Hierarchical renormalization goup fixed points

TL;DR

This paper explores the connection between hierarchical renormalization group transformations and non-associative algebras, demonstrating how fixed points are solutions to polynomial equations, with methods illustrated through a scalar model in multiple dimensions.

Contribution

It introduces a novel approach linking hierarchical RG fixed points to algebraic equations and presents solution methods using a scalar model in higher dimensions.

Findings

Fixed points correspond to solutions of polynomial equations

Methods for solving algebraic equations are demonstrated

Hierarchical RG transformations relate to non-associative algebras

Abstract

Hierarchical renormalization group transformations are related to non-associative algebras. Non-trivial infrared fixed points are shown to be solutions of polynomial equations. At the example of a scalar model in $d(\ge2)$ dimensions some methods for the solution of these algebraic equations are presented.