization and, finally, (iv) the possibility to form ${\chi }_{c,b}(1^{+})$ by the fusion of three gluons, where one extra gluon comes from another parton cascade, as in the Double Parton Scattering processes.