Using Analytical Solutions at Large Momentum Transfer to Obtain Zero-Energy, Bound-State, Bethe–Salpeter Solutions of a Scalar and Spin-1/2 Fermion Exchanging Photons

Abstract

The existence of families of leptons and quarks plus the properties of their mass and spin spectra suggest that leptons and quarks might be relativistic bound states of a scalar and a spin-1/2 fermion interacting via minimal electrodynamics. To begin exploring the properties of this bound-state system, the Bethe–Salpeter equation describing bound states of a minimally interacting scalar and spin-1/2 fermion with arbitrary masses is solved in the ladder approximation when the bound-state energy is zero. At large momentum transfer solutions are calculated analytically, yielding boundary conditions that are determined by the coupling constant. Zero-energy solutions, including the coupling constant that is calculated as an eigenvalue, are obtained by expanding the solution in terms of basis functions that obey the boundary conditions, discretizing the Bethe–Salpeter equation, and solving the resulting generalized matrix eigenvalue equation numerically. Since the coupling constant appears both in the Bethe–Salpeter equation as an eigenvalue and in the basis functions, the generalized matrix eigenvalue equation is nonlinear in the coupling constant and is solved iteratively. The spectrum of the coupling constant is discrete.