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

A sensitivity of the VEPP-2000 $e^{+}{e}^{-}$ collider in a search for the rare decay $\eta \to e^{+}{e}^{-}$ 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 $e^{+}{e}^{-}\to \eta$ in the decay mode $\eta \to {\pi }^0{\pi }^0{\pi }^0$. In the absence of background, a sensitivity to ${\cal B}(\eta \rightarrow e^+ e^-)$of$10^{-6}$ can be reached during two weeks of VEPP-2000 operation. Such a sensitivity is better than the current upper limit on $\mathcal{B}(\eta \to e^{+}{e}^{-})$ by a factor of 2.3.