Melonic Dominance in Subchromatic Sextic Tensor Models

TL;DR

This paper investigates sextic tensor models with specific symmetry and interaction structures, revealing that only three types of maximally-single-trace interactions dominate in the large N limit, leading to a summable class of melonic diagrams.

Contribution

It classifies all subchromatic sextic maximally-single-trace interactions and proves melonic dominance in their large N expansions.

Findings

Only three subchromatic sextic interactions exist: prism, wheel, and octahedron.

Melonic diagrams dominate the large N free energy contributions.

The dominant diagrams can be explicitly summed.

Abstract

We study tensor models based on $O(N)^r$ symmetry groups constructed out of rank-$r$ tensors with order-$q$ interaction vertices. We refer to those tensor models for which $r<q-1$ as \textit{subchromatic}. We focus most of our attention on sextic ($q=6$) models with maximally-single-trace interactions. We show that only three subchromatic sextic maximally-single-trace interaction vertices exist: these are the $r=3$ prism, the $r=3$ wheel (or $K_{3,3}$) and the $r=4$ octahedron. For theories based on these interactions we demonstrate that the set of Feynman diagrams that contribute to the free energy in the large $N$ limit are melonic (or closely related to melonic diagrams, in the case of the prism) and thus can be explicitly summed.