Bosonic tensor models at large N and small ϵ

Abstract

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 ¼ 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 ¼ 4 − ϵ has complex values of the couplings for some of the OðNÞ3 invariant operators. We show that a similar phenomenon holds in the OðNÞ2 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 ¼ 3 − ϵ.