Search for Invisible Decays of $\eta$ and $\eta^\prime$ in the Processes $J/\psi \to \phi\eta$ and $\phi \eta^\prime$

We present searching for the invisible decays of $\eta$ and $\eta^\prime$ in $J/\psi$ to $\phi\eta$ and $\phi\eta^\prime$. The $\phi$ signals are reconstructed in $K^+K^-$ final states which are used to tag the invisible decays of $\eta$ and $\eta^\prime$. The results are obtained from data sample of $58\times 10^6$ $J/\psi$ decays collected with the BES II detector at the BEPC. We find no signals for the invisible decays of both $\eta$ and $\eta^\prime$. The upper limits are established at the 90% confidence level of $1.65 \times 10^{-3}$ for the ratio $\frac{B(\eta\to invisible)}{B(\eta\to\gamma\gamma)}$ and $6.69\times 10^{-2}$ for the ratio $\frac{B(\eta^\prime\to invisible)}{B(\eta^\prime\to\gamma\gamma)}$. These are the first searches of the $\eta$ and $\eta^\prime$ decays into invisible final states in the world