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

Abstract : 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 $f_0(600)/\sigma$ 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 $\Dppp$, 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 $\Dppp$ 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 $D\to \sigma$ transition form factor at $q^2=m_\pi^2$.
Document type :
Journal articles

http://hal.in2p3.fr/in2p3-00358271
Contributor : Swarna Bassava <>
Submitted on : Tuesday, February 3, 2009 - 2:05:13 PM
Last modification on : Thursday, December 10, 2020 - 12:31:48 PM

### Citation

D. R. Boito, J. -P. Dedonder, B. El-Bennich, O. Leitner, B. Loiseau. Scalar resonances in a unitary $\pi-\pi$ $S$-wave model for $D^+ \to \pi^+ \pi^- \pi^+$. Physical Review D, American Physical Society, 2008, 79, pp.3. ⟨10.1103/PhysRevD.79.034020⟩. ⟨in2p3-00358271⟩

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