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We extend our analysis of transverse single spin asymmetry (SSA) inelectroproduction of $J/\psi$ to include the effect of the scale evolution ofthe transverse momentum dependent (TMD) parton distribution functions and gluonSivers function. We estimate SSA for JLab, HERMES, COMPASS, and eRHIC energiesusing the color evaporation model of charmonium production, using ananalytically obtained approximate solution of TMD evolution equations discussedin the literature. We find that there is a reduction in the asymmetry comparedwith our predictions for the earlier case considered by us, wherein the $Q^2$dependence came only from DGLAP evolution of the unpolarized gluon densitiesand a different parametrization of the TMD Sivers function was used.

Transverse Single Spin Asymmetry in $e+p^\uparrow \to e+J/\psi +X $ and Transverse Momentum Dependent Evolution of the Sivers Function