S-matrix renormalization in effective theories

Abstract

This is the fifth paper in the series devoted to explicit formulation of the rules needed to manage an effective field theory of strong interactions in $S$-matrix sector. We discuss the principles of constructing the meaningful perturbation series and formulate two basic ones: uniformity and summability. Relying on these principles, one obtains the bootstrap conditions which restrict the allowed values of the physical (observable) parameters appearing in the extended perturbation scheme built for a given localizable effective theory. The renormalization prescriptions needed to fix the finite parts of counterterms in such a scheme can be divided into two subsets: minimal, needed to fix the $S$-matrix, and nonminimal, for eventual calculation of Green functions; in this paper we consider only the minimal one. In particular, it is shown that, in theories with the asymptotic behavior governed by known Regge intercepts, the system of independent renormalization conditions only contains those fixing the counterterm vertices with $n\ensuremath{\le}3$ lines, while other prescriptions are determined by self-consistency requirements. Moreover, the prescriptions for $n\ensuremath{\le}3$ cannot be taken arbitrarily: an infinite number of bootstrap conditions should be respected. The concept of localizability, introduced and explained in this article, is closely connected with the notion of resonance in the framework of perturbative quantum field theory. We discuss this point and, finally, compare the cornerstones of our approach with the philosophy known as ``analytic $S$-matrix.''