A two-particle model for high-energy collisions

Summary  This paper studies a very simplified model of high-energy collisions of strongly interacting particles. It introduces two particles A and B, where A can be regarded as stable and B as an unstable resonance of A. It assumes that all high-energy collisions can be described as taking place between the three channels A+A, A+B and B+B, so that theS-matrix is 3×3. Taking further A and B to be self-conjugate, it enables one to express all consequences of crossing symmetry in terms of the nineS-matrix elements. From these crossing relations and from unitarity it is shown that for each orbital momentumI theS-matrix depends on a single real parametera l which has a simple physical significance in terms of theK-matrix. Constant high-energy cross-sections are obtained whenal is a functiona(ϱ) of the impact parameter. Whena(ϱ) is a smoothly decreasing function ofϱ approaching zero forϱ → ∞ more rapidly than any power ofϱ -1, and has moderately large values for smallϱ, one finds that the elastic differential cross-sections derived from the model agree with the experimental shape of the diffraction peak and are insensitive to the detailed form ofa(ϱ). Ifa(ϱ) is very large for allϱ for which it does not vanish the elastic angular distributions become of the black sphere type. This insensitivity is due to the strong damping imposed by unitarity of theS-matrix whenever theK-matrix elements are appreciate.