Identifying optimal large $N$ limits for marginal $\phi^4$ theory in 4d

TL;DR

This paper introduces an algebraic approach to identify optimal large N limits in 4d marginal $\,\phi^4$ theories, bypassing traditional diagrammatic methods and revealing new, potentially simpler limit models.

Contribution

The authors develop a novel algebraic method to determine large N limits in 4d scalar QFTs, enabling the discovery of new limits without diagrammatic analysis.

Findings

Algebraic approach successfully identifies optimal large N scalings.

New large N limits are proposed that are simpler than planar but more complex than vector models.

Method applies to various models, including bifundamental, trifundamental, and matrix-vector models.

Abstract

We apply our previously developed approach to marginal quartic interactions in multiscalar QFTs, which shows that one-loop RG flows can be described in terms of a commutative algebra, to various models in 4d. We show how the algebra can be used to identify optimal scalings of the couplings for taking large $N$ limits. The algebra identifies these limits without diagrammatic or combinatorial analysis. For several models this approach leads to new limits yet to be explored at higher loop orders. We consider the bifundamental and trifundamental models, as well as a matrix-vector model with an adjoint representation. Among the suggested new limit theories are some which appear to be less complex than general planar limits but more complex than ordinary vector models or melonic models.