On the size of the sigma meson and its nature

Abstract

In this work the nature of the \sigma or f_0(600) resonance is discussed by evaluating its quadratic scalar radius, \la r^2\ra_s^\sigma. This allows one to have a quantitative estimate for the size of this resonance. We obtain that the \sigma resonance is a compact object with $\la r^2\ra_s^\sigma=(0.19\pm0.02)-i(0.06\pm0.02)$ fm^2. Within our approach, employing unitary chiral perturbation theory, the \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 \sigma is then a spread \pi\pi bound state. These results are connected with other recent works that support a non standard nature of the \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 \pi\pi scattering data from where we extract our values for the threshold parameters of S-wave \pi\pi phase shifts, the {\cal O}(p^4) chiral perturbation theory low energy constants as well as the \sigma pole position. From the comparison with other accurate determinations in the literature we obtain the average values for the isospin 0 S-wave \pi\pi threshold parameters, $a_0^0=0.220\pm0.003$, $b_0^0=0.279\pm 0.003 M_\pi^{-2}$, and for the real and imaginary parts of the \sigma pole position in \sqrt{s}, $458\pm14-i(261\pm17)$ MeV. The quark mass dependence of the size of the \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 \sigma is a dynamically generated meson-meson resonance.