Study of Bound-State Solutions to the Pion-Nucleon Bethe-Salpeter Equation

The Bethe-Salpeter equation describing the interaction of pseudoscalar mesons and nucleons via pseudoscalar coupling is solved numerically for energies below the elastic threshold by use of variational techniques. We consider only the "ladder" approximation with a local potential corresponding to the exchange of an elementary nucleon. Simple generalizations of this form of the interaction are considered as well. In the absence of a cutoff, this leads to a marginally singular integral equation. We examine in detail the boundary conditions to be imposed on the solutions in order to lead to a discrete eigenvalue spectrum. The study of this problem is considerably simplified at zero total c.m. energy, where the (Wick-rotated) equation is invariant under four-dimensional rotations. In order to take full advantage of this symmetry, we construct a new set of spinor spherical harmonics belonging to the representations ($\frac{1}{2}(n\pm 1)$, $\frac{1}{2}n$) and ($\frac{1}{2}n,\frac{1}{2}(n\pm 1)$) of the four-dimensional rotation group. The discussion is then extended to the general case, in which we examine briefly the formal structure of the $E\ne 0$ solutions.