Renormalization on noncommutative torus

TL;DR

This paper investigates the renormalization of a scalar  theory on noncommutative tori, showing one-loop renormalizability in specific dimensions and discussing higher-loop effects, with implications for UV/IR mixing.

Contribution

It provides the first detailed analysis of renormalization for  theory on noncommutative tori, including counterterms and higher-loop considerations, indicating potential renormalizability.

Findings

One-loop renormalization is achievable in 2D and 4D cases.

Higher-loop effects do not introduce UV/IR mixing problems.

Two-point amplitude exhibits wild behavior depending on the noncommutativity matrix.

Abstract

We study a self-interacting scalar $\varphi^4$ theory on the $d$-dimensional noncommutative torus. We determine, for the particular cases $d=2$ and $d=4$, the counterterms required by one-loop renormalization. We discuss higher loops in two dimensions and two-loop contributions to the self-energy in four dimensions. Our analysis points toward the absence of any problems related to the UV/IR mixing and thus to renormalizability of the theory. However, we find another potentially troubling phenomenon which is a wild behavior of the two-point amplitude as a function of the noncommutativity matrix $\theta$.