Abstract
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. The possible origin of the absence of UV/IR mixing in noncommutative scalar field theories on $\mathbb{R}^3_\theta$ as well as on $\mathbb{R}^3_\lambda $, another deformation of $\mathbb{R}^3$, is discussed.
Highlights
JHEP05(2016)146 true for alternative approaches based on a IR damping mechanism aiming to neutralize the mixing [44, 45].2 Note that the matrix model interpretation of noncommutative gauge theories provided interesting results onclassical properties and/or one-loop computations
We find that the 2-point functions are UV finite with the deformation parameter θ playing the role of a natural UV cut-off
We have considered the noncommutative space R3θ defined in (2.1) for which the star product ⋆D is closed for the trace functional d3x
Summary
Where M(R3) is the multiplier algebra of S(R3) (the set of Schwartz functions on R3) for the star-product ⋆D defined by f ⋆D g =. For any (suitably behaving) functions f and g where “·” is the commutative product on R3 and dμ(x) is some integration measure that can be chosen in the following to be the usual Lebesgue measure (while dμ(x) · (f ⋆W g)(x) = dμ(x) · (f · g)(x)). Such a transformation T may not exist in general. For a discussion of this point, see [62] where it is pointed out that ⋆D is the Kontsevich product [70]
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