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 $U(x)=\mathrm{exp}[g\ensuremath{\varphi}(x)]$, where $g$ denotes the minor coupling constant and $\ensuremath{\varphi}(x)$ 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.