Production of J/$\psi+\eta_{c}$ vs. J/$\psi+J/\psi$ at the LHC: Importance of the Real $\alpha_s^5$ corrections

We proceed for the first time to the evaluation of the Born cross section for J/psi+eta(c) production, namely via g+g -> J/psi+eta(c)+g, and show that it has a harder P_T spectrum than the J/psi-pair yield at Born level. If one stuck to a comparison at Born level, one would conclude that J/psi+eta(c) production would surpass that of J/psi+J/psi at large P_T. This is nonetheless not the case since J/psi-pair production, as for single J/psi, receives leading-P_T contributions at higher orders in alpha_s. We also present the first evaluation of these leading-P_T next-to-leading order contributions. These are indeed significant for increasing P_T and are of essential relevance for comparison with forthcoming data. We also compute kinematic correlations relevant for double-parton-scattering studies. Finally, we evaluate the polarisation of a J/psi accompanied either by a eta(c) or a J/psi and another light parton. These results may be of great help to understand the polarisation of quarkonia produced at colliders.