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Abstract:
We introduce a tensorial group field theory endowed with weighted interaction terms of the form p2aϕ4. The model can be seen as a field theory over d=3,4 copies of U(1) where formal powers of Laplacian operators, namely Δa, a>0, act on tensorial ϕ4-interactions producing, after Fourier transform, p2aϕ4 interactions. Using multi-scale analysis, we provide a power counting theorem for this type of models. A new quantity depending on the incidence matrix between vertices and faces of Feynman graphs is invoked in the degree of divergence of amplitudes. As a result, generally, the divergence degree is enhanced compared to the divergence degree of models without weighted vertices. The subleading terms in the partition function of the ϕ4 tensorial models become, in some cases, the dominant ones in the p2aϕ4 models. Finally, we explore sufficient conditions on the parameter a yielding a list of potentially super-renormalizable p2aϕ4 models.
