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Masses and widths of scalar-isoscalar multi-channel resonances from data analysis

Peculiarities of obtaining parameters for broad multi-channel resonances from data are discussed analyzing the experimental data on processes $ππ\toππ,K\bar{K}$ in the $I^GJ^{PC}=0^+0^{++}$ channel in a model-independent approach based on analyticity and unitarity and using an uniformization procedure. We show that it is possible to obtain a good description of the $ππ$ scattering data from the threshold to 1.89 GeV with parameters of resonances cited in the PDG tables as preferred. However, in this case, first, representation of the $ππ$ background is unsatisfactory; second, the data on the coupled process $ππ\to K\bar{K}$ are not well described even qualitatively above 1.15 GeV when using the resonance parameters from the only $ππ$ scattering analysis. The combined analysis of these coupled processes is needed, which is carried out satisfactorily. Then both above-indicated flaws, related to the analysis of solely the $ππ$-scattering, are cured. The most remarkable change of parameters with respect to the values of only $ππ$ scattering analysis appears for the mass of the $f_0 (600)$ which is now in some accordance with the Weinberg prediction on the basis of mended symmetry and with an analysis using the large-$N_c$ consistency conditions between the unitarization and resonance saturation. The obtained $ππ$-scattering length $a_0^0$ in case when we restrict to the analysis of the $ππ$ scattering or consider so-called A-solution (with a lower mass and width of $f_0(600)$ meson) agrees well with prediction of chiral perturbation theory (ChPT) and with data extracted at CERN by the NA48/2 Collaboration from the analysis of the $K_{e4}$ decay and by the DIRAC Collaboration from the measurement of the $π^+π^-$ lifetime.