# RG stability of integrable fishnet models

## TL;DR

This paper investigates the perturbative stability of various scalar fishnet models across different dimensions, highlighting conditions under which these models remain consistent and free from RG flow corrections.

## Contribution

It demonstrates that 3D $ ext{phi}^6$ fishnet models are perturbatively stable at large N, and shows the 6D $ ext{phi}^3$ fishnet model is perturbatively consistent, with explicit anomalous dimension calculations.

## Key findings

- 3D $ ext{phi}^6$ fishnet model is RG stable at large N.
- 6D $ ext{phi}^3$ fishnet model is perturbatively consistent.
- Explicit anomalous dimensions computed for operators.

## Abstract

We address the question of perturbative consistency in the scalar fishnet models presented by Caetano, Gurdogan and Kazakov\cite{Gurdogan:2015csr, Caetano:2016ydc}. We argue that their 3-dimensional $\phi^{6}$ fishnet model becomes perturbatively stable under renormalization in the large $N$ limit, in contrast to what happens in their 4-dimensional $\phi^{4}$ fishnet model, in which double trace terms are known to be generated by the RG flow. We point out that there is a direct way to modify this second theory that protects it from such corrections. Additionally, we observe that the 6-dimensional $\phi^{3}$ Lagrangian that spans an hexagonal integrable scalar fishnet is consistent at the perturbative level as well. The nontriviality and simplicity of this last model is illustrated by computing the anomalous dimensions of its $\text{tr}\phi_i \phi_j$ operators to all perturbative orders.