  We update the study of the polarisation of J/psi produced in proton-proton
collisions at RHIC at sqrt(s)=200 GeV using the QCD-based Colour-Singlet Model
(CSM), including next-to-leading order partonic matrix elements from gluon and
light quark fusion and leading-order contributions from charm-quark initiated
processes. To do so, we also evaluate the corresponding cross section
differential in P_T which agrees qualitatively with the measurements of PHENIX
in the central and forward regions at low P_T -- for instance below 2 GeV --,
while emphasising the need for Initial State Radiation (ISR) resummation. At
mid P_T, we also compare the measurements from PHENIX and STAR with the same
evaluation complemented with the dominant alphaS^5 contributions (NNLO*). We
find a reasonable agreement with the data. Regarding the polarisation, as shown
for previous studies at larger sqrt(s) and P_T, the polarisation pattern from
gluon and light quark fusion in the helicity frame is drastically modified at
NLO and is shown to be increasingly longitudinal. The yield from charm-gluon
fusion is found to be slightly transversally polarised. Combining both these
contributions with a data-driven range for the polarisation of J/psi from
chi_c, we eventually provide an evaluation of the polarisation of the prompt
J/psi yield which is in a good agreement with the experimental data from PHENIX
both in the central and forward regions.
