Bound-state problem in a model nonpolynomial-Lagrangian theory

Abstract

We have shown that the bound-state problem in a nonpolynomial-Lagrangian theory can be solved as in the usual polynomial theory assuming that a Wick rotation is admissible. For definiteness we have assumed the interaction of two scalar fields interacting via the exchange of a superfield to be of the form U(x)=exp[gφ(x)], where g denotes the minor coupling constant and φ(x) is a massless neutral scalar field. The major coupling constant is introduced through the ladder diagrams in the Bethe-Salpeter formalism. We find that the Wick-rotated Bethe-Salpeter equation reduces to a standard Fredholm equation with a modified kernel corresponding to the exchange of the superfield U(x). To study the physical content in the theory we have investigated the equation in the instantaneous approximation. The resulting nonrelativistic equation is projected onto the surface of a four-dimensional sphere by using Fock's transformation variables. The bound-state eigenvalue problem is solved approximately in the weak-binding limit, using Hecke's theorem, leading to a Balmer-type formula. Finally, the fully relativistic equation at E=0 is considered by transforming it onto the surface of a five-dimensional Euclidean sphere. The approximate-symmetry property of the equation is studied, and the eigenvalue problem is solved in terms of the coupling constants of the theory.