Overlapping Resonances in Dispersion Theory

Abstract

The Khuri-Treiman dispersion representation is applied to the discussion of overlapping resonances among particles in production and decay final states. The kernel of the dynamical equation following from the Khuri-Treiman representation has branch points overlapping the integration contour, but recently reported work permits us to select the correct branch of the kernel. We thus eliminate all restrictions on the masses of the final-state particles or strengths of the resonances. An iteration procedure is developed for the solution of the dynamical equation when three spinless particles are present in the final state. There is no restriction on the angular momentum of the resonances, but for simplicity only $s$-wave resonances are considered here. Plausibility arguments are given which indicate that for narrow resonances the onceiterated approximation to the solution is a good approximation. A detailed study of all higher approximations supports this assertion. In the once-iterated approximation, one finds a branch point on the second sheet of the transition amplitude which may cause a characteristic variation of the amplitude near the low-energy boundaries of the physical region. This variation is studied quantitatively for the kinematically favorable reaction $N+N\to N+N+\pi$, and is found to be of negligible importance. The suppression of the variation is related to the threshold behavior of two-particle scattering amplitudes.