Renormalization Group Functional Equations

TL;DR

This paper employs functional conjugation methods to analyze the structure of renormalization group trajectories, revealing complex behaviors such as multiple branches and nuanced fixed point definitions.

Contribution

It introduces a framework that derives continuous flows from step-scaling functions and uncovers novel features of flow solutions, including exotic behaviors and non-traditional fixed points.

Findings

Flow solutions can have multiple branches.

Fixed points of step-scaling are not always true fixed points.

Zeroes of beta functions may indicate turning points, not fixed points.

Abstract

Functional conjugation methods are used to analyze the global structure of various renormalization group trajectories, and to gain insight into the interplay between continuous and discrete rescaling. With minimal assumptions, the methods produce continuous flows from step-scaling {\sigma} functions, and lead to exact functional relations for the local flow {\beta} functions, whose solutions may have novel, exotic features, including multiple branches. As a result, fixed points of {\sigma} are sometimes not true fixed points under continuous changes in scale, and zeroes of {\beta} do not necessarily signal fixed points of the flow, but instead may only indicate turning points of the trajectories.