Using $e^{+}{e}^{-}$ annihilation data corresponding to an integrated luminosity of $3.19{\mathrm{fb}}^{-1}$ collected at a center-of-mass energy of 4.178 GeV with the BESIII detector, we measure the absolute branching fractions ${\mathcal{B}}_{D_s^{+}\to \eta e^{+}{\nu }_e}=(2.323\pm {0.063}_{\text{stat}}\pm {0.063}_{\text{syst}})\%$ and ${\mathcal{B}}_{D_s^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_e}=(0.824\pm {0.073}_{\text{stat}}\pm {0.027}_{\text{syst}})\%$ via a tagged analysis technique, where one $D_s$ is fully reconstructed in a hadronic mode. Combining these measurements with previous BESIII measurements of ${\mathcal{B}}_{D^{+}\to {\eta }^{(\prime )}{e}^{+}{\nu }_e}$, the $\eta -{\eta }^{\prime }$ mixing angle in the quark flavor basis is determined to be ${\varphi }_P=(40.1\pm 2.1_{\text{stat}}\pm 0.7_{\text{syst}})°$. From the first measurements of the dynamics of $D_s^{+}\to {\eta }^{(\prime )}{e}^{+}{\nu }_e$ decays, the products of the hadronic form factors $f_{+}^{{\eta }^{(\prime )}}(0)$ and the Cabibbo-Kobayashi-Maskawa matrix element $|V_{\mathrm{cs}}|$ are determined with different form factor parametrizations. For the two-parameter series expansion, the results are $f_{+}^{\eta }(0)|V_{cs}|=0.4455\pm {0.0053}_{\text{stat}}\pm {0.0044}_{\text{syst}}$ and $f_{+}^{{\eta }^{\prime }}(0)|V_{\mathrm{cs}}|=0.477\pm {0.049}_{\text{stat}}\pm {0.011}_{\text{syst}}$.

• Received 8 January 2019
• Revised 27 February 2019

DOI:https://doi.org/10.1103/PhysRevLett.122.121801

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Published by the American Physical Society