Remarks on a melonic field theory with cubic interaction

TL;DR

This paper revisits the Amit-Roginsky model, a scalar field theory with cubic interactions related to tensor models and SYK, analyzing its fixed points, spectrum, and long-range variants across different dimensions.

Contribution

It provides a corrected and extended analysis of the AR model, exploring its conformal fixed points, spectrum, and a long-range version with exact marginality at large N.

Findings

For 5.74<d<6, the model defines a real CFT.

Below d<5.74, complex dimensions emerge due to operator merging.

A long-range version yields a real, unitary CFT for d<6, with both real and imaginary couplings.

Abstract

We revisit the Amit-Roginsky (AR) model in the light of recent studies on Sachdev-Ye-Kitaev (SYK) and tensor models, with which it shares some important features. It is a model of $N$ scalar fields transforming in an $N$-dimensional irreducible representation of $SO(3)$. The most relevant (in renormalization group sense) invariant interaction is cubic in the fields and mediated by a Wigner $3jm$ symbol. The latter can be viewed as a particular rank-3 tensor coupling, thus highlighting the similarity to the SYK model, in which the tensor coupling is however random and of even rank. As in the SYK and tensor models, in the large-$N$ limit the perturbative expansion is dominated by melonic diagrams. The lack of randomness, and the rapidly growing number of invariants that can be built with $n$ fields, makes the AR model somewhat closer to tensor models. We review the results from the old work of Amit and Roginsky with the hindsight of recent developments, correcting and completing some of their statements, in particular concerning the spectrum of the operator product expansion of two fundamental fields. For $5.74<d<6$ the fixed-point theory defines a real CFT, while for smaller $d$ complex dimensions appear, after a merging of the lowest dimension with its shadow. We also introduce and study a long-range version of the model, for which the cubic interaction is exactly marginal at large $N$, and we find a real and unitary CFT for any $d<6$, both for real and imaginary coupling constant, up to some critical coupling.