**5. Conclusions**

In this manuscript, we have studied with different methods the FRG applied to TGFT. First, we have derived the Wilson–Polchinski equation and given the perturbative solution. Secondly, we derived the Wetterich flow equation using the usual approximation, called truncation. The analytic solution of this equation was given. We obtained a fixed point denoted by *p*<sup>+</sup>. Then, we investigated the Ward identities as a new constraint along the flow and showed that the fixed point *p*+ violates this constraint. Finally, we improved the study of FRG by replacing the truncation method by the so-called EVE. The flow equation was improved, and the corresponding solution *p*˜+ was not so far from *p*<sup>+</sup>, i.e., *p*˜+ ≈ *p*<sup>+</sup>. However, the Ward identities are strongly violated at this fixed point, and therefore this unique fixed point seems to be unphysical. We have also showed the importance of the EVE method in the sense that, despite the fact that the fixed point *p*+ needs to be discarded, a first order phase transition exists very far from this point in the subspace E*C* of the theory space. We have showed that this new behavior cannot be observed using the truncation as the approximation.

In this review, we focused on the EVE method for the melonic approximation, and especially on the quartic melonic just-renormalizable sector. The complete quartic sector, including all the connected quartic bubbles, has already been considered in a complementary work [57], and the conclusion about the incompatibility with nonperturbative fixed points and Ward identities holds. The graphs added to the quartic melonic ones to complete the quartic sector have been called pseudo-melons due to the similarities of their respective leading order Feynman graphs. Finally, even if we expect that some aspects of the EVE method improve the standard truncation method, some limitations have to be addressed for future works. In particular, our investigations were limited to the symmetric phase, ensuring convergence of any expansion around the vanishing classical mean field. Moreover, we have retained only the first terms in the derivative expansion of the two-point function and only considered the local potential approximation, i.e., potentials that can be expanded as an infinite sum of connected melonic (and pseudo-melonic) interactions. Finally, a rigorous investigation of the behavior of the renormalization group flow into the physical phase space has to be addressed in the continuation of current works on this topic.

**Author Contributions:** Conceptualization, V.L. and D.O.-S.; methodology, V.L. and D.O.-S.; software, V.L. and D.O.-S.; validation, V.L. and D.O.-S.; formal analysis, V.L. and D.O.-S.; investigation, V.L. and D.O.-S.; resources, V.L. and D.O.-S.; data curation, V.L. and D.O.-S.; writing–original draft preparation, V.L. and D.O.-S.; writing–review and editing, V.L. and D.O.-S.; visualization, V.L. and D.O.-S.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
