Sextic tensor model in rank $3$ at next-to-leading order

TL;DR

This paper computes four-loop beta functions for sextic tensor models with $U(N)^3$ symmetry, analyzing their renormalization group flow at next-to-leading order in $N$ and small $psilon$, revealing a non-trivial IR fixed point in the short-range case.

Contribution

It provides the first detailed analysis of the renormalization group flow of rank-3 sextic tensor models at four loops, including $1/N$ corrections and stability properties.

Findings

Identifies a non-trivial IR stable fixed point in short-range models.

Finds $1/N$ corrections to the sextic tensor model.

Long-range case corrections are non-perturbative and unreliable.

Abstract

We compute the four-loop beta functions of short and long-range multi scalar models with general sextic interactions and complex fields. We then specialize the beta functions to a $U(N)^3$ symmetry and study the renormalization group at next-to-leading order in $N$ and small $\epsilon$. In the short-range case, $\epsilon$ is the deviation from the critical dimension while it is the deviation from the critical scaling of the free propagator in the long-range case. This allows us to find the $1/N$ corrections to the rank-3 sextic tensor model of arXiv:1912.06641. In the short-range case, we still find a non-trivial real IR stable fixed point, with a diagonalizable stability matrix. All couplings, except for the so-called wheel coupling, have terms of order $\epsilon^0$ at leading and next-to-leading order, which makes this fixed point different from the other melonic fixed points found in quartic models. In the long-range case, the corrections to the fixed point are instead not perturbative in $\epsilon$ and hence unreliable; we thus find no precursor of the large-$N$ fixed point.