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Semi-inclusive $J/\psi$ production in electron-proton collisions is apromising process to study gluon transverse momentum distributions (TMDs) atthe future Electron-Ion Collider. In this article, we improve on previousstudies of the $\cos 2 \phi$ azimuthal asymmetry that arises from the linearpolarization of gluons inside unpolarized protons by including TMD evolution.We find that in the TMD regime the asymmetry grows monotonically withincreasing transverse momentum of the outgoing $J/\psi$, in contrast to treelevel calculations with Gaussian TMDs. Our predictions for the asymmetry at EICcan become very large at larger $x$, $Q$, and transverse momenta, even largerthan the positivity bound. This problem stems from the very small $b$ regionand implies a range of validity of TMD factorization that is more restrictedthan usually expected. We also include an estimate of the nonperturbativeuncertainty from the large $b$ region and we conclude that it is smaller thanthe largest source of uncertainty, which stems from the choice of Color OctetLong-Distance Matrix Elements.

TMD Evolution Study of the $\cos 2 \phi$ Azimuthal Asymmetry in Unpolarized $J/\psi$ Production at EIC