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

Abstract

We propose a model for $D^+ \to π^+ π^- π^+$ decays following experimental results which indicate that the two-pion interaction in the $S$-wave is dominated by the scalar resonances $f_0(600)/σ$ and $f_0(980)$. The weak decay amplitude for $D^+\to R π^+$, where $R$ is a resonance that subsequently decays into $π^+π^-$, is constructed in a factorization approach. In the $S$-wave, we implement the strong decay $R\to π^-π^+$ by means of a scalar form factor. This provides a unitary description of the pion-pion interaction in the entire kinematically allowed mass range $m_{ππ}^2$ 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 $ρ(770)^0$, we use a Breit-Wigner description. Higher waves are accounted for by using the usual isobar prescription for the $f_2(1270)$ and $ρ(1450)^0$. The major achievement is a good reproduction of the experimental $m_{ππ}^2$ 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 $D\to σ$ transition form factor at $q^2=m_π^2$.

23 pages, 2 figures. Two new equations. The value for the strength of the contribution of the scalar form factor now agrees with other results in the literature. Main results unchanged. Version to appear in Phys. Rev. D