Phenomenologicalσmodels

The question of the phenomenological description of the broad scalar resonances compatible with chiral symmetry and unitarity is discussed within the framework of the linear $\sigma$ model. It is pointed out that a naive inclusion of the large decay widths in the scalar meson propagators is not an adequate approximation for physical amplitudes. It is shown that the simplest unitarization scheme for the real $\pi \pi \to \pi \pi$ Born amplitude with $l=I=0\mathrm{}$ leads to a broad resonance arising with a mass $m_{\mathrm{res}}\approx 420$ MeV and a width ${\Gamma }_{\mathrm{res}}\left(m_{\mathrm{res}}^2\right)\approx 740$ MeV at the bare $\sigma$ meson mass $m_{\sigma }\approx 1$ GeV. The resonance parameters are largely controlled by the nonresonant background amplitude. Some applications to other reactions are also discussed.