Bosonic tensor models at large $N$ and small $\epsilon$

We study the spectrum of the large $N$ quantum field theory of bosonic rank-3 tensors, the quartic interactions of which 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 $4-\epsilon$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator ${\varphi }^{abc}{\varphi }^{abc}$ is complex for $d<4$: the large $N$ fixed point in $d=4-\epsilon$ 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 ${\varphi }^{ab}$, where the double-trace operator has a complex coupling in $4-\epsilon$ dimensions. We also study the spectra of bosonic theories of rank-$q-1$ tensors with ${\varphi }^q$ 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 $d=3-\epsilon$.

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 $h_0=\frac{1}{2}+1.525i$.

Figure 2

Plot of $q_{\text{crit}}$ as a function of $d$. The orange region corresponds to $q>q_{\text{crit}}$, where ${\mathrm{\Delta }}_{{\varphi }^2}$ is real, and the theory is not obviously unstable. For integer dimensions, we obtained $q_{\text{crit}}(2)\approx 64.3$, $q_{\text{crit}}(3)\approx 5.9$, and $q_{\text{crit}}(4)=4$.

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 ${\epsilon }_{\text{crit}}\approx 0.02819$, they merge and go off to the complex plane.