New Scaling Limit for Fuzzy Spheres

Abstract

Using a new scaling limit as well as a new cut-off procedure, we show that ${\varphi }^4$ theory on noncommutative ${\mathbf{R}}^4$ can be obtained from the corresponding theory on fuzzy ${\mathbf{S}}^2\times {\mathbf{S}}^2$. The star-product on this noncommutative ${\mathbf{R}}^4$ is effectively local in the sense that the theory naturally has an ultra-violet cut-off $\Lambda$ which is inversely proportional to the noncommutativity $\theta$, i.e $\Lambda =\frac{2}{\theta }$. We show that the UV-IR mixing in this case is absent to one loop in the $2-$point function and also comment on the $4-$point function.

13 pages, latex