Renormalization of noncommutative phi 4-theory by multi-scale analysis

In this paper we give a much more efficient proof that the real Euclidean phi 4-model on the four-dimensional Moyal plane is renormalizable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalization proof based on renormalization group equations for non-local matrix models. On the other hand, our bounds permit a powerful multi-scale analysis of the resulting ribbon graphs. Here, the dual graphs play a particular r\^ole because the angular momentum conservation is conveniently represented in the dual picture. Choosing a spanning tree in the dual graph according to the scale attribution, we prove that the summation over the loop angular momenta can be performed at no cost so that the power-counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph.Comment: 34 page

Two and Three Loops Beta Function of Non Commutative Φ44\Phi^4_4 Theory

The simplest non commutative renormalizable field theory, the $\phi_4^4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe at one loop, as shown by H. Grosse and R. Wulkenhaar. We extend this result up to three loops. If this remains true at any loop, it should allow a full non perturbative construction of this model.Comment: 24 pages, 7 figure

EPRL/FK Group Field Theory

The purpose of this short note is to clarify the Group Field Theory vertex and propagators corresponding to the EPRL/FK spin foam models and to detail the subtraction of leading divergences of the model.Comment: 20 pages, 2 figure

Two and four-loop β\beta-functions of rank 4 renormalizable tensor field theories

A recent rank 4 tensor field model generating 4D simplicial manifolds has been proved to be renormalizable at all orders of perturbation theory [arXiv:1111.4997 [hep-th]]. The model is built out of $\phi^6$ ($\phi^6_{(1/2)}$), $\phi^4$ ($\phi^4_{(1)}$) interactions and an anomalous term ($\phi^4_{(2)}$). The $\beta$-functions of this model are evaluated at two and four loops. We find that the model is asymptotically free in the UV for both the main $\phi^6_{(1/2)}$ interactions whereas it is safe in the $\phi^4_{(1)}$ sector. The remaining anomalous term turns out to possess a Landau ghost.Comment: 31 pages, 31 figures; improved versio

Exorcizing the Landau Ghost in Non Commutative Quantum Field Theory

We show that the simplest non commutative renormalizable field theory, the $\phi^4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe to all orders in perturbation theor

One-loop β\beta functions of a translation-invariant renormalizable noncommutative scalar model

Recently, a new type of renormalizable $\phi^{\star 4}_{4}$ scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a $a/(\theta^2p^2)$ term. We calculate here the $\beta$ and $\gamma$ functions at one-loop level for this model. The coupling constant $\beta_\lambda$ function is proved to have the same behaviour as the one of the $\phi^4$ model on the commutative $\mathbb{R}^4$. The $\beta_a$ function of the new parameter $a$ is also calculated. Some interpretation of these results are done.Comment: 13 pages, 3 figure

The 1/N expansion of colored tensor models in arbitrary dimension

In this paper we extend the 1/N expansion introduced in [1] to group field theories in arbitrary dimension and prove that only graphs corresponding to spheres S^D contribute to the leading order in the large N limit.Comment: 4 pages, 3 figure

Overview of the parametric representation of renormalizable non-commutative field theory

We review here the parametric representation of Feynman amplitudes of renormalizable non-commutative quantum field models.Comment: 10 pages, 3 figures, to be published in "Journal of Physics: Conference Series

Commutative limit of a renormalizable noncommutative model

Renormalizable $\phi^{\star 4}_4$ models on Moyal space have been obtained by modifying the commutative propagator. But these models have a divergent "naive" commutative limit. We explain here how to obtain a coherent such commutative limit for a recently proposed translation-invariant model. The mechanism relies on the analysis of the uv/ir mixing in general Feynman graphs.Comment: 11 pages, 3 figures, minor misprints being correcte

Bounding bubbles: the vertex representation of 3d Group Field Theory and the suppression of pseudo-manifolds

Based on recent work on simplicial diffeomorphisms in colored group field theories, we develop a representation of the colored Boulatov model, in which the GFT fields depend on variables associated to vertices of the associated simplicial complex, as opposed to edges. On top of simplifying the action of diffeomorphisms, the main advantage of this representation is that the GFT Feynman graphs have a different stranded structure, which allows a direct identification of subgraphs associated to bubbles, and their evaluation is simplified drastically. As a first important application of this formulation, we derive new scaling bounds for the regularized amplitudes, organized in terms of the genera of the bubbles, and show how the pseudo-manifolds configurations appearing in the perturbative expansion are suppressed as compared to manifolds. Moreover, these bounds are proved to be optimal.Comment: 28 pages, 17 figures. Few typos fixed. Minor corrections in figure 6 and theorem