Abstract
We revisit the Amit-Roginsky (AR) model in the light of recent studies on Sachdev-Ye-Kitaev (SYK) and tensor models, with which it shares some important features. It is a model of N scalar fields transforming in an N-dimensional irreducible representation of SO(3). The most relevant (in renormalization group sense) invariant interaction is cubic in the fields and mediated by a Wigner 3jm symbol. The latter can be viewed as a particular rank-3 tensor coupling, thus highlighting the similarity to the SYK model, in which the tensor coupling is however random and of even rank. As in the SYK and tensor models, in the large-N limit the perturbative expansion is dominated by melonic diagrams. The lack of randomness, and the rapidly growing number of invariants that can be built with n fields, makes the AR model somewhat closer to tensor models. We review the results from the old work of Amit and Roginsky with the hindsight of recent developments, correcting and completing some of their statements, in particular concerning the spectrum of the operator product expansion of two fundamental fields. For 5.74 < d < 6 the fixed-point theory defines a real CFT, while for smaller d complex dimensions appear, after a merging of the lowest dimension with its shadow. We also introduce and study a long-range version of the model, for which the cubic interaction is exactly marginal at large N , and we find a real and unitary CFT for any d < 6, both for real and imaginary coupling constant, up to some critical coupling.
Highlights
The Amit-Roginsky (AR) model has a number of interesting characteristics
Theories with cubic interactions have been studied since the early days of the renormalization group: the beta functions for a generic multiscalar model with cubic interactions have been computed at one loop in [32], and for the case with a global symmetry such that there is a single coupling they have been computed at two loops in [33,34,35], at three loops in [36], and at four loops in [37]
Along the lines of long-range O(N )3 tensor model [19,20,21,22], we will introduce and study a long-range version of the AR model, for which the coupling is exactly marginal at large N, and we will find that in this case a real and unitary CFT can be identified at small coupling for either real or imaginary coupling, even at integer dimensions d < 6
Summary
The idea behind Amit and Roginsky’s work was to generalize the Potts model ( described by a multiscalar theory with cubic interaction [33, 38]) by endowing it with a continuous symmetry, in such a way to allow the introduction of a useful large-N limit. Another variant can be considered along the lines of the bi-adjoint model of [48, 49], writing a two-index field φmn. This could be interpreted as an SO(4) invariant model, since SO(4) is isomorphic to SU(2) × SU(2)
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