Closed star product on noncommutative ${\mathbb{R}}^3$ and scalar field dynamics

Abstract

We consider the noncommutative space ${\mathbb{R}}_{\theta }^3$, a deformation of ${\mathbb{R}}^3$ for which the star product is closed for the trace functional. We study one-loop IR and UV properties of the 2-point function for real and complex noncommutative scalar field theories with quartic interactions and Laplacian on ${\mathbb{R}}^3$ as kinetic operator. We find that the 2-point functions for these noncommutative scalar field theories have no IR singularities in the external momenta, indicating the absence of UV/IR mixing. We also find that the 2-point functions are UV finite with the deformation parameter $\theta$ playing the role of a natural UV cut-off. The possible origin of the absence of UV/IR mixing in noncommutative scalar field theories on ${\mathbb{R}}_{\theta }^3$ as well as on ${\mathbb{R}}_{\lambda }^3$, another deformation of ${\mathbb{R}}^3$, is discussed.

arXiv comment19 pages

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