The understanding of the gluon Sivers effect is a challenging problem of future experiments on the nucleon structure. The single transverse-spin asymmetries(SSAs) in heavy quarkonium productions have been
recently studied as ideal observables to investigate the gluon Sivers effect within TMD
factorization framework.
We here show our first study on the SSA in $J/\psi$ production within the collinear twist-3 framework
combined with NRQCD framework for the description of the hadronization mechanism of $J/\psi$.
Our result shows that the $J/\psi$ SSA is an ideal observable to pin down the $C$-even type twist-3
gluon distribution that has a direct relationship with the gluon Sivers TMD function.
We also perform some numerical simulations of the $J/\psi$ SSA
for the kinematics accessible at the future electron-ion-collider(EIC) experiment.
If we only take into account the color-singlet contribution,
all the nonperturbative effects except the $C$-even twist-3 gluon
distribution are canceled, which means the $J/\psi$ SSA allows us to directly
investigate the gluon Sivers effect at the future EIC.