A combined study of hadronic $D^+\to K^-π^+π^+$ and $D_s^+\to K^+K^+π^-$ decays by means of the analysis of semileptonic $D^+\to K^-π^+ \ell^+ν_\ell$ decays

We perform a combined study of the two hadronic decays $D^+\to K^-\pi^+\pi^+$ and $D_s^+\to K^+K^+\pi^-$ using a detailed analysis of the semileptonic decays $D^+\to K^-\pi^+ \ell^+\nu_\ell$ ($\ell=e, \mu$) thanks to the high-statistics $D^+\to K^-\pi^+ e^+\nu_e$ dataset provided by the BESIII Collaboration. We propose simple and suitable amplitude parametrizations of the studied reactions that shall be of interest to experimentalists for upcoming analyses. These new parametrizations are based on the na\"ive factorization hypothesis and the description of the resulting matrix elements in terms of well-known hadronic form factors, with special emphasis on the $K\pi$ scalar and vector cases. Such form factors account for final state interactions which fulfil analyticity, unitarity and chiral symmetry constraints. As a result of our study, we find a good prediction for the $P$-wave BR in $D^+\to K^-\pi^+\pi^+$ decays, while adopting a global phase and moderate rescaling for the $S$-wave, that we ascribe to three-body effects, we find a reasonable description. The resulting model is confronted against $D_s^+\to K^+K^+\pi^-$ decays, obtaining again a good agreement with data.