# Line of fixed points in a bosonic tensor model

## TL;DR

This paper analyzes the renormalization group flow of a bosonic tensor model with an $O(N)^3$ symmetry, revealing a line of fixed points and an infrared attractive fixed point for imaginary tetrahedral couplings, using a non-perturbative large-$N$ approach.

## Contribution

It provides a non-perturbative large-$N$ analysis of the RG flow in a bosonic tensor model, identifying a line of fixed points and the conditions for an infrared attractive fixed point.

## Key findings

- Four fixed points depending on the tetrahedral coupling.
- Existence of an infrared attractive fixed point for imaginary tetrahedral couplings.
- Beta functions are quadratic in pillow and double-trace couplings, independent of the tetrahedral coupling at leading order.

## Abstract

We consider the $O(N)^3$ tensor model of Klebanov and Tarnopolsky \cite{Klebanov:2016xxf} in $d<4$ with a free covariance modified to fit the infrared conformal scaling. We study the renormalization group flow of the model using a Wilsonian approach valid in any $d$ (notably we do not require $d=4-\epsilon$ with small $\epsilon$). At large $N$, the tetrahedral coupling has a finite flow, hence it becomes a free parameter. The remaining flow can be parameterized by two couplings which do not mix. We show that, at leading order in $1/N$ but non perturbatively in the couplings, the beta functions stop at quadratic order in the pillow and double-trace couplings. We find four fixed points which depend parametrically on the tetrahedral coupling. For purely imaginary values of the latter we identify a real and \emph{infrared attractive} fixed point. We remark that an imaginary tetrahedral coupling is in fact natural from the onset as the tetrahedral invariant does not have any positivity property, and moreover in the large-$N$ limit beta functions depend on the square of the tetrahedral coupling, thus they remain real, as long as the other couplings stay real.