Noncommutative field theories on $R_{\lambda }^3$: Towards UV/IR mixing freedom

Abstract

We consider the noncommutative space ${\mathbb{R}}_{\lambda }^3$, a deformation of the algebra of functions on ${\mathbb{R}}^3$ which yields a "foliation" of ${\mathbb{R}}^3$ into fuzzy spheres. We first construct a natural matrix base adapted to ${\mathbb{R}}_{\lambda }^3$. We then apply this general framework to the one-loop study of a two-parameter family of real-valued scalar noncommutative field theories with quartic polynomial interaction, which becomes a non-local matrix model when expressed in the above matrix base. The kinetic operator involves a part related to dynamics on the fuzzy sphere supplemented by a term reproducing radial dynamics. We then compute the planar and non-planar 1-loop contributions to the 2-point correlation function. We find that these diagrams are both finite in the matrix base. We find no singularity of IR type, which signals very likely the absence of UV/IR mixing. We also consider the case of a kinetic operator with only the radial part. We find that the resulting theory is finite to all orders in perturbation expansion.

arXiv comment31 pages, 4 figures. Improved version. Sections 5.1 and 5.2 have been clarified. A minor error corrected. References added

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