Sextic tensor field theories in rank $3$ and $5$

TL;DR

This paper investigates bosonic sextic tensor field theories in low dimensions, identifying fixed points and spectra for rank-3 and rank-5 models with different propagator types, revealing melonic fixed points only in rank-3 cases.

Contribution

It provides the first explicit four-loop beta function calculations and fixed point analysis for sextic tensor models with rank-3 and rank-5 tensors, including the discovery of melonic fixed points in rank-3 models.

Findings

Rank-3 models have melonic fixed points with real couplings.

Short-range model exhibits a Wilson-Fisher fixed point in d=3-ε.

Long-range model has a line of fixed points parametrized by a real coupling.

Abstract

We study bosonic tensor field theories with sextic interactions in $d<3$ dimensions. We consider two models, with rank-3 and rank-5 tensors, and $U(N)^3$ and $O(N)^5$ symmetry, respectively. For both of them we consider two variations: one with standard short-range free propagator, and one with critical long-range propagator, such that the sextic interactions are marginal in any $d<3$. We derive the set of beta functions at large $N$, compute them explicitly at four loops, and identify the respective fixed points. We find that only the rank-3 models admit a melonic interacting fixed points, with real couplings and critical exponents: for the short-range model, we have a Wilson-Fisher fixed point with couplings of order $\sqrt{\epsilon}$, in $d=3-\epsilon$; for the long-range model, instead we have for any $d<3$ a line of fixed points, parametrized by a real coupling $g_1$ (associated to the so-called wheel interaction). By standard conformal field theory methods, we then study the spectrum of bilinear operators associated to such interacting fixed points, and we find a real spectrum for small $\epsilon$ or small $g_1$.