Roles of $f_{0}(500)$ and $f_{0}(980)$ in the $D_{(s)}^{+}\rightarrow\pi^{+}\pi^{+}\pi^{-}$ decays

TL;DR

This paper investigates the roles of the $f_{0}(500)$ and $f_{0}(980)$ resonances in $D_{(s)}^{+} ightarrow ext{pi}^+ ext{pi}^+ ext{pi}^-$ decays, explaining their different contributions through a chiral unitary approach aligned with recent experimental data.

Contribution

The study provides a theoretical explanation for the distinct resonance contributions in $D_{s}^{+}$ and $D^{+}$ decays using a chiral unitary framework, aligning with recent LHCb measurements.

Findings

The model reproduces the experimental magnitudes and phases of $ ext{pi}^+ ext{pi}^-$ S-wave amplitudes.

It confirms $f_{0}(500)$ as a $ ext{pi} ext{pi}$ resonance and $f_{0}(980)$ as a $Kar{K}$ resonance.

The results support the dynamical generation of these resonances from meson interactions.

Abstract

With the recent measurements of the $D_{s}^{+}\rightarrow \pi^{+}\pi^{+}\pi^{-}$ and $D^{+}\rightarrow \pi^{+}\pi^{+}\pi^{-}$ decays by the LHCb Collaboration, we study these two decay processes by considering the final state interaction formalism. Taking into account the external and internal $W$-emission dominant mechanisms at the quark level, our model can naturally explain why the $D_{s}^{+}$ decay only has a contribution from the $f_{0}(980)$ resonance, while the $D^{+}$ decay has contributions from both the $f_{0}(500)$ and $f_{0}(980)$ states. The magnitudes and phases of $\pi^{+}\pi^{-}$ for $S$-wave amplitudes in our model are in agreement with the experimental measurements. These results support the interpretations of $f_{0}(500)$ and $f_{0}(980)$ as the $\pi\pi$ and $K\bar{K}$ resonances, respectively, where they are dynamically generated from the $S$-wave pseudoscalar-pseudoscalar meson interactions within the chiral unitary approach.