On Large $N$ Limit of Symmetric Traceless Tensor Models

TL;DR

The paper investigates the large-$N$ behavior of rank-$3$ tensor models with tetrahedral quartic interaction, focusing on the fully symmetric traceless tensor with a single $O(N)$ symmetry. By explicit diagrammatic combinatorics and scaling arguments, the authors show that in the melonic large-$N$ limit with $g^2 N^3$ fixed, melonic diagrams dominate and non-melonic contributions remain subleading, with odd-vertex graphs suppressed by a power of $N$. They compare the $O(N)$ symmetric-traceless case to the $O(N)^3$ tri-fundamental model, finding stronger but still insufficient enhancements for non-melonic graphs to compete, and conjecture a smooth melonic large-$N$ limit for the symmetric traceless model. The results bridge known melonic limits in multi-symmetry tensor theories and support the universality of melonic dominance across tensor representations, with potential implications for solvable higher-dimensional SYK-like systems and quantum gravity connections.

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.

Figures (12)

• Figure 1: All the melonic vacuum diagrams up to order $g^6$.
• Figure 2: Leading melonic propagator corrections in the $O(N)^3$ and $O(N)$ theories.
• Figure 3: Order $g^{3}$ stranded diagrams for $O(N)^{3}$ and $O(N)$ theories. The diagram in the $O(N)^3$ theory has 6 loops and scales as $g^3 N^6$; the diagram in the $O(N)$ theory has 7 loops and scales as $g^3 N^7$.
• Figure 4: All vacuum diagrams of order $g^4$. The upper integer, shown in black, gives the leading power of $N$ in the model of a symmetric traceless rank-$3$ tensor of $O(N)$; the lower integer, shown in blue, gives the leading power in the model of a tri-fundamental of $O(N)^3$. The letter B labels the bi-partite diagrams.
• Figure 5: All vacuum diagrams of order $g^5$.