### 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.

Original language | English (US) |
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Article number | 37 |

Journal | Journal of High Energy Physics |

Volume | 2017 |

Issue number | 10 |

DOIs | |

State | Published - Oct 1 2017 |

### All Science Journal Classification (ASJC) codes

- Nuclear and High Energy Physics

### Keywords

- 1/N Expansion
- Conformal Field Theory
- Nonperturbative Effects

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## Cite this

*Journal of High Energy Physics*,

*2017*(10), [37]. https://doi.org/10.1007/JHEP10(2017)037