A Renormalizable 4-Dimensional Tensor Field Theory (1111.4997v3)

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 $\phi^6$ rather than of the $\phi^4$ type, since two different $\phi^6$-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 $(\int \phi^2)^2$ term, which can be interpreted as the generation of a scalar matter field out of pure gravity.

• The paper constructs a renormalizable 4D tensor model by introducing unique φ⁶ interactions and counterterms based on a new locality principle.
• It employs a multiscale analysis with a power counting theorem to identify the melonic divergent graphs essential for renormalization.
• An unexpected log-divergent term hints at the generation of a scalar matter field from pure gravity, opening new avenues for quantum gravity research.

A Renormalizable 4-Dimensional Tensor Field Theory

The paper "A Renormalizable 4-Dimensional Tensor Field Theory" by Joseph Ben Geloun and Vincent Rivasseau introduces a novel four-dimensional tensor model capable of renormalization to all orders in perturbation theory. The work adapts the Gurau colored tensor model, a construct pivotal for developing quantum gravity models in higher dimensions, and supplements it with a U(1)^4 Bosonic propagator. This paper provides the first example of such a renormalizable model, aligning with the goal of quantizing space-time in four-dimensional Euclidean gravity.

Key Contributions

1. Model Construction: The paper constructs a four-dimensional quantum field model featuring a unique $\phi^6$ interaction type instead of the conventional $\phi^4$. Notably, two distinct $\phi^6$ type interactions are identified as logarithmically divergent, becoming marginal in the renormalization group sense.
2. Multiscale Analysis: The renormalization process leverages a multiscale analysis, employing a power counting theorem to identify all divergent graphs. This method centers on discerning the contribution of graphs with "melonic" internal and external structures, which dominate in the 1/N expansion of colored tensor models, analogous to planar ribbon graphs in matrix models.
3. New Locality Principle: A groundbreaking locality principle is introduced, permitting the renormalization of divergences in melonic graphs via counterterms reflecting the form of bare Lagrangian interactions. This principle is crucial for ensuring that the divergences can be counteracted appropriately within the theory.
4. Anomalous Log-Divergent Term: An unexpected log-divergent term, $(\int \phi^2)^2$, emerges, potentially indicating the generation of a scalar matter field from pure gravity. This anomaly suggests new pathways for interpreting the interactions within tensorial models devoid of traditional locality.

Theoretical and Practical Implications

The presented model steps into a broader context where conventional approaches to quantum gravity have encountered hurdles, particularly regarding a renormalizable Einstein-Hilbert action modeled around a flat space. By integrating tensor and group field theories more abstractly, these models avoid some of the pitfalls like non-renormalizability.

• Quantum Gravity Modeling: By establishing a renormalizable framework, this model provides a foundation for more feasibly combining quantum field theories with theories of space-time quantization. The tensor field theory offers a potential path away from traditional locality and symmetry assumptions, suggesting Euclidean quantum field theories as effective phases that emerge from such non-traditional backgrounds.
• Path for Future Research: The success of the 4D tensor model invites further scrutiny and excitation in the paper of tensor field theories across dimensions greater than two. The paper hints at a potential future where tensor models themselves comprise the next foothold in the quest for a coherent quantum gravity theory, potentially also encompassing and explaining known physics phenomena in novel ways.

Overall, the work embodies a significant stride in advancing the theory surrounding rank-higher-than-2 tensor fields, offering a blend between the renormalization group formalism and modern approaches to quantum gravity. The results show promise in providing a framework for constructing new models that are simple, renormalizable, and possibly more reflective of the complex dynamics of quantum space-time.