Study of pion-hyperon resonances by dispersion relations in the unphysical sheet

Abstract

A procedure has been developed for the study of resonances in multichannel two-particle systems. It is shown that the S matrix in the angular-momentum representation can be diagonalized by a real orthogonal transformation in the physical energy region, where all channels under consideration are open. The analytic properties of the transformation matrix and of the eigenamplitudes are examined in detail. The scattering matrix is continued across the physical section of the unitarity cut into the unphysical sheet, and the condition for the occurrence of resonance poles in that sheet is determined in general. Using the pion-hyperon scattering problem as an illustration of the calculational procedure, we find the locations of the singularities of the partial-wave amplitudes from the double-spectral representation and calculate explicity the nearby singularities, which are used to determine the analytic structure of the eigen-amplitudes in the unphysical sheet. We put in a conjugate pair of resonance poles, the real part of whose positions is regarded as a parameter of the theory; the imaginary part, being the half-width of the resonance, is related to the coupling constants by dispersion relations. The branching ratio is inferred from the orthogonal matrix that effects the diagonalization.