We show that rational Lagrangians of the type G@(l + ,kj-yl can be renormalized by introducing a finite class of infinite counter terms providing that the Dyson index v0 -"I < 3. The form of the counter terms is explicitly exhibited. The theories become unrenormalizable when Y 0 -v1 > 3; we discuss, in particular, the case q, -y1 = 4, which resembles a g#" theory for $ --, co, and is nonrenormalizable, contrary to what one may have naively expected.
1. Introduction
In two earlier papers [l], the problem of infinities associated with nonpolynomial Lagrangians, in general, and rational Lagrangians, in particular, was considered. In this paper, we study the problem of absorbing these infinities into a renormalization of constants in the theory by a small number of counterterms. Our results can be stated for the typical Lagrangian V = G@(l + h~$-~l[~ 2 1, V, integer] and sums thereof:
(i) When V, -+ < 1 all S-matrix elements are completely free of infinities and the theory is superrenormalizable.
(ii) When v0 -vr = 2 and 3, there are a finite number of distinct types of infinities in the S-matrix elements which can be absorbed into a finite class of counterterms that serve to renormalize the constants in the theory.
(iii) When v0 -v1 > 4, the theory is nonrenormalizable. This conclusion is especially interesting for the case v,, -v1 = 4; thus, although the polynomial interaction gqS4 is renormalizable by itself, we find that the rational interaction G#'/(l + hq5) which possesses the same limit when 4 -+ co is not.