Properties of the Salpeter-Bethe Two-Nucleon Equation

Abstract

The relativistic two-nucleon equation of Salpeter and Bethe is examined in the ladder approximation for large binding energy, with an invariant interaction function. The binding energy is considered an adjustable parameter and the coupling constant, $\frac{g^2}{4\pi }$, is taken as the eigenvalue of the problem. As a starting point for this study, the special case of two equal masses and binding energy equal to the total mass is considered. It is found that in this case the equation may be simplified to a remarkable degree. For zero quantum mass, a one-dimensional integral equation in momentum space is obtained, and solved, in closed form. The solutions can also be displayed in closed form in configuration space. Solutions exist corresponding to this binding energy for all positive $g^2$. Requiring normalizability on a space-like surface in configuration space eliminates solutions for sufficiently small $g^2$, but a normalizable continuum remains. Arguments are presented to show that this continuum is not due to the choice of binding energy, but is, in fact, characteristic of the invariant equation. It is shown that by introducing a high frequency cutoff into the particle propagators and then going to the limit of infinite cutoff, the remaining continuum is reduced to a single physically sensible solution.