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

We consider the noncommutative space $\mathbb{R}^3_θ$, 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 $θ$ 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 $\mathbb{R}^3_θ$ as well as on $\mathbb{R}^3_λ$, another deformation of $\mathbb{R}^3$, is discussed.

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