Prediction of charmoniumlike structures in the hidden-charm di-eta decays of higher charmonia

Considering the situation that a single chiral partilce, $\eta$, is initially emitted, we study the hidden-charm di-eta decays of charmoniumlike state $Y(4660)$ and the predicted charmonium $\psi(4790)$, i.e., $Y(4660)/\psi(4790) \to J/\psi\eta\eta$ through the inetermediates, $\eta\left[D^{(*)}\bar{D}^{(*)}\right]$ and/or $\eta\ \left[D_s^{+(*)}{D}_s^{-(*)}\right]$, and answer to the important question whether there exist isoscalar charmoniumlike structures in the $D^{(*)}\bar{D}^{(*)}$ and/or $D_s^{+(*)}{D}_s^{-(*)}$ channels. Our results predict that there will be enhancement structures near $D\bar{D}^*$, $D^*\bar{D}^*$ and $D_s\bar{D}_s^*$ thresholds for $Y(4660)$ and near $D^*\bar{D}^*$, $D_s\bar{D}_s^*$ and $D_s^*\bar{D}_s^*$ thresholds for $\psi(4790)$ in the $M_{\mathrm{max}}({J/\psi \eta})$ distributions of $Y(4660)/\psi(4790) \to \eta \eta J/\psi$, respectively. These peaks are accessible at future experiments, especially BESIII, Belle, BaBar and forthcoming BelleII.