Momentum-dependence in the infinitesimal Wilsonian renormalization group

Moritz Helias

doi:10.1088/1751-8121/abb169

Abstract

Wilson’s original formulation of the renormalization group is perturbative in nature. We here present an alternative derivation of the infinitesimal momentum shell renormalization group, akin to the Wegner and Houghton scheme, that is a priori exact. We show that the momentum-dependence of vertices is key to obtain a diagrammatic framework that has the same one-loop structure as the vertex expansion of the Wetterich equation. Momentum dependence leads to a delayed functional differential equation in the cutoff parameter. Approximations are then made at two points: truncation of the vertex expansion and approximating the functional form of the momentum dependence by a momentum-scale expansion. We exemplify the method on the scalar φ 4-theory, computing analytically the Wilson–Fisher fixed point, its anomalous dimension η(d) and the critical exponent ν(d) non-perturbatively in d ∈ [3, 4] dimensions. The results are in reasonable agreement with the known values, despite the simplicity of the method.

Highlights

The renormalization group (RG) is a standard tool to study phase transitions in statistical physics as well as to investigate renormalizable field theories in high energy physics
Its field-theoretical formulation has initially been applied to renormalizable theories only [1]; those, in which all physical quantities can be expressed in terms of a few renormalized parameters
The initial sections set up the notation and present a coherent exposure of the basic idea of the renormalization group: In Section 2.1 we introduce the notation for the field theory to be considered and the φ4-theory as a particular example

Summary

Introduction

The renormalization group (RG) is a standard tool to study phase transitions in statistical physics as well as to investigate renormalizable field theories in high energy physics. Its field-theoretical formulation has initially been applied to renormalizable theories only [1]; those, in which all physical quantities can be expressed in terms of a few renormalized parameters. It was realized later that the field-theoretical formulation is applicable to non-renormalizable theories [5]. Recent examples of non-renormalizable models describe erosion of landscapes [6, 7]. These models are structurally similar to a variant of the KPZ equation [8], the model by [9], whose RG flow was first shown by Antonov and Vasiliev to require arbitrary many couplings [10]

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Discussion

Conclusion