An Impact Parameter Formalism

Abstract

A relativistic impact-parameter formalism is introduced without any approximation. It is applicable for all physical values of energy and scattering angle. An impact parameter amplitude is defined from an invariant scattering amplitude by the integral with the Bessel function, in place of the Legendre function in the definition of a partial-wave amplitude. We find that this impact parameter amplitude is an example of a new integral expansion. It is proved that one of the necessary and sufficient conditions that this new expansion is permissible, is that the amplitude should satisfy the Kapteyn equation, which is required in the Webb-Kapteyn theory of the Neumann series expansion of an odd function. The physical interpretation of the amplitude is discussed and cross sections are expressed by simple integrals over the impact parameter without any approximation. The unitarity relation for the amplitude is expressed in the simpler form. Comparison is made with three other relativistic formalisms proposed by Blankenbecler and Goldberger, Ida, and Cottingham and Peiels. In contrast with our formalism, the applications of these three formalisms are limited either to the extremely high energy region or to only the forward angle scattering.