# On Large $N$ Limit of Symmetric Traceless Tensor Models

## TL;DR

This paper investigates the large $N$ limit of a symmetric traceless rank-3 tensor model with tetrahedral interactions, finding that melonic diagrams dominate and proposing a smooth large $N$ limit where only these diagrams contribute.

## Contribution

It introduces the combinatorial analysis of a symmetric traceless tensor model, showing melonic diagrams dominate in the large $N$ limit, and conjectures a simplified large $N$ behavior.

## Key findings

- Melonic diagrams dominate in the large $N$ limit.
- Non-melonic diagrams are suppressed or not dominant.
- Proposes a smooth large $N$ limit with only melonic diagrams.

## Abstract

For some theories where the degrees of freedom are tensors of rank $3$ or higher, there exist solvable large $N$ limits dominated by the melonic diagrams. Simple examples are provided by models containing one rank-$3$ tensor in the tri-fundamental representation of the $O(N)^3$ symmetry group. When the quartic interaction is assumed to have a special tetrahedral index structure, the coupling constant $g$ must be scaled as $N^{-3/2}$ in the melonic large $N$ limit. In this paper we consider the combinatorics of a large $N$ theory of one fully symmetric and traceless rank-$3$ tensor with the tetrahedral quartic interaction; this model has a single $O(N)$ symmetry group. We explicitly calculate all the vacuum diagrams up to order $g^8$, as well as some diagrams of higher order, and find that in the large $N$ limit where $g^2 N^3$ is held fixed only the melonic diagrams survive. While some non-melonic diagrams are enhanced in the $O(N)$ symmetric theory compared to the $O(N)^3$ one, we have not found any diagrams where this enhancement is strong enough to make them comparable with the melonic ones. Motivated by these results, we conjecture that the model of a real rank-$3$ symmetric traceless tensor possesses a smooth large $N$ limit where $g^2 N^3$ is held fixed and all the contributing diagrams are melonic. A feature of the symmetric traceless tensor models is that some vacuum diagrams containing odd numbers of vertices are suppressed only by $N^{-1/2}$ relative to the melonic graphs.