Size of the σ meson and its nature

In this work the nature of the sigma or f(0)(500) resonance is discussed by evaluating its quadratic scalar radius, < r(2)>(sigma)(s). 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 < r(2)>(sigma)(s) = (0.19 +/- 0.02) - i(0.06 +/- 0.02) 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 nonstandard 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 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 0S-wave pi pi threshold parameters, a(0)(0) = 0.220 +/- 0.003, b(0)(0) = 0.279 +/- 0.003M(pi)(-2), and for the real and imaginary parts of the sigma pole position in root s, 458 +/- 14 - i261 +/- 17 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.