On Renormalization Group Flows and Polymer Algebras

TL;DR

This paper discusses a rigorous approach to controlling the renormalization group flow in field theories using polymer algebras, simplifying analysis and solving large volume and field problems.

Contribution

It introduces polymer algebras and recursive functions to replace cluster expansions, providing a new framework for RG analysis in various field theories.

Findings

Polymer activities effectively handle large volume and field problems.

Polymer algebras simplify RG flow analysis.

Application examples include $4$-model, $O(N)$ $$-model, and hierarchical scalar field theory.

Abstract

In this talk methods for a rigorous control of the renormalization group (RG) flow of field theories are discussed. The RG equations involve the flow of an infinite number of local partition functions. By the method of exact beta-function the RG equations are reduced to flow equations of a finite number of coupling constants. Generating functions of Greens functions are expressed by polymer activities. Polymer activities are useful for solving the large volume and large field problem in field theory. The RG flow of the polymer activities is studied by the introduction of polymer algebras. The definition of products and recursive functions replaces cluster expansion techniques. Norms of these products and recursive functions are basic tools and simplify a RG analysis for field theories. The methods will be discussed at examples of the $\Phi^4$-model, the $O(N)$ $\sigma$-model and hierarchical scalar field theory (infrared fixed points).