Numerical Method for Solving Two-Body Bound-State Bethe-Salpeter Equations for Unequal Constituent Masses

 A theoretical technique is developed for obtaining finite-energy numerical solutions to a class of two-body, bound-state Bethe-Salpeter equations in the ladder approximation when the constituent masses are unequal. The class of equations is restricted to those for which the Bethe-Salpeter equation can be written as a differential equation and to situations where the coupling constant is real. Such equations can result when the binding force is created by the exchange of a massless quanta. The theoretical technique is tested numerically by obtaining finite-energy solutions of the partially-separated Bethe-Salpeter equation describing the unequal-mass Wick-Cutkosky model in the ladder approximation. Received February 19, 1997; Revised April 2, 1998; accepted for publication October 30, 1998     

Charge-magnetic-dipole bound states using the Klein-Gordon equation

We investigate bound states of a composite system consisting of a charged particle orbiting a neutral, stationary magnetic dipole. We find all bound states are metastable and none exist with angular momentum less than eleven. Our calculations is performed in two space dimensions.     

The Importance of Boundary Conditions in Solving the Bound-State Bethe-Salpeter Equation

The characteristic of bound-state, Bethe-Salpeter equations that makes them so difficult to solve numerically can be overcome, in some if not many cases, by expanding solutions in terms of basis functions that obey the boundary conditions that are satisfied by the solutions. The utility of such basis functions is demonstrated by calculating the zero-energy, bound-state solutions of a spin-0 boson and a spin-&frac; fermion with unequal masses. The constituents interact via scalar electrodynamics and are described by the Bethe-Salpeter equation in the ladder approximation. Although the Bethe-Salpeter equation that is solved is separable in the zero-energy limit, the feature that typically prevents solutions from being obtained numerically is still present. A technique for calculating boundary conditions, which is readily generalized to other Bethe-Salpeter equations, is discussed in detail.Supported by a grant from the Ohio Supercomputer CenterReceived January 31, 2003; accepted April 4, 2003 Published online August 25, 2003     

Are Leptons, Quarks or Both Highly Relativistic Bound States of a Minimally Interacting Fermion and Scalar?

The possibility that leptons, quarks or both might be highly relativistic bound states of a spin-0 and spin-1/2 constituent bound by minimal electrodynamics is discussed in the context of the two-body, bound-state Bethe?CSalpeter equation. For most interactions, strongly bound solutions exist only when the coupling constant is on the order of or greater than unity. However, for the bound-state system discussed here, in the strong-binding limit there exist two classes of boundary conditions that could yield solutions with coupling constants on the order of the fine structure constant. In both classes only bound states with spin 1/2 can exist, thus providing a possible explanation for the absence of higher spin leptons and quarks. Also, a mechanism for the suppression of the decay??? ?? e?+??? exists.     

Numerical Solutions to the Partially Separated,Equal-Mass,Wick-Cutkosky Model

We discuss a numerical technique for solving four-dimensional, relativistic, bound-state, two-body equations that have not been completely separated. The angular variables are first separated what is always possible for a rotationally invariant system. The resulting partially separated equation is, in general, a set of coupled integral or partial differential equations in two variables that is solved numerically by expressing the solutions in terms of B-splines. We demonstrate the efficacy of the method by solving the partially separated Bethe-Salpeter equation for the equal-mass, Wick-Cutkosky model in the ladder approximation. Received January 21, 1994; revised September 25, 1994; accepted for publication October 15, 1994     

General Relativistic Bound States of a Fermion and a Scalar Interacting via a Massive Scalar

A complete set of numerical solutions is obtained, in the ladder approximation, to the Bethe-Salpeter equation describing bound states of a spin- fermion and spin-0 boson with arbitrary masses that interact via the exchange of a massive, spin-0 boson. The equation has been used previously, without solutions actually being calculated, to derive some properties of nucleons by treating the physical nucleon as a bound state of a “bare” nucleon and a “bare” meson. It is likely that most, if not all, two-body, bound-state Bethe-Salpeter equations can be solved in the ladder approximation using the method discussed here.     

Fixed point theorem for the Poincaré group

The fixed points for an arbitrary proper Poincaré transformation are found by exploiting the homomorphism between the Lorentz group andSL(2,C).     

Cabibbo current and CP violation in a six quark gauge model

The extended Cabibbo current with CP-violation in a six quark gauge model is obtained in terms of quark mass ratios and phases in the quark mass matrices. The model predicts a long-lived heavy quark and its selection rule.