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Bosonic Colored Group Field Theory

Joseph Ben Geloun, Jacques Magnen, Vincent Rivasseau

TL;DR

This work analyzes perturbative bounds for bosonic colored group field theories, focusing on the 4D colored Ooguri (BF) model and proving optimal large-spin bounds that scale as $|\mathcal{A}_\mathcal{G}| \le K^n \Lambda^{9n/2+9}$ and, in general dimension $D$, as $|\mathcal{A}_\mathcal{G}| \le K^n \Lambda^{3(D-1)(D-2)n/4 + 3(D-1)}$. A key result is the absence of generalized tadpoles in the colored Ooguri model, which enables controlled power counting and the construction of a two-color Matthews–Salam representation that is positive for imaginary coupling. By integrating two colors and applying a cactus expansion with the Brydges–Kennedy forest formula, the authors establish a pathway toward constructive analysis and potential renormalization of colored GFTs in any dimension, with extensions toward more physically realistic models such as EPRL-FK. The work thus strengthens the renormalization program for GFTs and provides rigorous bounds and reformulations that facilitate nonperturbative approaches and multiscale analyses.

Abstract

Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the "ultraspin" (large spin) limit. The results are generalized in any dimension. Finally integrating out two colors we write a new representation which could be useful for the constructive analysis of this type of models.

Bosonic Colored Group Field Theory

TL;DR

This work analyzes perturbative bounds for bosonic colored group field theories, focusing on the 4D colored Ooguri (BF) model and proving optimal large-spin bounds that scale as and, in general dimension , as . A key result is the absence of generalized tadpoles in the colored Ooguri model, which enables controlled power counting and the construction of a two-color Matthews–Salam representation that is positive for imaginary coupling. By integrating two colors and applying a cactus expansion with the Brydges–Kennedy forest formula, the authors establish a pathway toward constructive analysis and potential renormalization of colored GFTs in any dimension, with extensions toward more physically realistic models such as EPRL-FK. The work thus strengthens the renormalization program for GFTs and provides rigorous bounds and reformulations that facilitate nonperturbative approaches and multiscale analyses.

Abstract

Bosonic colored group field theory is considered. Focusing first on dimension four, namely the colored Ooguri group field model, the main properties of Feynman graphs are studied. This leads to a theorem on optimal perturbative bounds of Feynman amplitudes in the "ultraspin" (large spin) limit. The results are generalized in any dimension. Finally integrating out two colors we write a new representation which could be useful for the constructive analysis of this type of models.

Paper Structure

This paper contains 9 sections, 5 theorems, 35 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Perturbative bounds of colored models
  3. The colored Ooguri model
  4. Properties of Feynman diagrams
  5. Perturbative bounds
  6. D dimensional colored GFT
  7. Integration of two fields: first steps of constructive analysis
  8. Conclusion
  9. Appendix: The case of tadpoles

Key Result

Lemma 2.1

Given a $N$-point graph with $n$ internal vertices, if one color is missing on the external legs, then

Figures (9)

  • Figure 1: The propagator or covariance of the colored model.
  • Figure 2: The vertices of the colored model: Pentachores I and II are associated with interactions of the form $\phi^5$ and $\bar{\phi}^5$, respectively.
  • Figure 3: Generalized tadpoles.
  • Figure 4: The vertex operators.
  • Figure 5: The chains of graphs $G^{23}_{n}$ and $G^{14}_{n}$ with $2n$ vertices.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Lemma 2.1
  • Definition 2.1
  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.2
  • Theorem 2.2
  • Theorem 2.3