Size of theσmeson and its nature

Abstract

In this work the nature of the $\ensuremath{\sigma}$ or ${f}_{0}(500)$ resonance is discussed by evaluating its quadratic scalar radius, $⟨{r}^{2}{⟩}_{s}^{\ensuremath{\sigma}}$. This allows one to have a quantitative estimate for the size of this resonance. We obtain that the $\ensuremath{\sigma}$ resonance is a compact object with $⟨{r}^{2}{⟩}_{s}^{\ensuremath{\sigma}}=(0.19\ifmmode\pm\else\textpm\fi{}0.02)\ensuremath{-}i(0.06\ifmmode\pm\else\textpm\fi{}0.02)\text{ }\text{ }{\mathrm{fm}}^{2}$. Within our approach, employing unitary chiral perturbation theory, the $\ensuremath{\sigma}$ is a dynamically generated resonance that stems from the pion-pion interactions. Given its small size we conclude that the two pions inside the resonance are merged. A four-quark picture is then more appropriate. However, when the pion mass increases, for pion masses somewhat above 400 MeV, the picture of a two-pion molecule is the appropriate one. The $\ensuremath{\sigma}$ is then a spread $\ensuremath{\pi}\ensuremath{\pi}$ bound state. These results are connected with other recent works that support a nonstandard nature of the $\ensuremath{\sigma}$ as well, while fulfilling strong QCD constraints, as well as with lattice QCD. We offer a detailed study of the low-energy $S$-wave $\ensuremath{\pi}\ensuremath{\pi}$ scattering data from where we extract our values for the threshold parameters of $S$-wave $\ensuremath{\pi}\ensuremath{\pi}$ phase shifts, the $\mathcal{O}({p}^{4})$ chiral perturbation theory low-energy constants as well as the $\ensuremath{\sigma}$ pole position. From the comparison with other accurate determinations in the literature we obtain the average values for the isospin $0S$-wave $\ensuremath{\pi}\ensuremath{\pi}$ threshold parameters, ${a}_{0}^{0}=0.220\ifmmode\pm\else\textpm\fi{}0.003$, ${b}_{0}^{0}=0.279\ifmmode\pm\else\textpm\fi{}0.003{M}_{\ensuremath{\pi}}^{\ensuremath{-}2}$, and for the real and imaginary parts of the $\ensuremath{\sigma}$ pole position in $\sqrt{s}$, $458\ifmmode\pm\else\textpm\fi{}14\ensuremath{-}i261\ifmmode\pm\else\textpm\fi{}17\text{ }\text{ }\mathrm{MeV}$. The quark mass dependence of the size of the $\ensuremath{\sigma}$, its mass and width are considered too. The latter agree accurately with a previous lattice QCD calculation. The fact that the mass of this resonance tends to follow the threshold of two pions is a clear indication that the $\ensuremath{\sigma}$ is a dynamically generated meson-meson resonance.