Noncommutative field theories on $ \mathbb{R}_{\lambda}^3 $: towards UV/IR mixing freedom

Abstract

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.