We calculate the cos 2ϕ asymmetry in J/ψ 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/ψ satisfies z<1 and the dominating subprocess is γ∗+g→c+cˉ+g. We use non-relativistic QCD based color singlet
model for J/ψ 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.Comment: Version to appear in Phys. Rev.