Accessing Linearly Polarized Gluon Distribution in $J/\psi$ Production at the Electron-Ion Collider

We calculate the $cos~2 \phi$ asymmetry in $J/\psi$ production in electron-proton collision for the kinematics of the planned electron-ion collider (EIC). This directly probes the Weisz\"acker-Williams (WW) type linearly polarized gluon distribution. Assuming generalized factorization, we calculate the asymmetry at next-to-leading-order (NLO) when the energy fraction of the $J/\psi$ satisfies $z<1$ and the dominating subprocess is $\gamma^* +g \rightarrow c + {\bar c}+g$. We use non-relativistic QCD based color singlet model for $J/\psi$ production. We investigate the small $x$ region which will be accessible at the EIC. We present the upper bound of the asymmetry, as well as estimate it using a (i) Gaussian type parametrization for the TMDs and (ii) McLerran-Venugopalan model at small $x$. We find small but sizable asymmetry in all the three cases.