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Outer automorphisms are sufficient conditions for RG fixed points

Thede de Boer, Andreas Trautner

Abstract

We point out that the existence of an outer automorphism (Out) is a sufficient condition for the existence of a fixed hyperplane (fixed point, separatrix) in the renormalization group (RG) flow of a Quantum Field Theory (QFT). The corresponding RG fixed hyperplane is determined by a symmetry argument and can be computed without resorting to perturbation theory. This provides the mathematical underpinning of 't Hooft's technical naturalness argument, and results in a systematic way to derive non-perturbative all-order constraints on the RG beta functions. If an Out exists, the symmetry of the fully coupled system of beta functions is larger than the symmetry of the action. We also stress the importance of including goofy transformations in these considerations.

Outer automorphisms are sufficient conditions for RG fixed points

Abstract

We point out that the existence of an outer automorphism (Out) is a sufficient condition for the existence of a fixed hyperplane (fixed point, separatrix) in the renormalization group (RG) flow of a Quantum Field Theory (QFT). The corresponding RG fixed hyperplane is determined by a symmetry argument and can be computed without resorting to perturbation theory. This provides the mathematical underpinning of 't Hooft's technical naturalness argument, and results in a systematic way to derive non-perturbative all-order constraints on the RG beta functions. If an Out exists, the symmetry of the fully coupled system of beta functions is larger than the symmetry of the action. We also stress the importance of including goofy transformations in these considerations.
Paper Structure (9 sections, 2 theorems, 40 equations, 2 figures)

This paper contains 9 sections, 2 theorems, 40 equations, 2 figures.

Table of Contents

  1. Introduction
  2. General Argument (in Words)
  3. Example I
  4. Example II
  5. General Argument (in formulae)
  6. Discussion
  7. Conclusions
  8. Beta functions for Example II
  9. Motivating Eq.(23)

Key Result

Corollary 1

If the couplings fulfill $F_n(\lambda)=\lambda_n$, then the transformation $\phi\mapsto\tilde{\phi}$ leaves the Lagrangian invariant. If this holds for a subgroup of $\mathrm{Out}(G)$ then this subgroup is a symmetry of the theory.

Figures (2)

  • Figure 1: Example II: RG flow in the $\lambda$-$\lambda_p$ plane. The red and green colored lines correspond to the (partial) fixed points listed in Eq. \ref{['eq:D8FPs']}. The blue arrows illustrate the action of the $\mathds{Z}_{2}$ outer automorphism transformation $u$, Eq. \ref{['eq:D8Out']}, as mapping on the parameter space and RG flow.
  • Figure 2: Example diagrams for contributions to $\beta_\lambda$ (left) and $\beta_{\lambda_p}$ (right) at the two- and three-loop order. $\phi_1$ ($\phi_2$) propagators are drawn as solid (dashed) lines. We indicate the parametric order as well as the sign-flips of $\kappa_2$, corresponding to $\phi_2$-propagator sign-flips under the goofy transformation $w$.

Theorems & Definitions (2)

  • Corollary 1
  • Corollary 2