  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 in 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).
