Coupled Dynamical Equations Incorporating Unitarity and Crossing. I. ππ and πK Scattering

An equation proposed by Zimmermann, which takes account of the normal thresholds in s, t, and u channels, is generalized to two-body reactions involving particles of any spin or isospin, and extended to include a number of coupled processes in each channel. In this form, the resulting equations, besides satisfying the requirements of low-energy unitarity and crossing symmetry, take some account of the effect of all reactions in each channel, as well as the detailed energy-dependent effects of one or two low-mass processes. The coupled ππ and πK system is considered, and approximate solutions are found which give the ρ and K^* resonance parameters within 15% of the accepted experimental values, as well as a κ resonance in the I=12 s-wave πK partial wave. The ππ s-wave phase shifts in this case are found to be very negative, decreasing rapidly to -π but with the coupling of the ππ-->πω reaction to this system; a solution is obtained giving an I=0, ππ s-wave which, at low energies, is in agreement with some recent phenomenological analyses.