On the nature of the lightest scalar resonances

We briefly review the recent progresses in the new unitarization approach being developed by us. Especially we discuss the large Nc ππ scatterings by making use of the partial wave S matrix parametrization form. We find that the σ pole may move to the negative real axis on the second sheet of the complex s plane, therefore it raises the interesting question that this ‘σ ’ pole may be related to the σ in the linear σ model. The problem of how to restore unitarity and meanwhile respecting chiral perturbation amplitudes at low energies is very interesting and also difficult. A simple solution one has when dealing with such a difficult problem is the Padé approximation and its variations, which achieved some phenomenological success. Nevertheless, the Padé approximation encounters serious problems [1] which can hardly be resolved within the method itself. For this reason, it is worthwhile to make further efforts to study the problem from a more rigorous and different point of view. In Refs. [2, 3, 4], a new parametrization form – which we call as the ‘PKU’ parametrization form – for partial wave S matrices in the elastic channel is developed, which, when combined with chiral symmetry, has been proven useful in probing the resonance structure of low energy strong interaction dynamics. For example, it reveals that the existence of the σ [5, 4] meson is fully consistent with chiral symmetry. Combining with crossing symmetry, it further predicts the σ pole mass and width to be Mσ = 470± 50MeV, Γσ = 570± 50MeV. [4] Also it is shown that there should exist the κ resonance if the πK scattering length in the I,J=1/2,0 channel does not deviate much from the value predicted by chiral perturbation theory. [3] The PKU parametrization form is the following, Sphy. = ∏ i S i ·Scut , (1) where S i denote various kinds of poles: resonance, bound state and virtual bound state. For resonance poles we have SR(s) = M(z0)− s+ iρ(s)sG[z0] M2(z0)− s− iρ(s)sG[z0] , (2) 1 Talk presented by Zheng at “Quark Confinement and Hadron Spectroscopy VI”, 21–25 Sept. 2004, Cagliari, Italy