2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/51512Using a new scaling limit as well as a new cut-off procedure, we show that $ϕ^4$ theory on noncommutative ${\bf R}^4$ can be obtained from the corresponding theory on fuzzy ${\bf S}^2 \times {\bf S}^2$. The star-product on this noncommutative ${\bf R}^4$ is effectively local in the sense that the theory naturally has an ultra-violet cut-off $Λ$ which is inversely proportional to the noncommutativity $θ$, i.e $ Λ= \frac{2}θ$. 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, latexHigh Energy Physics - Theorytext