Exact Renormalization Group Equations and the Field Theoretical Approach to Critical Phenomena

Abstract

After a brief presentation of the exact renormalization group equation, we illustrate how the field theoretical (perturbative) approach to critical phenomena takes place in the more general Wilson (nonperturbative) approach. Notions such as the continuum limit and the renormalizability and the presence of singularities in the perturbative series are discussed.This paper has two parts. In the first part we restrict ourselves to the presentation of some selected issues taken from a recent review on the exact renormalization group (RG) equation (ERGE) in the pure scalar case.[1] In the second part we illustrate the Wilson continuum limit of field theory (or, equivalently, what may be understood as the nonperturbative renormalizability). More generally we would like to indicate how the historical first version of the RG, based on a perturbative approach, takes place in the more general nonperturbative framework developed by Wilson. The illustration will be done in the light of actual RG trajectories obtained from a numerical study of the ERGE in the local potential approximation. [2] 1 Selected issues What does "exact" mean ?The word "exact" means: a continuous (i.e. not discrete) realization of the Wilson RG transformation of the action in which no approximation is made and no expansion is involved with respect to some small parameter. We are here only interested in the differential form of the ERGE. One could think that the word exact is not adapted to the Wilson renormalization because, compared to the standard version of renormalization 1 , it involves a finite cutoff and it is only a semi-group. Nevertheless the historical first version is entirely contained in the Wilson theory (see part 2).

An Introduction to the Nonperturbative Renormalization Group

“…Bagnuls and Bervillier have used the following metaphor: [49,51] the RG trajectories on the critical surface are like rivers in the mountains. In the valley, there is a large river L along which the flow is slow.…”

A holographic form for Wilson's RG

“…Another instance of this was in the loop variable approach[5] where an extra string coordinate was introduced whose zero mode was dual to the scale. The scale also appears as a dimension in[6] in brane engineered QCD.2 Further developments are reviewed in, for instance,[11][12][13][14][15][16] …”

Gradient flow and the renormalization group

“…It is tempting to regard our equation (2.5) as a sort of exact renormalization group [13,16,17,18] (see [19,20,21] for a nice review on this subject). However, one must be careful in establishing this relationship, because the RG interpretation of (2.5) is possible only when we make a field-variable transformation at every step such that the kinetic term is kept in the canonical form [see discussions below (3.1)].…”