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 $\phi (1020)$, $0.98<M(e^{+}{e}^{-})<1.04$ ~GeV/$c^2$, the decay $D_s^{+}\to {\pi }^{+}\phi ,\phi \to e^{+}{e}^{-}$ is observed with a statistical significance of 7.8$\sigma$, and evidence for the decay $D_s^{+}\to {\rho }^{+}\phi ,\phi \to e^{+}{e}^{-}$ is found for the first time with a statistical significance of 4.4$\sigma$. The decay branching fractions are measured to be $\mathcal{B}(D_s^{+}\to {\pi }^{+}\phi ,\phi \to e^{+}{e}^{-})=(1.17_{-0.21}^{+0.23}\pm 0.03)\times {10}^{-5}$, and $\mathcal{B}(D_s^{+}\to {\rho }^{+}\phi ,\phi \to e^{+}{e}^{-})=(2.44_{-0.62}^{+0.67}\pm 0.16)\times {10}^{-5}$, where the first uncertainties are statistical and the second systematic. No significant signal for the three four-body decays of $D_s^{+}\to {\pi }^{+}{\pi }^0{e}^{+}{e}^{-},D_s^{+}\to K^{+}{\pi }^0{e}^{+}{e}^{-}$, and $D_s^{+}\to K_S^0{\pi }^{+}{e}^{+}{e}^{-}$ is observed. For $D_s^{+}\to {\pi }^{+}{\pi }^0{e}^{+}{e}^{-}$, the $\phi$ mass region is vetoed to minimize the long-distance effects. The 90$\mathrm{\%}$ confidence level upper limits set on the branching fractions of these decays are in the range of $(7.0-8.1)\times {10}^{-5}$.