A Renormalizable 4-Dimensional Tensor Field Theory

TL;DR

This work constructs and proves the perturbative renormalizability to all orders of a four-dimensional tensor field theory built from a Gurau colored tensor model with a standard $(-\Delta+m^2)^{-1}$ propagator on $U(1)^4$. The analysis relies on a multiscale decomposition, leading to a precise power-counting bound controlled by melonic (rank-4 tensor) graphs and their jackets, with a new locality principle enabling renormalization via a finite set of marginal/relevant interactions up to order six. A notable outcome is the emergence of an anomalous log-divergent term $\big(\int \phi^2\big)^2$, interpreted as scalar matter generation from gravity, which is absorbed by an additional fourth-order counterterm. The results establish a consistent renormalization program for a tensor-type RG in 4D, providing a concrete step toward a quantum-gravity framework based on pregeometric tensor dynamics and inviting further study of RG flows, symmetries, and potential phase transitions in this class of models.

Abstract

We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on $U(1)^4$ is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the $φ^6$ rather than of the $φ^4$ type, since two different $φ^6$-type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent $(\int φ^2)^2$ term, which can be interpreted as the generation of a scalar matter field out of pure gravity.

Figures (11)

• Figure 1: The propagator.
• Figure 2: Vertices of the type $V_{6;1}$ (left) and $V_{6;2}$ (right).
• Figure 3: Vertices of the type $V_{4;1}$ (left) and $V_{4;2}$ (right).
• Figure 4: A graph with a given scale attribution: lines of the graph are $\{L1,L2,L3,L4\}$ with scale $\{10,15,8,11\}$; the face $f1$ (in red) is open and formed by the strands $l1,l4,l3$; the face $f2$ (in green) is closed and formed by $l2,l3'$. All $G^{(k=1)}_i$ have a unique connected component. The trees $T_{f1}$ and $T_{f2}$ decomposed on scales appear as $T_{f1, i}$ and $T_{f2, i}$ the set of which forms a spanning forest of $f1$ and $f2$, respectively.
• Figure 5: A graph ${\mathcal{G}}$, its colored extension $\mathcal{G}_{\text{color}}$ (five valence vertices with colored (half-)lines), the jacket subgraph $J$$(01234)$ of $\mathcal{G}_{\text{color}}$ and its associated pinched jacket ${\widetilde{J}}$.

Theorems & Definitions (17)

• Theorem 1
• Lemma 1
• Lemma 2
• Definition 1
• Theorem 2
• Definition 2: $0k$-dipole and contraction
• Lemma 3: Graph contraction
• Lemma 4
• Definition 3: Jacket inclusion
• Lemma 5: Genus bounds