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On the size of the sigma meson and its nature

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