Renormalization of noncommutative $\phi^4$-theory by multi-scale analysis

Vincent Rivasseau (1) , F. Vignes-Tourneret (1) , Raimar Wulkenhaar

1 LPT - Laboratoire de Physique Théorique d'Orsay [Orsay]

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\ôle 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.

Mots clés: Noncommutative geometry; Renormalisation; Quantum field theory

Domaines: Physique mathématique [math-ph]; Physique des Hautes Energies - Théorie [hep-th]

Vincent Rivasseau, F. Vignes-Tourneret, Raimar Wulkenhaar. Renormalization of noncommutative $\phi^4$-theory by multi-scale analysis. Communications in Mathematical Physics, 2005, 262, pp.565-594. ⟨10.1007/s00220-005-1440-4⟩. ⟨hal-00020032⟩