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We carry out the first calculation for the twist-3 gluon contribution to thesingle transverse-spin asymmetry(SSA) in $J/\psi$ production in semi-inclusivedeep inelastic scattering. Our result shows that the $J/\psi$ SSA is an idealobservable to pin down the $C$-even type twist-3 gluon distribution that has adirect relationship with the gluon transverse-momentum-dependent distributionfunction. We also perform some numerical simulations of the $J/\psi$ SSA forthe kinematics accessible at the future electron-ion-collider experiment. Forcolor-singlet contribution, the hadronization effect of $J/\psi$ is completelycanceled at the level of the SSA and the spin-dependent structure functionsdirectly reflect the behavior of the $C$-even twist-3 gluon distribution.

The twist-3 gluon contribution to Sivers asymmetry in $J/\psi$ production in semi-inclusive deep inelastic scattering