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Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions

Sylvain Carrozza, Daniele Oriti, Vincent Rivasseau

TL;DR

The paper addresses renormalizability of Abelian TGFTs in four dimensions with gauge invariance by developing a multi-scale renormalization framework tailored to TGFTs. It introduces generalized notions of connectedness, locality (traciality), and subgraph contraction, along with melordering—an analogue of Wick ordering for melopoles—that renders the amplitudes finite. The main result is that the $\mathrm{U}(1)$ 4D TGFT with polynomial invariant interactions is super-renormalizable and perturbatively finite after melordering, with melopoles as the only divergences. The work also outlines a path toward non-Abelian generalizations and constructive approaches, aiming to connect TGFT renormalization with quantum gravity dynamics through geometrogenesis and RG flow analyses.

Abstract

We tackle the issue of renormalizability for Tensorial Group Field Theories (TGFT) including gauge invariance conditions, with the rigorous tool of multi-scale analysis, to prepare the ground for applications to quantum gravity models. In the process, we define the appropriate generalization of some key QFT notions, including: connectedness, locality and contraction of (high) subgraphs. We also define a new notion of Wick ordering, corresponding to the subtraction of (maximal) melonic tadpoles. We then consider the simplest examples of dynamical 4-dimensional TGFT with gauge invariance conditions for the Abelian U(1) case. We prove that they are super-renormalizable for any polynomial interaction.

Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions

TL;DR

The paper addresses renormalizability of Abelian TGFTs in four dimensions with gauge invariance by developing a multi-scale renormalization framework tailored to TGFTs. It introduces generalized notions of connectedness, locality (traciality), and subgraph contraction, along with melordering—an analogue of Wick ordering for melopoles—that renders the amplitudes finite. The main result is that the 4D TGFT with polynomial invariant interactions is super-renormalizable and perturbatively finite after melordering, with melopoles as the only divergences. The work also outlines a path toward non-Abelian generalizations and constructive approaches, aiming to connect TGFT renormalization with quantum gravity dynamics through geometrogenesis and RG flow analyses.

Abstract

We tackle the issue of renormalizability for Tensorial Group Field Theories (TGFT) including gauge invariance conditions, with the rigorous tool of multi-scale analysis, to prepare the ground for applications to quantum gravity models. In the process, we define the appropriate generalization of some key QFT notions, including: connectedness, locality and contraction of (high) subgraphs. We also define a new notion of Wick ordering, corresponding to the subtraction of (maximal) melonic tadpoles. We then consider the simplest examples of dynamical 4-dimensional TGFT with gauge invariance conditions for the Abelian U(1) case. We prove that they are super-renormalizable for any polynomial interaction.

Paper Structure

This paper contains 25 sections, 17 theorems, 84 equations, 9 figures.

Table of Contents

  1. Definition of the models
  2. Formal definition
  3. Regularization and the question of renormalizability
  4. Amplitudes
  5. Feynman graphs and gauge symmetry
  6. Multi-scale decomposition of the amplitudes
  7. Dipole moves and reduced graphs
  8. Connectedness and quasi-locality
  9. Contraction of connected subgraphs
  10. High subgraphs, contractiblity, and traciality
  11. Multi-scale analysis of Abelian models
  12. Propagator bounds
  13. Power-counting
  14. Power-counting in a slice
  15. Multi-scale power-counting
  16. ...and 10 more sections

Key Result

Proposition 1

Let ${\mathcal{H}}$ be a subgraph of ${\mathcal{G}}$, and ${\mathcal{H}}_c$ its colored extension. The contracted graph ${\mathcal{G}} / {\mathcal{H}}$ is obtained by:

Figures (9)

  • Figure 1: Some connected tensor invariants in $d = 4$
  • Figure 2: A graph with three vertices, six (internal) lines, and four external legs
  • Figure 3: $k$-dipoles from $k = 1$ (top, left) to $k = 4$ (top, right) in $d = 4$, and their faces (below)
  • Figure 4: A single-line melopole (left), and the result of its contraction
  • Figure 5: Two vacuum graphs with $D_2 (\widetilde{{\mathcal{H}}}) = F_k (\widetilde{{\mathcal{H}}} ) = 0$ for any $k \geq 3$.
  • ...and 4 more figures

Theorems & Definitions (42)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • ...and 32 more