Analysis on the composite nature of the light scalar mesons $f_{0}(980)$ and $a_0(980)$

TL;DR

This paper investigates the internal meson-meson structure of the $f_0(980)$ and $a_0(980)$ resonances, revealing dominant components and emphasizing the importance of proper parameterization and coupling distinctions.

Contribution

It introduces a comparative analysis of compositeness methods for these resonances, highlighting the meson-meson dominance in $f_0(980)$ and subdominance in $a_0(980)$, with detailed decay and compositeness coefficients.

Findings

$f_0(980)$ is predominantly meson-meson composite.

$a_0(980)$ has subdominant meson-meson components.

Proper distinction between bare and dressed couplings is crucial.

Abstract

We study the weight or compositeness of the $\pi\pi$-$K\bar{K}$ and $\pi\eta$-$K\bar{K}$ in the composition of the $f_0(980)$ and $a_0(980)$ resonances, respectively. Either we use the saturation of the total width and compositeness, or we use a Flatt\'e parameterization taking also into account the spectral function of a near-threshold resonance. We make connections and compare between these two methods. We take input values for the pole mass and width from several determinations in the literature. In addition, we take as third input either the total compositeness or the decay-width branching ratio to the lighter channel for each resonance. It turns out that for the poles considered the meson-meson components are dominant for the $f_0(980)$, while for the $a_0(980)$ resonance they are subdominant. We also provide partial decay widths and partial compositeness coefficients, so that the $K\bar{K}$ component is the most important one for the $f_0(980)$. Additionally, this study stresses the need to distinguish between the bare and dressed couplings and widths in a Flatt\'e parameterization. We elaborate on the connection between the partial-decay widths calculated in terms of the dressed couplings and the actual measured ones. Due to the coupled-channel dynamics when the pole lies near the heavier threshold in the second Riemann sheet some changes are needed with respect to standard relations.