Roles of $f_0(500)$ and $f_0(980)$ in the $D_{(s)}^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ decays

Abstract

With the recent measurements of the $D_s^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ and $D^{+}\to {\pi }^{+}{\pi }^{+}{\pi }^{-}$ 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.

Comments: 10 pages, 9 figures,