3D Tensor Field Theory: Renormalization and One-loop $β$-functions

TL;DR

This work establishes the perturbative renormalizability at all orders for a rank-3 tensor field theory on $U(1)^3$, proven via a momentum-space multiscale analysis that exploits a locality principle and topological power-counting. It computes the one-loop $\gamma$- and $\beta$-functions, showing that a model with a single coupling and a single wave-function renormalization is asymptotically free in the UV, and identifies several renormalizable anisotropic extensions that preserve the essential melonic structure. The analysis combines slice-decomposed propagators, face-factorized amplitudes, and dipole-contraction techniques to classify divergent graphs and implement renormalization in momentum space, aligning tensor GFT methods with standard RG concepts. The results support the tensor-model approach as a promising framework for emergent geometry and quantum gravity, with clear avenues for generalization to other groups and higher ranks.

Abstract

We prove that the rank 3 analogue of the tensor model defined in [arXiv:1111.4997 [hep-th]] is renormalizable at all orders of perturbation. The proof is given in the momentum space. The one-loop $γ$- and $β$-functions of the model are also determined. We find that the model with a unique coupling constant for all interactions and a unique wave function renormalization is asymptotically free in the UV.

Figures (13)

• Figure 1: Propagator.
• Figure 2: Vertices of the type $V_{4}$.
• Figure 3: A graph with a multiscale expansion ($L1$ is at scale $i=15$, etc.) and its Gallavotti-Nicolò tree; the face $f1$ (in red) is external and the face $f2$ (in green) is internal. $l_f$ is the strand element at given scale $i$ which should optimize the bound.
• Figure 4: A graph ${\mathcal{G}}$ (left) and its color extension $\mathcal{G}_{\text{color}}$ (right) in simplified notations: each line of color $\alpha=0,1,2,3$ corresponds to a propagator in the colored theory, i.e. $\int d\mu_C(\varphi) \bar{\varphi}^\alpha_{123}\varphi^\alpha_{123}$, and vertices are defined by Eq.\ref{['vertexcolo']}.
• Figure 6: The boundary ${\partial\mathcal{G}}$ of ${\mathcal{G}}$ and its rank two or ribbon structure.

Theorems & Definitions (14)

• Theorem 1
• Theorem 2
• Lemma 1
• Lemma 2: Prime power-counting
• Definition 1
• Theorem 3
• Definition 2: $0k$-dipole and contraction
• Proposition 1: Graph contraction
• Lemma 3: Decreasing genera
• Lemma 4: Genus bounds