I will discuss a new type of Tensorial Field Theory on copies of U(1), incorporating a Laplacian on the φ^4 interaction. A power counting theorem follows a multi-scale analysis and, clearly, we can realize that the terms which were power-counting suppressed in the usual theory become, in some cases, the dominant contributions. This is encouraging for identifying new types of tensor models having a scaling limit different from the so-called branched polymers.