$J/\psi$ polarization in large-$P_T$ semi-inclusive deep-inelastic scattering at the EIC

We present a detailed phenomenological study of $J/\psi$ polarization in semi-inclusive deep inelastic scattering processes, focusing on the kinematics accessible at the future Electron-Ion Collider. We show theoretical estimates for the standard polarization parameters for different frames usually adopted in the literature, in the large $P_T$ region, namely $P_T\gg \Lambda_\text{QCD}$, where collinear factorization is expected to hold. We adopt both the Color Singlet Model and the Nonrelativistic QCD approach, paying special attention to the role of different sets of Long Distance Matrix Elements. Finally we present a preliminary analysis of some frame independent polarization invariants.