On large N limit of symmetric traceless tensor models

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 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.