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Structural stability of the RG flow in the Gross-Neveu model

J. Dimock, Cheng Yuan

Abstract

We study flow of renormalization group (RG) transformations for the massless Gross-Neveu model in a non-perturbative formulation. The model is defined on a d=2 dimensional Euclidean space with a finite volume. The quadratic approximation to the flow stays bounded after suitable renormalization. We show that for weak coupling this property also is true for the complete flow. As an application we prove an ultraviolet stability bound for the model. Our treatment is an application of a method of Bauerschmidt, Brydges, and Slade. The method was developed for an infrared problem, and is now applied to an ultraviolet problem.

Structural stability of the RG flow in the Gross-Neveu model

Abstract

We study flow of renormalization group (RG) transformations for the massless Gross-Neveu model in a non-perturbative formulation. The model is defined on a d=2 dimensional Euclidean space with a finite volume. The quadratic approximation to the flow stays bounded after suitable renormalization. We show that for weak coupling this property also is true for the complete flow. As an application we prove an ultraviolet stability bound for the model. Our treatment is an application of a method of Bauerschmidt, Brydges, and Slade. The method was developed for an infrared problem, and is now applied to an ultraviolet problem.
Paper Structure (33 sections, 40 theorems, 499 equations)

This paper contains 33 sections, 40 theorems, 499 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Renormalization group
  5. The Grassmann algebra
  6. Localized functions and norms
  7. Symmetries
  8. Scaling
  9. Gaussian integrals
  10. A single RG transformation
  11. Some definitions
  12. Statement of the theorem
  13. Start of the proof
  14. Extraction
  15. Fluctuation integral
  16. ...and 18 more sections

Key Result

Lemma 1

Suppose that Then if $F \in {\cal G}_h$ and $G \in {\cal G}_h$ then $FG \in {\cal G}_h$ and $\| FG \|_h \leq \| F\|_h \|G\|_h$

Theorems & Definitions (41)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 1
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 31 more