The Renormalization of Non-Commutative Field Theories in the Limit of Large Non-Commutativity

Abstract

We show that renormalized non-commutative scalar field theories do not reduce to their planar sector in the limit of large non-commutativity. This follows from the fact that the RG equation of the Wilson-Polchinski type which describes the genus zero sector of non-commutative field theories couples generic planar amplitudes with non-planar amplitudes at exceptional values of the external momenta. We prove that the renormalization problem can be consistently restricted to this set of amplitudes. In the resulting renormalized theory non-planar divergences are treated as UV divergences requiring appropriate non-local counterterms. In 4 dimensions the model turns out to have one more relevant (non-planar) coupling than its commutative counterpart. This non-planar coupling is ``evanescent'': although in the massive (but not in the massless) case its contribution to planar amplitudes vanishes when the floating cut-off equals the renormalization scale, this coupling is needed to make the Wilsonian effective action UV finite at all values of the floating cut-off.

arXiv comment35 pages, 8 figures; typos corrected

References in corpus (6)

• Noncommutative Perturbative Dynamics
• Comments on Noncommutative Perturbative Dynamics
• Two-Loop Diagrams in Noncommutative ${\varphi }_4^4$ theory
• Wilsonian Renormalization Group and the Non-Commutative IR/UV Connection
• Nonplanar Anomalies in Noncommutative Theories and the Green-Schwarz Mechanism
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• The fate of the Wilson-Fisher fixed point in non-commutative ϕ^4
• Matrix Model Fixed Point of Noncommutative Phi-Four
• Quantized Noncommutative Geometry from Multitrace Matrix Models
• String Theory and the Fuzzy Torus
• Wilson RG of Noncommutative ${\Phi }_4^4$