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

TL;DR

This paper develops a unitarized scalar resonance model for the decay D+ to pi+ pi- pi+, accurately reproducing the pi-pi mass distribution and branching ratios by combining scalar form factors with isobar models for higher waves.

Contribution

It introduces a unitarized scalar form factor approach for the S-wave pion-pion interaction in D+ decays, improving the description of scalar resonances and decay distributions.

Findings

Successfully reproduces the pi-pi mass distribution and branching ratios.

Provides a prediction for the D to sigma transition form factor.

Highlights discrepancies with experimental branching ratios, suggesting areas for further refinement.

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 $q^2=m_\pi^2$.