Existence and Analyticity insof the Solution to Non-Fredholm Bethe-Salpeter Equations for Scattering of Spin-½ Particles

Abstract

A large class of non-Fredholm Bethe-Salpeter equations for the scattering Green's function of spin-½ particles is considered. Many commonly treated equations belong to this class; an example is the quantum-electrodynamical (QED) electron-positron Bethe-Salpeter (BS) equation in the ladder approximation. It is proven that a solution of these equations by successive iterations exists. A domain of analyticity of the solution in the $s$ plane is derived. For small coupling constant, this domain is very large. The domain shrinks to the point $s=0\mathrm{}$ when the coupling constant is increased to a limiting value (of order unity). A simple and general inequality is derived for the binding energy of any bound state in terms of the coupling constant. It is proven that Goldstein's pseudoscalar homogeneous solution to the QED equation at $s=0\mathrm{}$ does not correspond to a bound state when the fine-structure constant is less than of order unity. The method of proof of the existence and analyticity is to use majorization to show that an infinite series of analytic functions is uniformly convergent. No Hilbert-space techniques are used, so no difficulties arise from the non-Fredholm nature of the BS equations.