Large-$d$ behavior of the Feynman amplitudes for a just-renormalizable tensorial group field theory

This paper aims at giving a novel approach to investigate the behavior of the renormalization group flow for tensorial group field theories to all orders of the perturbation theory. From an appropriate choice of the kinetic kernel, we build an infinite family of just-renormalizable models, for tensor fields with arbitrary rank $d$. Investigating the large $ d$ limit, we show that the self-energy melonic amplitude is decomposed as a product of loop-vertex functions, depending only on dimensionless mass. The corresponding melonic amplitudes may be mapped as trees in the so-called Hubbard-Stratonivich representation, and we show that only trees with edges of different colors survive in the large $d$-limit. These two key features allow us to resum the perturbative expansion for self-energy, providing an explicit expression for arbitrary external momenta in terms of Lambert function. Finally, inserting this resumed solution into the Callan-Symanzik equations, and taking into account the strong relation between two and four-point functions arising from melonic Ward-Takahashi identities, we then deduce an explicit expression for relevant and marginal $\beta$-functions, valid to all orders of the perturbative expansion. By investigating the solutions of the resulting flow, we conclude about the non-existence of any fixed point in the investigated region of the full phase space.