Semileptonic $D_{e4}$ decays: hadronic dynamics and the determination of $|V_{cs}|$

The four-body decays $D^+ \to K^-\pi^+e^+\nu_e$ ($D_{e4}^+$) and $D^0\to \overline{K^0}\pi^-e^+\nu_e$ ($D^0_{e4}$) are studied in a model where the momentum-dependence of the hadronic matrix elements are described in terms of $K^*(892)$ and $D^*(2010)$ pole contributions. From fits to the recent data of the BESIII collaboration we find that the $D^*$-pole can mimic the effect of the $S$-wave contribution of the $K\pi$ system to the branching fraction. Implications for the determination of the $|V_{cs}|$ quark mixing matrix element are discussed.