Noncommutative field theories on $R^3_\lambda$: Towards UV/IR mixing freedom

TL;DR

This paper studies scalar field theories on a noncommutative space $R^3_lambda$ with fuzzy spheres, demonstrating finiteness at one-loop and likely absence of UV/IR mixing, indicating improved quantum behavior.

Contribution

It constructs a natural matrix base for $R^3_lambda$ and analyzes one-loop quantum corrections, showing finiteness and absence of UV/IR mixing in these noncommutative field theories.

Findings

One-loop planar and non-planar diagrams are finite in the matrix base.

No IR singularities observed, suggesting no UV/IR mixing.

Theories with only radial kinetic terms are finite at all orders.

Abstract

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