# Measurement of the Dynamics of the Decays ${ D_s^+ \rightarrow \eta^{(\prime)} e^{+} \nu_e}$

## TL;DR

This study measures the decay rates and form factors of $D_s^+$ meson decays to $ a^{("}e^{+}
u_e$ and determines the $ a- a'$ mixing angle, providing new insights into charm meson decay dynamics.

## Contribution

First measurements of the decay dynamics of $D_s^+ o a^{("}e^{+}
u_e$ decays and the $ a- a'$ mixing angle in the quark flavor basis.

## Key findings

- Branching fractions: $ a e^{+}
u_e$ = (2.323 ± 0.063 ± 0.063)% and $ a' e^{+}
u_e$ = (0.824 ± 0.073 ± 0.027)%
- Mixing angle $ a_{
m P} = (40.1 ± 2.1 ± 0.7)^ ext{o}$
- Product of form factors and CKM matrix element: $f_+^{ a}(0)|V_{cs}| = 0.4455 ± 0.0053 ± 0.0044$ and $f_+^{ a'}(0)|V_{cs}| = 0.477 ± 0.049 ± 0.011$

## Abstract

Using $e^+e^-$ annihilation data corresponding to an integrated luminosity of 3.19\,fb$^{-1}$ collected at a center-of-mass energy of 4.178~GeV with the BESIII detector, we measure the absolute branching fractions $\mathcal{B}_{D_s^+ \rightarrow \eta e^{+} \nu_e }$ = $(2.323\pm0.063_{\rm stat}\pm0.063_{\rm syst})\%$ and $\mathcal{B}_{D_s^+ \rightarrow \eta^{\prime} e^{+} \nu_e}$ = $(0.824\pm0.073_{\rm stat}\pm0.027_{\rm syst})\%$ 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 flavour basis is determined to be $\phi_{\rm P} = (40.1\pm2.1_{\rm stat}\pm0.7_{\rm syst})^\circ$. 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 $|V_{cs}|$ are determined with different form factor parameterizations. For the two-parameter series expansion, the results are $f^{\eta}_+(0)|V_{cs}| = 0.4455\pm0.0053_{\rm stat}\pm0.0044_{\rm syst}$ and $f^{\eta^{\prime}}_+(0)|V_{cs}| = 0.477\pm0.049_{\rm stat}\pm0.011_{\rm syst}$.