By applying copositivity criterion to the scalar potential of the economical 3 − 3 − 1 model, we derive necessary and sufficient bounded-from-below conditions at tree level. Although these are a large number of intricate inequalities for the dimensionless parameters of the scalar potential, we present general enlightening relations in this work. Additionally, we use constraints coming from the minimization of the scalar potential by means of the orbit space method, the positivity of the squared masses of the extra scalars, the Higgs boson mass, the Z gauge boson mass and its mixing angle with the SM Z boson in order to further restrict the parameter space of this model.

We analyze the dark matter (DM) puzzle in a class of models with gauge symmetry SU(3) C ⊗ SU(3) L ⊗ U(1) N , taking into account the constraints coming from Planck observations of the DM relic density, direct detection searches with Xenon1T, and indirect detection using Fermi-LAT data. The model has two possible candidates, a fermion and a scalar, which require, to evade the constraints set by Xenon1T and Fermi-LAT, and have relic densities in the range measured by the Planck collaboration, a minimum vacuum expectation value to break the SU(3) L symmetry of 10 and 15 TeV, respectively. In this case, these particles have masses of order 1 TeV, for the fermion, and between 500 and 600 GeV, for the scalar. KEYWORDS dark matter -elementary particles 1

Dark matter in the form of axions can be included in models with SU(3) C ⊗ SU(3) L ⊗ U(1) X symmetry with right-handed neutrinos, where the strong charge-parity (CP) problem can be solved. Different versions of the model have the appealing characteristic of giving the observed dark matter abundance, measured by the Planck collaboration, for suitable values of the parameters. We studied the constraints on the parameter space of the model obtained when the axion mass is calculated taking into account both quantum chromodynamics (QCD) effects and gravitationally induced operators, making a comparison of three different models. KEYWORDS dark matter -elementary particles 1

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