Search for the Rare Decays $D_s^+\to h^+(h^{0})e^+e^-$
Using 7.33~fb$^{-1}$ of $e^{+}e^{-}$ collision data collected by the BESIII detector at center-of-mass energies in the range of $\sqrt{s}=4.128 - 4.226$~GeV, we search for the rare decays $D_{s}^+\to h^+(h^{0})e^{+}e^{-}$, where $h$ represents a kaon or pion. By requiring the $e^{+}e^{-}$ invariant mass to be consistent with a $ϕ(1020)$, $0.98<M(e^{+}e^{-})<1.04$ ~GeV/$c^2$, the decay $D_s^+\toπ^+ϕ,ϕ\to e^{+}e^{-}$ is observed with a statistical significance of 7.8$σ$, and evidence for the decay $D_s^+\toρ^+ϕ,ϕ\to e^{+}e^{-}$ is found for the first time with a statistical significance of 4.4$σ$. The decay branching fractions are measured to be $\mathcal{B}(D_s^+\toπ^+ϕ, ϕ\to e^{+}e^{-} )=(1.17^{+0.23}_{-0.21}\pm0.03)\times 10^{-5}$, and $\mathcal{B}(D_s^+\toρ^+ϕ, ϕ\to e^{+}e^{-} )=(2.44^{+0.67}_{-0.62}\pm 0.16)\times 10^{-5}$, where the first uncertainties are statistical and the second systematic. No significant signal for the three four-body decays of $D_{s}^{+}\to π^{+}π^{0}e^{+}e^{-},\ D_{s}^{+}\to K^{+}π^{0}e^{+}e^{-}$, and $D_{s}^{+}\to K_{S}^{0}π^{+}e^{+}e^{-}$ is observed. For $D_{s}^{+}\to π^{+}π^{0}e^{+}e^{-}$, the $ϕ$ mass region is vetoed to minimize the long-distance effects. The 90$\%$ confidence level upper limits set on the branching fractions of these decays are in the range of $(7.0-8.1)\times 10^{-5}$.