Size of theσmeson and its nature

In this work the nature of the σ or f₀(500) resonance is discussed by evaluating its quadratic scalar radius, ⟨r²⟩ₛ^σ. This allows one to have a quantitative estimate for the size of this resonance. We obtain that the σ resonance is a compact object with ⟨r²⟩ₛ^σ=(0.19±0.02)-i(0.06±0.02) fm². Within our approach, employing unitary chiral perturbation theory, the σ 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 σ is then a spread ππ bound state. These results are connected with other recent works that support a nonstandard nature of the σ as well, while fulfilling strong QCD constraints, as well as with lattice QCD. We offer a detailed study of the low-energy S-wave ππ scattering data from where we extract our values for the threshold parameters of S-wave ππ phase shifts, the O(p⁴) chiral perturbation theory low-energy constants as well as the σ pole position. From the comparison with other accurate determinations in the literature we obtain the average values for the isospin $0S$-wave ππ threshold parameters, a₀⁰=0.220±0.003, b₀⁰=0.279±0.003M_π^-2, and for the real and imaginary parts of the σ pole position in √s, 458±14-i261±17 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 σ is a dynamically generated meson-meson resonance.