Novel approach to investigate $\eta$ decays via $\eta'\rightarrow\pi\pi\eta$

To avoid the impact from the background events directly from $e^+e^-$ annihilations or $J/\psi$ decays, we propose a novel approach to investigate $\eta$ decays, in particular for its rare or forbidden decays, by using $\eta^\prime\rightarrow\pi\pi\eta$ produced in $J/\psi$ decays at the $\tau-$charm factories. Based on the MC studies of a few typical decays, $\eta\rightarrow \pi\pi$, $\gamma l^+l^- (l= e, \mu)$, $l^+l^-$, as well as $l^+l^-\pi^0$, the sensitivities could be obviously improved by taking advantage of the extra constraint of $\eta^\prime$. Using one trillion $J/\psi$ events accumulated at the Super $\tau$-Charm facility, the precision on the investigation of $\eta$ decays could be improved significantly and the observation of the rare decay $\eta\rightarrow e^+e^-$ is even accessable.