Scalar resonances in a unitary $\pi -\pi$ $S$-wave model for $D^{+}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}$

We propose a model for $D^{+}\to {\pi }^{+}{\pi }^{-}{\pi }^{+}$ decays following experimental results which indicate that the two-pion interaction in the $S$-wave is dominated by the scalar resonances $f_0(600)\mathrm{/}\sigma$ and $f_0(980)$. The weak decay amplitude for $D^{+}\to R{\pi }^{+}$, where $R$ is a resonance that subsequently decays into ${\pi }^{+}{\pi }^{-}$, is constructed in a factorization approach. In the $S$-wave, we implement the strong decay $R\to \pi^-\pi^+$ by means of a scalar form factor. This provides a unitary description of the pion-pion interaction in the entire kinematically allowed mass range $m_{\pi \pi }^2$ from threshold to about 3 GeV${}^2$. In order to reproduce the experimental Dalitz plot for $\text{Dppp}$, we include contributions beyond the $S$-wave. For the $P$-wave, dominated by the $\rho (770{)}^0$, we use a Breit-Wigner description. Higher waves are accounted for by using the usual isobar prescription for the $f_2(1270)$ and $\rho (1450{)}^0$. The major achievement is a good reproduction of the experimental $m_{\pi \pi }^2$ distribution, and of the partial as well as the total $\text{Dppp}$ branching ratios. Our values are generally smaller than the experimental ones. We discuss this shortcoming and, as a byproduct, we predict a value for the poorly known $D\to \sigma$transitionformfactorat$q^2=m_\pi^2$.