Theoretical support for the $\pi (1300)$ and the recently claimed $f_0(1790)$ as molecular resonances

Abstract

A study of three-pseudoscalar $\pi K\bar{K}$ and $\pi \pi \eta$ coupled system is made by solving the Faddeev equations within an approach based on unitary chiral dynamics. A resonance with total isospin one and spin-parity $J^{\pi }=0^{-}$ is found with mass $\sim$ 1400 MeV when the $K\bar{K}$ system gets reorganized as the $f_0(980)$. This resonance is identified with the $\pi (1300)$ listed by the Particle Data Group. Further, the two-body amplitude which describes the interaction between a $\pi$ and the $f_0(980)$ is extracted from the study of the $\pi K\bar{K}$ and $\pi \pi \eta$ system and is then employed to study the $f_0(980)\pi \pi$ system. As a result, a scalar resonance is found near 1790 MeV which drives the two $f_0(980)\pi$ systems to resonate as the $\pi (1300)$ while the invariant mass of the two pions falls in the mass region of the scalar $\sigma (600)$. These findings support the existence of a new $f_0$ resonance near 1790 MeV, as found by the BES and Crystal Barrel collaborations. Our results show that this $f_0(1790)$ is definitely distinct to $f_0(1710)$, the latter of which seems to possess a glueball structure dominantly.