Phenomenological description of the $\pi^-\pi^+$ $S$-waves in $D^+\to\pi^-\pi^+\pi^+$ and $D^+_s\to\pi^-\pi^+\pi^+$ decays: The problem of phases

We present a phenomenological description of the LHCb data for the magnitudes and phases of the $\pi^-\pi^+$ $S$-wave amplitudes in the $D^+\to\pi^-\pi^+\pi^+$ and $D^+_s\to\pi^-\pi^+\pi^+$ decays. We operate within a simple model that takes into account the known pair interactions of particles in coupled channels. The seed complex amplitudes for various intermediate state production are assumed to be independent of the energy. Their values are determined by fitting. This model gives the satisfactorily description of virtually all features of the energy dependence of the experimentally measured $S$-wave amplitudes in the $D^+\to\pi^-\pi^+ \pi^+$ and $D^+_s\to \pi^-\pi^+\pi^+$ decays in the regions $2m_\pi<m_{\pi^-\pi^+}<1.39\mbox{ GeV}$ and $2m_\pi<m_{\pi^-\pi^+} <1.29\mbox{ GeV}$, respectively.