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We tackle the issue of renormalizability for Tensorial Group Field Theories(TGFT) including gauge invariance conditions, with the rigorous tool ofmulti-scale analysis, to prepare the ground for applications to quantum gravitymodels. In the process, we define the appropriate generalization of some keyQFT notions, including: connectedness, locality and contraction of (high)subgraphs. We also define a new notion of Wick ordering, corresponding to thesubtraction of (maximal) melonic tadpoles. We then consider the simplestexamples of dynamical 4-dimensional TGFT with gauge invariance conditions forthe Abelian U(1) case. We prove that they are super-renormalizable for anypolynomial interaction.

Renormalization of Tensorial Group Field Theories: Abelian U(1) Models in Four Dimensions