Bosonic tensor models at large N and small ϵ

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-ϵ expansion, finding perfect agreement. This helps elucidate why the dimension of operator φabcφabc 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 φab, 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 φq 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≈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.