Final state interactions in hadronicDdecays

We show that the large corrections due to final-state interactions in the $D^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+},$ $D_s^{+}\to {\pi }^{-}{\pi }^{+}{\pi }^{+}$, and $D^{+}\to K^{-}{\pi }^{+}{\pi }^{+}$ decays can be accounted for by invoking scattering amplitudes in agreement with those derived from phase shifts studies. In this way, broad/overlapping resonances in S-waves are properly treated and the phase motions of the transition amplitudes are driven by the scattering matrix elements determined in many other experiments. This is an important step forward in resolving the puzzle of the final-state interactions in these decays. We also discuss why the $\sigma$ and $\kappa$ resonances, hardly visible in scattering experiments, are much more prominent and clearly visible in these decays, without destroying the agreement with the experimental $\pi \pi$ and $K\pi$ low energy S-wave phase shifts.