Higher-order corrections in nonrenormalizable quantum field theories

Abstract

Within the framework of Lagrangian quantum field theory, the perturbative expansion of the S-matrix leads to a power series in the coupling constant with diverging coefficients. These infinities can be properly removed, through the renormalization program, only in a few cases, while in all the other cases a general reliable prescription is not yet available. However, it is possible that finite S-matrix elements exist also for nonrenormalizable theories, even if the single terms of the perturbative expansion are diverging. This assumption is supported by some specific field theoretical models (1-3) and enables us to infer that nonrenormalizable theories may be physically meaningful, if we are able to sum the whole perturbative series. For instance, in the intermediate boson theory of weak interactions, it has been shown that the sum of an infinite subset of Feynman graphs, the so-called (( uncrossed ladder graphs ~>, can be obtained by solving an integral equation and gives finite results. This procedure, however, is not completely justified because the contributions coming from the missed diagrams are not, a pr ior i , negligible. In this paper we want to propose a general summation procedure for perturbative series with divergent terms, arising from nonrenormalizable Lagrangians. Let us start from the complete set of Feynman graphs relative to some physical process and let us regularize the divergent integrals by means of a proper cut-off parameter A s in order to make the series finite term by term. We are then faced with the infinite sum