Precision Measurements of $D_s^+ \to ηe^+ ν_e$ and $D_s^+ \to η^\prime e^+ ν_e$
Precision measurements of the semileptonic decays $D_s^+ \to ηe^+ ν_e$ and $D_s^+ \to η^\prime e^+ ν_e$ are performed with 7.33\,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 $\mathcal{B}(D_s^+ \to ηe^{+} ν_e)$ = $(2.255\pm0.039_{\rm stat}\pm 0.051_{\rm syst})\%$ and $\mathcal{B}(D_s^+ \to η^{\prime} e^{+} ν_e)$ = $(0.810\pm0.038_{\rm stat}\pm 0.024_{\rm syst})\%$. Combining these results with the $\mathcal{B}(D^+\toηe^+ ν_e)$ and $\mathcal{B}(D^+\toη^\prime e^+ ν_e)$ obtained from previous BESIII measurements, the $η-η^\prime$ mixing angle in the quark flavor basis is determined to be $ϕ_{\rm P} = (40.0\pm2.0_{\rm stat}\pm0.6_{\rm syst})^\circ$. Moreover, from the fits to the partial decay rates of $D_s^+ \to ηe^+ ν_e$ and $D_s^+ \to η^\prime e^+ ν_e$, the products of the hadronic transition form factors $f_+^{η^{(\prime)}}(0)$ 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 $f^η_+(0)|V_{cs}| = 0.4519\pm0.0071_{\rm stat}\pm0.0065_{\rm syst}$ and $f^{η^\prime}_+(0)|V_{cs}| = 0.525\pm0.024_{\rm stat}\pm0.009_{\rm syst}$ are extracted. All results determined in this work supersede those measured in the previous BESIII analyses based on the 3.19 fb$^{-1}$ subsample of data at 4.178 GeV.