We study bosonic tensor field theories with sextic interactions in d < 3 dimensions. We consider two models, with rank-3 and rank-5 tensors, and U(N) 3 and O(N) 5 symmetry, respectively. For both of them we consider two variations: one with standard short-range free propagator, and one with critical long-range propagator, such that the sextic interactions are marginal in any d < 3. We derive the set of beta functions at large N , compute them explicitly at four loops, and identify the respective fixed points. We find that only the rank-3 models admit melonic interacting fixed points, with real couplings and critical exponents: for the short-range model, we have a Wilson-Fisher fixed point with couplings of order √ , in d = 3 − ; for the long-range model, instead we have for any d < 3 a line of fixed points, parametrized by a real coupling g 1 (associated to the so-called wheel interaction). By standard conformal field theory methods, we then study the spectrum of bilinear operators associated to such interacting fixed points, and we find a real spectrum for small or small g 1 .

Perturbing the standard Gross-Neveu model for N 3 fermions by quartic interactions with the appropriate tensorial contraction patterns, we reduce the original U (N 3 ) symmetry to either U (N ) × U (N 2 ) or U (N ) × U (N ) × U (N ). In the large-N limit, we show that in three dimensions such models admit new ultraviolet fixed points with reduced symmetry, besides the well-known one with maximal symmetry. The phase diagram notably presents a new phase with spontaneous symmetry breaking of one U (N ) component of the symmetry group. arXiv:1810.04583v2 [hep-th] 16 Nov 2018 1 Most commonly matrix models are presented as describing fields transforming in the adjoint representation, e.g. of U (N ). Such point of view is natural when introducing them as gauge fields, and wishing to discuss the different behavior of fields in the fundamental representation (flavor fields) and in the adjoint (connection fields) of the same group. Here we discuss them instead as fields in the fundamental representation of a smaller group, because we want to highlight the role of the smaller symmetry group for a given set of fields. Furthermore, the existence of a melonic large-N limit for tensors transforming in an irreducible representation of U (N ) (or O(N )) has been shown only very recently [7,8,9], and such models are less understood.2 Another way to obtain similar equations is to consider multi-matrix models at large N and in the limit of large number of matrices (see [10] and references therein).