*Review* **Progress in Solving the Nonperturbative Renormalization Group for Tensorial Group Field Theory**

### **Vincent Lahoche 1,\* and Dine Ousmane Samary 1,2,\***


Received: 12 December 2018; Accepted: 18 March 2019; Published: 26 March 2019

**Abstract:** This manuscript aims at giving new advances on the functional renormalization group applied to the tensorial group field theory. It is based on the series of our three papers (Lahoche, et al., Class. Quantum Gravity 2018, 35, 19), (Lahoche, et al., Phys. Rev. D 2018, 98, 126010) and (Lahoche, et al., Nucl. Phys. B, 2019, 940, 190–213). We consider the polynomial Abelian *U*(1)*<sup>d</sup>* models without the closure constraint. More specifically, we discuss the case of the quartic melonic interaction. We present a new approach, namely the effective vertex expansion method, to solve the exact Wetterich flow equation and investigate the resulting flow equations, especially regarding the existence of non-Gaussian fixed points for their connection with phase transitions. To complete this method, we consider a non-trivial constraint arising from the Ward–Takahashi identities and discuss the disappearance of the global non-trivial fixed points taking into account this constraint. Finally, we argue in favor of an alternative scenario involving a first order phase transition into the reduced phase space given by the Ward constraint.

**Keywords:** nonperturbative renormalization group; quantum gravity; random geometry
