Equivalence of nonrenormalizable theories to renormalizable theories in a composite ($Z=0\mathrm{}$) limit

A graphical parallelism between $\left(\overline{\psi }\psi \right){\phi }^2$ and $\left(\overline{\psi }\chi \right)\phi$ interaction theories is investigated by means of skeleton diagrams. It is shown that the nonrenormalizable $\left(\overline{\psi }\psi \right){\phi }^2$ theory can be made renormalizable, provided it contains a bound state expressed by the composite field $\chi =\psi \phi$. In order for the quartic interaction theory to be renormalizable it is necessary that the renormalization constant for the $\chi$ field vanishes in the corresponding Yukawa-type theory. A concrete example in which such a situation is realized is given by a new soluble model—a hybrid of the Lee model and the Ruijgrok-Van Hove model. The corresponding quartic interaction model is seen to be renormalizable due to the renormalization scheme presented here.