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