Renormalisation of Noncommutative ϕ4-Theory by Multi-Scale Analysis

In this paper we give a much more efficient proof that the real Euclidean ϕ4-model on the four-dimensional Moyal plane is renormalisable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalisation proof based on renormalisation 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 rôle 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.

[1]  Renormalisation of Φ4-Theory on Non-Commutative $$\mathbb{R}^{4}$$ to All Orders , 2004, hep-th/0403232.

[2]  Non-commutative Renormalization , 2004, 0705.0705.

[3]  Alain Connes,et al.  Noncommutative Geometry and Matrix Theory: Compactification on Tori , 1997, hep-th/9711162.

[4]  Power-Counting Theorem for Non-Local Matrix Models and Renormalisation , 2003, hep-th/0305066.

[5]  Shiraz Minwalla,et al.  Noncommutative perturbative dynamics , 2000 .

[6]  Nathan Seiberg,et al.  String theory and noncommutative geometry , 1999 .

[7]  Convergence theorem for non-commutative Feynman graphs and renormalization , 2000, hep-th/0008090.

[8]  Grosse,et al.  The β-function in duality-covariant noncommutative φ 4-theory , 2008 .

[9]  H. Grosse,et al.  Renormalisation of φ^4-theory on noncommutative R^4 to all orders , 2004 .

[10]  Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces , 2003, hep-th/0308043.

[11]  V. Schomerus.  D-branes and Deformation Quantization , 1999, hep-th/9903205.

[12]  The $\beta$-function in duality-covariant non-commutative $\phi^4$-theory , 2004, hep-th/0402093.

[13]  Vincent Rivasseau,et al.  From Perturbative to Constructive Renormalization , 1991 .

[14]  Joseph Polchinski,et al.  Renormalization and effective lagrangians , 1984 .

[15]  H. Grosse,et al.  Renormalisation of ϕ4-Theory on Noncommutative ℝ4 in the Matrix Base , 2004, hep-th/0401128.

[16]  K. Wilson,et al.  The Renormalization group and the epsilon expansion , 1973 .