Study the molecular nature of $\sigma$, $f_0(980)$, and $a_0(980)$ states

Abstract

We investigate the characteristics of $\sigma$, $f_0(980)$, and $a_0(980)$ with the formalism of chiral unitary approach. With the dynamical generation of them, we make a further study of their properties by evaluating the couplings, the compositeness, the wave functions and the radii. We also research their properties in the single channel interactions, where the $a_0(980)$ can not be reproduced in the $K\bar{K}$ interactions with isospin $I=1$ since the potential is too weak. In our results, the states of $\sigma$ and $f_0(980)$ can be dynamically reproduced stably with varying cutoffs both in the coupled channel and the single channel cases. We find that the $\pi \eta$ components is much important in the coupled channel interactions to dynamically reproduce the $a_0(980)$ state, which means that $a_0(980)$ state can not be a pure $K\bar{K}$ molecular state. We obtain their radii as: $|⟨r^2⟩{|}_{f_0(980)}=1.80\pm 0.35$ fm, $|⟨r^2⟩{|}_{\sigma }=0.68\pm 0.05$ fm and $|⟨r^2⟩{|}_{a_0(980)}=0.94\pm 0.09$ fm. Based on our investigation results, we conclude that the $f_0(980)$ state is mainly a $K\bar{K}$ bound state, the $\sigma$ state a resonance of $\pi \pi$ and the $a_0(980)$ state a loose $K\bar{K}$ bound state. From the results of the compositeness, they are not pure molecular states and have something non-molecular components, especially for the $\sigma$ state.

Comments: Comments and acknowledgments added