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One-loop \(\beta \) functions of a translation-invariant renormalizable noncommutative scalar model. (English) Zbl 1175.81175

Summary: 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 \(a/(\theta^{2} p^{2})\) 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 behavior 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.

MSC:

81T75 Noncommutative geometry methods in quantum field theory
81T10 Model quantum field theories
46L65 Quantizations, deformations for selfadjoint operator algebras
53D55 Deformation quantization, star products
33B15 Gamma, beta and polygamma functions
46F10 Operations with distributions and generalized functions
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