Why Two Renormalization Groups are Better than One

TL;DR

This paper advocates for using multiple renormalization groups to better analyze complex systems with multiple length scales, providing more accessible calculations and fewer required inputs.

Contribution

It introduces the concept of employing more than one RG to handle systems with multiple diverging length scales, demonstrating advantages over single RG approaches.

Findings

Multiple RGs provide complementary insights.

Using multiple RGs reduces the physical input needed.

It enables solving problems with multiple diverging length scales.

Abstract

The advantages of using more than one renormalization group (RG) in problems with more than one important length scale are discussed. It is shown that: i) using different RG's can lead to complementary information, i.e. what is very difficult to calculate with an RG based on one flow parameter may be much more accessible using another; ii) using more than one RG requires less physical input in order to describe via RG methods the theory as a function of its parameters; iii) using more than one RG allows one to solve problems with more than one diverging length scale. The above points are illustrated concretely in the context of both particle physics and statistical physics using the techniques of environmentally friendly renormalization. Specifically, finite temperature $\lambda\phi^4$ theory, an Ising-type system in a film geometry, an Ising-type system in a transverse magnetic field, the QCD coupling constant at finite temperature and the crossover between bulk and surface critical behaviour in a semi-infinite geometry are considered.