Convergence of Yukawa Theories with a Finite Number of Interacting Boson Modes

Abstract

Convergence of the perturbation expansion of the S matrix is demonstrated for a field theory where a quantized fermion field with regularized propagator interacts with a finite number of quantized boson modes. Conventional graphical techniques and combinatorial analysis are used to establish this result, derived earlier by Edwards, but we believe not properly clarified. A zero radius of convergence is demonstrated for the theory where a Yukawa-type interaction takes place between one boson field interacting in only a finite number of modes with another boson field coupled to it bilinearly. The relation of this result to the convergence of infinite coupled mode theories is discussed.