Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Discrete Renormalization Group for SU(2) Tensorial Group Field Theory

Sylvain Carrozza

TL;DR

This work develops a Wilsonian, perturbative renormalization group for a rank-3 SU(2) tensorial group field theory and introduces a discrete RG using dimensionless couplings guided by power counting. It shows that melonic graphs dominate UV divergences and derives melonic flow equations, revealing two marginal directions corresponding to the $u_{6,1}$ and $u_{6,2}$ couplings and two unstable directions around the Gaussian fixed point. Second-order corrections demonstrate that positive marginal perturbations do not yield asymptotic freedom, while negative perturbations render the Gaussian fixed point UV-attractive, albeit in a regime that may be non-perturbatively delicate. The results clarify the RG structure of TGFTs and point to potential nontrivial UV behavior, Landau-pole scenarios, or nonperturbative fixed points, motivating future investigations via epsilon-expansion and alternative RG methods.

Abstract

This article provides a Wilsonian description of the perturbatively renormalizable Tensorial Group Field Theory introduced in arXiv:1303.6772 [hep-th] (Commun. Math. Phys. 330, 581-637). It is a rank-3 model based on the gauge group SU(2), and as such is expected to be related to Euclidean quantum gravity in three dimensions. By means of a power-counting argument, we introduce a notion of dimensionality of the free parameters defining the action. General flow equations for the dimensionless bare coupling constants can then be derived, in terms of a discretely varying cut-off, and in which all the so-called melonic Feynman diagrams contribute. Linearizing around the Gaussian fixed point allows to recover the splitting between relevant, irrelevant, and marginal coupling constants. Pushing the perturbative expansion to second order for the marginal parameters, we are able to determine their behaviour in the vicinity of the Gaussian fixed point. Along the way, several technical tools are reviewed, including a discussion of combinatorial factors and of the Laplace approximation, which reduces the evaluation of the amplitudes in the UV limit to that of Gaussian integrals.

Discrete Renormalization Group for SU(2) Tensorial Group Field Theory

TL;DR

This work develops a Wilsonian, perturbative renormalization group for a rank-3 SU(2) tensorial group field theory and introduces a discrete RG using dimensionless couplings guided by power counting. It shows that melonic graphs dominate UV divergences and derives melonic flow equations, revealing two marginal directions corresponding to the and couplings and two unstable directions around the Gaussian fixed point. Second-order corrections demonstrate that positive marginal perturbations do not yield asymptotic freedom, while negative perturbations render the Gaussian fixed point UV-attractive, albeit in a regime that may be non-perturbatively delicate. The results clarify the RG structure of TGFTs and point to potential nontrivial UV behavior, Landau-pole scenarios, or nonperturbative fixed points, motivating future investigations via epsilon-expansion and alternative RG methods.

Abstract

This article provides a Wilsonian description of the perturbatively renormalizable Tensorial Group Field Theory introduced in arXiv:1303.6772 [hep-th] (Commun. Math. Phys. 330, 581-637). It is a rank-3 model based on the gauge group SU(2), and as such is expected to be related to Euclidean quantum gravity in three dimensions. By means of a power-counting argument, we introduce a notion of dimensionality of the free parameters defining the action. General flow equations for the dimensionless bare coupling constants can then be derived, in terms of a discretely varying cut-off, and in which all the so-called melonic Feynman diagrams contribute. Linearizing around the Gaussian fixed point allows to recover the splitting between relevant, irrelevant, and marginal coupling constants. Pushing the perturbative expansion to second order for the marginal parameters, we are able to determine their behaviour in the vicinity of the Gaussian fixed point. Along the way, several technical tools are reviewed, including a discussion of combinatorial factors and of the Laplace approximation, which reduces the evaluation of the amplitudes in the UV limit to that of Gaussian integrals.

Paper Structure

This paper contains 24 sections, 14 theorems, 171 equations, 22 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model and its divergences
  3. Definition and Feynman expansion
  4. Divergences and power-counting
  5. Contraction operators
  6. Renormalization group flow
  7. Wilson's effective action and dimensionless coupling constants
  8. Discrete renormalization group flow
  9. Reducible graphs
  10. Melonic flow equations
  11. Relevant and irrelevant directions around the Gaussian fixed point
  12. Linearized flow equations
  13. Irrelevant, marginal and relevant directions
  14. Computation of the relevant and marginal eigendirections
  15. Second order contributions to the flow of marginal coupling constants
  16. ...and 9 more sections

Key Result

Proposition 1

The symmetry factor $k({\mathcal{G}})$ associated to an arbitrary Feynman graph ${\mathcal{G}}$ is the order of its group of automorphisms.

Figures (22)

  • Figure 1: Examples of $3$-colored graphs.
  • Figure 2: Three graphs with different symmetry factors: $k({\mathcal{G}}_1) = 1$, $k({\mathcal{G}}_2) = 2$ and $k({\mathcal{G}}_3) = 3$.
  • Figure 3: From left to right: $1$-, $2$-, $3$- and $4$-dipoles, together with their contractions.
  • Figure 4: A melopole with two lines. $\{ l_1 \}$ and $\{l_1 , l_2 \} / \{l_1 \}$ are $3$-dipoles, as illustrated by the successive contractions of $l_1$ and $l_2$.
  • Figure 5: A melonic graph which reduces to a melopole after contraction of a single tree line.
  • ...and 17 more figures

Theorems & Definitions (35)

  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 25 more