Measurement of the Dynamics of the Decays ${D}_{s}^{+}\to {\eta }^{\left(\prime \right)}{e}^{+}{\nu }_{e}$

Using ${e}^{+}{e}^{-}$ annihilation data corresponding to an integrated luminosity of $3.19\text{}\text{}{\mathrm{fb}}^{-1}$ collected at a center-of-mass energy of 4.178 GeV with the BESIII detector, we measure the absolute branching fractions ${B}_{{D}_{s}^{+}\to \eta {e}^{+}{\nu }_{e}}=\left(2.323±{0.063}_{\text{stat}}±{0.063}_{\text{syst}}\right)%$ and ${B}_{{D}_{s}^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_{e}}=\left(0.824±{0.073}_{\text{stat}}±{0.027}_{\text{syst}}\right)%$ via a tagged analysis technique, where one ${D}_{s}$ is fully reconstructed in a hadronic mode. Combining these measurements with previous BESIII measurements of ${B}_{{D}^{+}\to {\eta }^{\left(\prime \right)}{e}^{+}{\nu }_{e}}$, the $\eta -{\eta }^{\prime }$ mixing angle in the quark flavor basis is determined to be ${\varphi }_{P}=\left(40.1±2.{1}_{\text{stat}}±0.{7}_{\text{syst}}\right)°$. From the first measurements of the dynamics of ${D}_{s}^{+}\to {\eta }^{\left(\prime \right)}{e}^{+}{\nu }_{e}$ decays, the products of the hadronic form factors ${f}_{+}^{{\eta }^{\left(\prime \right)}}\left(0\right)$ 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 ${f}_{+}^{\eta }\left(0\right)|{V}_{cs}|=0.4455±{0.0053}_{\text{stat}}±{0.0044}_{\text{syst}}$ and ${f}_{+}^{{\eta }^{\prime }}\left(0\right)|{V}_{\mathrm{cs}}|=0.477±{0.049}_{\text{stat}}±{0.011}_{\text{syst}}$.