Study of the $f_{0}(980)$ and $f_{0}(500)$ Scalar Mesons through the Decay $D_{s}^{+} \to π^{+} π^{-} e^{+} ν_{e}$

Using $e^+e^-$ collision data corresponding to an integrated luminosity of 7.33~${\rm fb^{-1}}$ recorded by the BESIII detector at center-of-mass energies between 4.128 and 4.226~${\rm GeV}$, we present an analysis of the decay $D_{s}^{+} \to π^{+}π^{-} e^{+}ν_{e}$, where the $D_s^+$ is produced via the process $e^+e^- \to D_{s}^{*\pm}D_{s}^{\mp}$. We observe the $f_{0}(980)$ in the $π^+π^-$ system and the branching fraction of the decay $D_{s}^{+} \to f_{0}(980)e^{+}ν_{e}$ with $f_0(980)\toπ^+π^-$ measured to be $(1.72 \pm 0.13_{\rm stat} \pm 0.10_{\rm syst}) \times10^{-3}$, where the uncertainties are statistical and systematic, respectively. The dynamics of the $D_{s}^{+} \to f_{0}(980)e^{+}ν_{e}$ decay are studied with the simple pole parametrization of the hadronic form factor and the Flatté formula describing the $f_0(980)$ in the differential decay rate, and the product of the form factor $f^{f_0}_{+}(0)$ and the $c\to s$ Cabibbo-Kobayashi-Maskawa matrix element $|V_{cs}|$ is determined for the first time to be $f^{f_0}_+(0)|V_{cs}|=0.504\pm0.017_{\rm stat}\pm0.035_{\rm syst}$. Furthermore, the decay $D^+_s\to f_0(500) e^+ν_e$ is searched for the first time but no signal is found. The upper limit on the branching fraction of $D^+_s\to f_0(500) e^+ν_e,~f_0(500)\toπ^+π^-$ decay is set to be $3.3 \times 10^{-4}$ at 90\% confidence level.