On large N limit of symmetric traceless tensor models

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 g⁸, as well as some diagrams of higher order, and find that in the large N limit where g²N³ 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 g²N³ 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.