On a search for the $\eta \rightarrow e^+ e^-$ decay at the VEPP-2000 $e^+e^-$ collider

A sensitivity of the VEPP-2000 $e^+e^-$ collider in a search for the rare decay $\eta \rightarrow e^+ e^-$ has been studied. The inverse reaction $e^+ e^- \rightarrow \eta$ is proposed for this search. We have analyzed a data sample with an integrated luminosity of 108 nb$^{-1}$ collected with the SND detector in the center-of-mass energy range $520{-}580$ MeV and found no background events for the reaction $e^+ e^- \rightarrow \eta$ in the decay mode $\eta\to\pi^0\pi^0\pi^0$. In the absence of background, a sensitivity to $\mathcal{B}(\eta \rightarrow e^+ e^-)$ of $10^{-6}$ can be reached during two weeks of VEPP-2000 operation. Such a sensitivity is better than the current upper limit on $\mathcal{B}(\eta \rightarrow e^+ e^-)$ by a factor of 2.3.