Higher melonic theories

TL;DR

This paper classifies a broad class of melonic theories with various symmetries, revealing their connection to graph factorization problems and analyzing their impact on interaction strengths and correlation functions.

Contribution

It provides a comprehensive classification of melonic theories with arbitrary interactions and explores their symmetry structures and implications for Schwinger-Dyson equations.

Findings

Interaction vertices exhibit $ ext{Z}_2^n$ symmetries.

Number of theories grows rapidly with $q$.

Symmetries influence effective interaction strength.

Abstract

We classify a large set of melonic theories with arbitrary $q$-fold interactions, demonstrating that the interaction vertices exhibit a range of symmetries, always of the form $\mathbb{Z}_2^n$ for some $n$, which may be $0$. The number of different theories proliferates quickly as $q$ increases above $8$ and is related to the problem of counting one-factorizations of complete graphs. The symmetries of the interaction vertex lead to an effective interaction strength that enters into the Schwinger-Dyson equation for the two-point function as well as the kernel used for constructing higher-point functions.