A MODEL FOR THE CHARMED $D_s^{+}$ MESON DECAYS INTO THREE PIONS

We propose a phenomenological three component model describing the decay amplitude of the process $D_{s}^{+} \ra 3\pi$. The first component is a constant background, the second one is a Breit-Wigner amplitude associated to a quasi two-body $f_{0}(980) \pi^{+}$ state and the third one another Breit-Wigner amplitude corresponding to a possible quasi two-body $\rho(770) \pi$ state. We show that it is possible to reproduce the observed total rate for $D_{s}^{+} \ra \pi^{+}\pi^{+}\pi^{-}$ as well as the two other measured branching ratios for the non resonant part and the resonant $f_{0}\pi^{+}$ one. Implication of the $\rho\pi^{+}$ state, of which an experimental limit has been given, is discussed. An upper bound in the 10 MeV range for the decay constant $f_{\pi'}$ of the $\pi(1300)$ meson is obtained . Predictions are given for the $D_{s}^{+} \ra \pi^{0}\pi^{0}\pi^{+}$ rate as well as for the $\pi^{+}$ and $\pi^{-}$ ($\pi^{0}$ and $\pi^{+}$) energy distributions for these two decay modes $D_{s}^{+} \ra \pi^{+}\pi^{+}\pi^{-}$ ($D_{s}^{+} \ra \pi^{0}\pi^{0}\pi^{+}$) respectively.