Remarks on a melonic field theory with cubic interaction

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.

Spectroscopy of light baryons in a semi-relativistic constituent three-quark model

We studied the non-strange baryon spectroscopy by presenting a simple semi-relativistic constituent three-quark model. Assuming a separation of the interaction potential in terms of a leading SU(6) symmetric component and a subleading SU(6) breaking term, we treated the baryons as a spin-independent three-quark system and presented the analytical solution for the problem. Using perturbative and approximative approaches in order to deal with problematic linear confining term in SU(6)-invariant interaction, we obtained analytical formulas for energy levels and the hyperradial wave functions and the average energy values of the nonstrange resonances are reproduced. To describe the hyperfine structure of the baryon, the splittings within the SU(6)-multiplets are produced by the perturbative spin- and isospin-dependent terms. The resulting description of the baryon spectrum for both approaches are given and compared with the experimental spectrum.

CONFORMAL INVARIANT INTERACTION OF A SCALAR FIELD WITH HIGHER SPIN FIELD IN AdSD

The explicit form of linearized gauge invariant interactions of scalar and general higher even spin fields in the AdSD space is obtained. In the case of general spin-ℓ a generalized "Weyl" transformation is proposed and the corresponding "Weyl" invariant action is constructed. In both cases the invariant actions of the interacting higher even spin gauge field and the scalar field include the whole tower of invariant actions for couplings of the same scalar with all gauge fields of smaller even spin. For the particular value of ℓ = 4, all results are in exact agreement with Ref. 1.

PHASE SHIFT ANALYSIS OF PT-SYMMETRIC NONHERMITIAN EXTENSION OF AN-1 CALOGERO MODEL WITHOUT CONFINING INTERACTION

We discuss a manyparticle quantum system, which is obtained by adding some nonhermitian but PT (i.e. combined parity and time reversal) invariant interaction to the AN-1 rational Calogero model without confining potential. This model gives rise to scattering states with continuous real spectrum. The scattering phase shift is determined through the exchange statistics parameter. We find that, unlike the case of the usual Calogero model, the exclusion and exchange statistics parameter differ from each other in the presence of PT invariant interaction.