  We compute the $J/\psi$ polarization observables $\lambda_\theta$,
$\lambda_\phi$, $\lambda_{\theta\phi}$ in a Color Glass Condensate (CGC) +
nonrelativistic QCQ (NRQCD) formalism that includes contributions from both
color singlet and color octet intermediate states. Our results are compared to
low $p_T$ data on $J/\psi$ polarization from the LHCb and ALICE experiments on
proton-proton collisions at center-of-mass energies of $\sqrt{s}=7$ TeV and 8
TeV. Our CGC+NRQCD computation provides a better description of data for $p_T
\leq 15$ GeV relative to extant next-to-leading (NLO) calculations within the
collinear factorization framework. These results suggest that higher order
computations in the CGC+NRQCD framework have the potential to greatly improve
the accuracy of extracted values of the NRQCD universal long distance matrix
elements.
