On the S-matrix renormalization in effective theories

TL;DR

This paper develops principles and bootstrap conditions for managing effective field theories of strong interactions, focusing on S-matrix renormalization and the role of localizability and Regge behavior.

Contribution

It introduces a systematic approach to S-matrix renormalization in effective theories, emphasizing bootstrap conditions and the concept of localizability related to resonances.

Findings

Renormalization conditions for n ≤ 3 lines are sufficient in Regge-governed theories.

Bootstrap conditions impose infinite constraints on counterterm prescriptions.

The approach connects localizability with the resonance concept in perturbative QFT.

Abstract

This is the 5-th paper in the series devoted to explicit formulating 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 non-minimal -- 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 amplitudes which asymptotic behavior is governed by known Regge intercepts, the system of independent renormalization conditions only contains those fixing the counterterm vertices with $n \leq 3$ lines, while other prescriptions are determined by self-consistency requirements. Moreover, the prescriptions for $n \leq 3$ cannot be taken arbitrary: 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 QFT. We discuss this point and, finally, compare the corner stones of our approach with the philosophy known as ``analytic S-matrix''.