We study J/psi production in pp collisions at sqrt(s)=1.96 and 7 TeV using the Colour-Singlet Model (CSM), including next-to-leading order (NLO) corrections and dominant alphaS^5 contributions (NNLO*). We find that the CSM reproduces 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 and NNLO* is increasingly longitudinal in the helicity frame when P_T gets larger. Contrary to what is sometimes claimed

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... the literature, the prompt J/psi yield polarisation in the CSM is compatible with the experimental data from the CDF collaboration, when one combines the direct yield with a data-driven range for the polarisation of J/psi from chi(c).