Bosonic Tensor Models at Large $N$ and Small $ε$

TL;DR

This work analyzes large-N bosonic tensor theories with melonic dominance, using Schwinger-Dyson equations and the 4-ε framework to determine the spectrum of bilinear operators across spins and dimensions. A key result is the emergence of complex scaling dimensions for certain operators (notably $\phi^{abc}\phi^{abc}$) at the IR fixed point, explained by complex fixed-point couplings; similar complex dimensions appear in the $O(N)^2$ matrix case from double-trace terms. The authors extend the analysis to rank $(q-1)$ tensors with $\phi^q$ interactions, identifying a critical dimension $d_{cr}$ below which complex towers persist and showing how $q_{\rm crit}(d)$ behaves, with implications for IR stability. They also explore a melonic $\phi^6$ theory near $d=3$, finding a narrow real-spectrum window ($\epsilon<0.02819$) and conjecturing a perturbatively stable fixed point for $2.97<d<3$, potentially accessible by $d=3-\epsilon$ perturbation theory.

Abstract

We study the spectrum of the large $N$ quantum field theory of bosonic rank-$3$ tensors, whose quartic interactions are such that the perturbative expansion is dominated by the melonic diagrams. We use the Schwinger-Dyson equations to determine the scaling dimensions of the bilinear operators of arbitrary spin. Using the fact that the theory is renormalizable in $d=4$, we compare some of these results with the $4-ε$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator $φ^{abc}φ^{abc}$ is complex for $d<4$: the large $N$ fixed point in $d=4-ε$ has complex values of the couplings for some of the $O(N)^3$ invariant operators. We show that a similar phenomenon holds in the $O(N)^2$ symmetric theory of a matrix field $φ^{ab}$, where the double-trace operator has a complex coupling in $4-ε$ dimensions. We also study the spectra of bosonic theories of rank $q-1$ tensors with $φ^q$ interactions. In dimensions $d>1.93$ there is a critical value of $q$, above which we have not found any complex scaling dimensions. The critical value is a decreasing function of $d$, and it becomes $6$ in $d\approx 2.97$. This raises a possibility that the large $N$ theory of rank-$5$ tensors with sextic potential has an IR fixed point which is free of perturbative instabilities for $2.97<d<3$. This theory may be studied using renormalized perturbation theory in $d=3-ε$.

Figures (3)

• Figure 1: The graphical solution of the eigenvalue equation $g(h)=1$ in $d=1$. This method works for finding the real solutions only; it misses the complex solution $h_0= \frac{1}{2} + 1.525 i$.
• Figure 2: Plot of $q_{\textrm{crit}}$ as a function of $d$. The orange region corresponds to $q>q_{\textrm{crit}}$, where $\Delta_{\phi^{2}}$ is real and the theory is not obviously unstable. For integer dimensions we obtained $q_{\textrm{crit}}(2)\approx 64.3$, $q_{\textrm{crit}}(3)\approx 5.9$ and $q_{\textrm{crit}}(4) = 4$.
• Figure 3: Plot of the two lowest operator dimensions $h_{-}$ and $h_{+}$ as a function of $\epsilon$. As $\epsilon$ increases, $h_{-}$ approaches $h_{+}$, and at $\epsilon_{\textrm{crit}} \approx 0.02819$ they merge and go off to complex plane.