Chapter 12 Solution of the Bethesalpeter Equation by Means of Padé Approximants

Abstract

This chapter discusses the solution of the Bethe–Salpeter (BS) equation by means of Pade approximants. During the past few years there has been some interest in the solution of the BS equation in the ladder approximation as a relativistic model for two-body scattering. The study of the case of scalar particles with equal masses is useful for understanding the general properties of the BS equations and their solutions. This chapter presents the integral equations for πN scattering using the ladder approximation with nucleon exchange as potential. Any attempt to reproduce the main features of low-energy πN scattering would require a more sophisticated potential. Two features may appear in the πN system: the nucleon spin and the unequal masses of the external particles. The unequal masses change the locations of the singularities, but do not affect the method of solution in an essential way. However, it turns out that the presence of spin makes it technically difficult to solve the integral equations by matrix inversion because of the dimensions of the kernel.