  We calculate the polarization of prompt $J/\psi$ production in the improved
color evaporation model at leading order employing the $k_T$-factorization
approach. In this paper, we present the polarization parameter
$\lambda_\vartheta$ of prompt $J/\psi$ as a function of transverse momentum in
$p+p$ and $p$ + A collisions to compare with data in the helicity,
Collins-Soper and Gottfried-Jackson frames. We also present calculations of the
charmonium production cross sections as a function of rapidity and transverse
momentum. This is the first $p_T$-dependent calculation of charmonium
polarization in the improved color evaporation model. We find agreement with
both charmonium cross sections and polarization measurements.
