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, 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 $4-\epsilon$ expansion, finding perfect agreement. This helps elucidate why the dimension of operator $\phi^{abc}\phi^{abc}$ 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 $\phi^{ab}$, 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 $\phi^q$ 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 $d=3-\epsilon$.