Bootstrap equations in effective theories

TL;DR

This paper explores the constraints on effective field theories imposed by bootstrap equations, emphasizing the limited freedom in choosing renormalization conditions to ensure perturbative correctness.

Contribution

It provides a detailed analysis of how bootstrap equations restrict the parameters and renormalization conditions in localizable effective field theories.

Findings

Severe restrictions on renormalization conditions for effective theories.

Conditions for the correctness of tree-level amplitude construction.

Derivation and analysis of bootstrap equations in effective theories.

Abstract

We consider the general properties of effective field theories. We note that the freedom to fix the renormalization conditions in the effective field theory is not as great as it seems. The consideration of minimal requirements of correctness of the perturbative scheme based on Dyson's formula for S-matrix leads to severe restriction on the essential parameters of a theory and, hence, on the allowed set of renormalization conditions. In the first part we make a short review of the structure of localizable effective field theories. We discuss the conditions needed to ensure the correctness of the very first step of perturbative calculation of the S-matrix -- the construction of the tree-level amplitudes. In the second part we consider the examples demonstrating the main stages of derivation and analysis of the system of bootstrap equations.