  We calculate the production and polarization of direct $J/\psi$ in the
improved color evaporation model at $\mathcal{O}(\alpha_s^3)$ in the collinear
factorization approach. We present the first calculation of polarization
parameters $\lambda_\vartheta$, $\lambda_{\phi}$, and $\lambda_{\vartheta
\phi}$ in the helicity and the Collins-Soper frames, as well as the
frame-invariant polarization parameter $\tilde{\lambda}$ as a function of
transverse momentum. We find agreement with both $J/\psi$ cross sections and
the invariant polarization parameters at small and moderate $p_T$.
