Asymptotic restrictions and sum rules for the characteristics of. pi sigma. resonances

Sum rules for the characteristics of ..pi sigma.. resonances are derived from the condition of an asymptotic restriction (S. Weinberg, Phys. Rev. 177, 2604 (1969); V.V. Vereshchagin, Nucl. Phys. B55, 621 (1973)) on the amplitude for ..pi sigma.. scattering in chiral dynamics. Those of the sum rules which can be checked numerically on the basis of decay data are found to be in agreement with the available data.