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We study J/psi production in pp collisions at sqrt(s)=1.96 and 7 TeV usingthe Colour-Singlet Model (CSM), including next-to-leading order (NLO)corrections and dominant alphaS^5 contributions (NNLO*). We find that the CSMreproduces the existing data if the upper range of the NNLO* is near the actual--but presently unknown-- NNLO. The direct yield polarisation for the NLO andNNLO* is increasingly longitudinal in the helicity frame when P_T gets larger.Contrary to what is sometimes claimed in the literature, the prompt J/psi yieldpolarisation in the CSM is compatible with the experimental data from the CDFcollaboration, when one combines the direct yield with a data-driven range forthe polarisation of J/psi from chi(c).

J/psi production at sqrt(s)=1.96 and 7 TeV: Color-Singlet Model, NNLO* and polarisation