The light scalars: four- vs. two-quark states in the complex energy plane from Bethe-Salpeter equations

Abstract

We study the dynamical generation of scalar mesons in the light quark sector ($q\in\{u,d,s\}$) and calculate the masses and widths for the $f_0(500),a_0(980)$ and $f_0(980)$. To this end we study the mixing of conventional $q\bar{q}$ and `exotic' $q\bar{q}q\bar{q}$ states via a coupled set of two-body Bethe-Salpeter equations based on a symmetry-preserving truncation of the underlying Dyson-Schwinger equations. This allows us to determine the dominant components of each state. Furthermore, we extend our previous framework into the complex energy plane such that we can study the analytic structure of the states in question and extract their width. At the physical point of small quark masses, the $σ$ meson is predominantly a $ππ$ resonance. Consequently, its mass and width is driven by the effects of chiral symmetry breaking. At larger quark masses, however, the conventional $q\bar{q}$ components take over. Furthermore, we find a strong molecular $K\bar{K}$ component for both, the strange-light $f_0(980)$ and the $a_0(980)$.