Zero-Energy Solutions of the Bethe-Salpeter Equation for a Spinor-Scalar System Exchanging Photons

Abstract

Zero-energy, bound-state solutions are obtained in the ladder approximation to the Bethe-Salpeter equation describing a spinor and a scalar with arbitrary masses that interact via quantum electrodynamics. The coupling constant is found to be almost independent of the masses of the two constituents whereas the solutions themselves depend significantly on the constituent masses. Even though the Bethe-Salpeter equation is separable in the zeroenergy limit, the feature that makes obtaining numerical solutions difficult is still present: it does not appear to be possible to discretize the Bethe-Salpeter equation in such a way that the coupling constant, which is real in the Lagrangian, is also always real when calculated as an eigenvalue of the discretized Bethe-Salpeter equation. Nevertheless, solutions with real coupling constants are obtained using a systematic, numerical method that yields both zero- and finite-energy solutions to many, if not all, two-body, bound-state Bethe-Salpeter equations. As part of the process of obtaining solutions, boundary conditions satisfied by the solutions are determined analytically. At large momenta such boundary conditions are often determined only within ranges. Because eigenvalues are sensitive to the behavior of solutions at large momenta, an iterative, numerical procedure is used to ensure that the input boundary conditions and the boundary conditions actually obeyed by each numerical solution agree. Among the solutions that are found, those that are the zero-energy limit of normalizable, finite-energy solutions are identified.