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We investigate the feasibility of exploring the gluontransverse-momentum-dependent distribution functions (TMDs) inside alongitudinally polarized nucleon. We utilize quarkonium production via thecolor-octet mechanism combined with TMD formalism in semi-inclusive deeplyinelastic scattering (SIDIS) at low transverse momentum as a tool to accesspolarized gluon TMDs. The corresponding cross-section of the process isexpressed in terms of gluon TMDs and non-relativistic QCD matrix elements. Weprovide the expressions for the $\sin2\phi$ azimuthal asymmetries of $J/\psi$production in SIDIS with a unpolarized beam colliding on longitudinallypolarized nucleon. The asymmetry is contributed by the time-reversal-odd gluonTMD $h_{1L}^{\perp \,g}(x,\boldsymbol{p}_T^2)$. The maximum possible asymmetrydeduced from the positivity bound is sizable and could be measured. We alsoestimate the double longitudinal spin asymmetry $A_{LL}$ of $J/\psi$ productionusing a spectator model result for $g_{1L}^g(x,\boldsymbol{p}_T^2)$.

Gluon TMDs from J/$\psi$ production in longitudinally polarized deeply inelastic scattering