Unitarity Sufficiency Condition in Some Concrete Cases

Abstract

We have examined numerically the question whether the sufficiency conditions of the existence and uniqueness of the solution for the unitarity equation are met in e-H and rc+p scattering. Neither case is completely realistic, but in the absence of better examples they provide the closest cases for testing these conditions. In the first instance the closed form of the Born amplitude for elastic scattering from the ground state has been used. Even though the spin is neglected and the Born approximation is not realistic for the case at hand this example was chosen to investigate for the first time the Newton condition on the unitarity integral in a multi-channel problem. It is found that in the elastic region this condition is met at all energies and it looks like that it may even be met in the low inelastic region. We also study the case of rc+ p scattering neglecting the spin for which the differential cross section is taken from experiments but reconstructed from Lovelace phase shifts. The sufficiency conditions of solubility and uniqueness for the unitarity equation have been studied by several authorsll~n in the last decade. Comparison of the results with experiments is realistically not possible. The reason for this is that even in the simplest case which is scalar particle scattering (spinless particles) one has to know the modulus of the amplitude either as an analytic expression from theory or as a real function from experiment or phenomenology. Unfortuna­ tely no such case is known. Suppose now there is a scalar case for which the amplitude is known from phase shifts. The choice of such a set of phase shifts makes the very question one wants to investigate redundant. Because of the ambi­ guities in the phase shifts there may be one or more sets of phase shifts which fit the same differential cross sections (leaving aside also the experimental uncer­ tainties). Those phase shifts determine not only the modulus of the amplitude, but also the phase. Thus there is no need left to test the existence and uniqueness conditions for the solutions of the unitarity integral. For these reasons any test of those conditions is bound to be more or less unrealistic but will serve to give an idea about the class of' functions which can represent modulus functions. In a few cases such tests were made; either the spin was neglected and the scattering was considered to be represented by a scalar