On the Origin of the UV-IR Mixing in Noncommutative Matrix Geometry

TL;DR

This paper investigates the origin of UV-IR mixing in noncommutative scalar field theories on fuzzy spheres, introducing a new planar limit that avoids UV-IR mixing by controlling the noncommutativity scale.

Contribution

It proposes a novel planar limit of fuzzy spheres where the noncommutativity parameter also acts as a momentum cutoff, eliminating UV-IR mixing in the resulting noncommutative field theory.

Findings

Standard continuum limit reproduces known UV-IR mixing.

New limit introduces a cutoff that prevents UV-IR mixing.

Analysis of 4-point function in the new regime.

Abstract

Scalar field theories with quartic interaction are quantized on fuzzy $S^2$ and fuzzy $S^2\times S^2$ to obtain the 2- and 4-point correlation functions at one-loop. Different continuum limits of these noncommutative matrix spheres are then taken to recover the quantum noncommutative field theories on the noncommutative planes ${\mathbb R}^2$ and ${\mathbb R}^4$ respectively. The canonical limit of large stereographic projection leads to the usual theory on the noncommutative plane with the well-known singular UV-IR mixing. A new planar limit of the fuzzy sphere is defined in which the noncommutativity parameter ${\theta}$, beside acting as a short distance cut-off, acts also as a conventional cut-off ${\Lambda}=\frac{2}{\theta}$ in the momentum space. This noncommutative theory is characterized by absence of UV-IR mixing. The new scaling is implemented through the use of an intermediate scale that demarcates the boundary between commutative and noncommutative regimes of the scalar theory. We also comment on the continuum limit of the $4-$point function.