Precision measurements of ${D}_{s}^{+}\to \eta {e}^{+}{\nu }_{e}$ and ${D}_{s}^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_{e}$

Precision measurements of the semileptonic decays ${D}_{s}^{+}\to \eta {e}^{+}{\nu }_{e}$ and ${D}_{s}^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_{e}$ are performed with $7.33\text{}\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ collision data collected at center-of-mass energies between 4.128 and 4.226 GeV with the BESIII detector. The branching fractions obtained are $B\left({D}_{s}^{+}\to \eta {e}^{+}{\nu }_{e}\right)=\left(2.255±0.03{9}_{\mathrm{stat}}±0.05{1}_{\mathrm{syst}}\right)%$ and $B\left({D}_{s}^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_{e}\right)=\left(0.810±0.03{8}_{\mathrm{stat}}±0.02{4}_{\mathrm{syst}}\right)%$. Combining these results with the $B\left({D}^{+}\to \eta {e}^{+}{\nu }_{e}\right)$ and $B\left({D}^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_{e}\right)$ obtained from previous BESIII measurements, the $\eta -{\eta }^{\prime }$ mixing angle in the quark flavor basis is determined to be ${\varphi }_{P}=\left(40.0±{2.0}_{\mathrm{stat}}±{0.6}_{\mathrm{syst}}\right)°$. Moreover, from the fits to the partial decay rates of ${D}_{s}^{+}\to \eta {e}^{+}{\nu }_{e}$ and ${D}_{s}^{+}\to {\eta }^{\prime }{e}^{+}{\nu }_{e}$, the products of the hadronic transition form factors ${f}_{+}^{{\eta }^{\left(\prime \right)}}\left(0\right)$ and the modulus of the $c\to s$ Cabibbo-Kobayashi-Maskawa matrix element $|{V}_{cs}|$ are determined by using different hadronic transition form factor parametrizations. Based on the two-parameter series expansion, the products ${f}_{+}^{\eta }\left(0\right)|{V}_{cs}|=0.4519±0.007{1}_{\mathrm{stat}}±0.006{5}_{\mathrm{syst}}$ and ${f}_{+}^{{\eta }^{\prime }}\left(0\right)|{V}_{cs}|=0.525±0.02{4}_{\mathrm{stat}}±0.00{9}_{\mathrm{syst}}$ are extracted. All results determined in this work supersede those measured in the previous BESIII analyses based on the $3.19\text{}\text{}{\mathrm{fb}}^{-1}$ subsample of data at 4.178 GeV.