Internal Symmetries in a Coupled-Channel Soluble Model with Inelasticity
James T. Cushing · Physical Review · 1966
The conjecture that an internal symmetry group may be selected by a bootstrap mechanism is studied within the framework of a closed, exactly soluble model. In order to have two-body unitarity and crossing without necessarily identically zero amplitudes, two types of particles of different mass (each possessing internal quantum numbers corresponding to an unspecified internal symmetry group and each assigned to irreducible representations of this group) are allowed to scatter in a two-dimensional world. The symmetry group makes its appearance explicitly only through the crossing matrix, characterized by a parameter which is to be determined self-consistently. All two-body channels, both elastic and inelastic, are treated exactly. Unitary, crossing-symmetric, analytic scattering amplitudes corresponding to the various processes are constructed for continuous ranges of the parameter of the crossing matrix. Even with the additional constraint of a self-consistency requirement in the form of Levinson's theorem, an internal symmetry group is not selected by the bootstrap mechanism in this model. Also, a technique is developed for converting a coupled set of singular, linear Cauchy integral equations into an equivalent uncoupled Fredholm set.