Real Next-to-Next-to-Leading Order QCD Corrections to J/psi and Upsilon Hadroproductiom in Association with a Photon

We update the study of the QCD corrections to direct J/{psi} and {Upsilon} hadroproduction in association with a photon in the QCD-based approach of the Colour-Singlet (CS) Model. After comparison with the recent full next-to-leading-order (NLO) computation for this process, we provide an independent confirmation to the inclusive case that NLO QCD corrections to quarkonium-production processes whose LO exhibits a non-leading P{sub T} behavior can be reliably computed at mid and large P{sub T} by considering only the real emission contributions accompanied with a kinematical cut. In turn, we evaluate the leading part of the {alpha}{sub S}{sup 4}{alpha} contributions, namely those coming from (J/{psi}, {Upsilon}) + {gamma} associated with two light partons. We find that they are dominant at mid and large P{sub T}. This confirms our expectations from the leading P{sub T} scaling of the new topologies appearing at NNLO. We obtain that the yield from the CS becomes one order of magnitude larger than the upper value of the potential Colour-Octet yield. The polarization of the {sup 3}S{sub 1} quarkonia produced in association with a photon is confirmed to be longitudinal at mid and large P{sub T}