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

Understand

We consider the noncommutative space $\mathbb{R}^3_\theta$, 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.