Another proof of the $1/N$ expansion of the rank three tensor model with tetrahedral interaction

The rank three tensor model with tetrahedral interaction was shown by Carrozza and Tanasa to admit a $1/N$ expansion, dominated by melonic diagrams, and double tadpoles decorated with melons at next-to-leading order. This model has generated a renewed interest in tensor models because it has the same large $N$ limit as the SYK model. In contrast with matrix models, there is no method which would be able to prove the existence of $1/N$ expansions in arbitrary tensor models. The method used by Carrozza and Tanasa proves the existence of the $1/N$ expansion using two-dimensional topology, before identifying the leading order and next-to-leading graphs. However, another method was required for complex, rank three tensor models with planar interactions, which is based on flips. The latter are moves which cut two propagators of Feynman graphs and reglue them differently. They allow transforming graphs while tracking their orders in the $1/N$ expansion. Here we use this method to re-prove the results of Carrozza and Tanasa, thereby proving the existence of the $1/N$ expansion, the melonic dominance at leading order and the melon-decorated double tadpoles at next-to-leading order, all in one go.