Fermion Mass Hierarchy and Double Seesaw Mechanism in a 3-3-1 Model with an Axion

We present a model based on the $SU(3)_C\otimes SU(3)_L\otimes U(1)_X$ gauge symmetry that relates the mass hierarchy of the fermions with the solution to the strong CP problem through the $U(1)_{PQ}$ Peccei-Quinn symmetry. This last symmetry arises accidentally with the imposition of a discrete $Z_9$ symmetry, which also secludes the different scales in the double seesaw mechanism taking place in the neutrino sector. The symmetry breakdown is performed by three scalar triplets plus a scalar singlet hosting an axion field, whose particle excitation can be a component of dark matter. We show a mechanism where a small effective vev is generated for a scalar triplet which is supposed to have a bare mass above the energy scale where the $SU(3)_L\otimes U(1)_X$ symmetry is broken. Combined with the energy scale in which the $U(1)_{PQ}$ is broken, such a mechanism gives rise to a natural hierarchy to the fermions. Beyond the Standard Model particle content, the model predicts an invisible axion, $a$, three GeV neutrinos, $N_{iL}$, plus several new particles at the TeV scale which are: five vector bosons, $U^\pm$, $V^0$, $V^{0\dagger}$, and $Z^\prime$; one up-type $U$, and two down-type $D_a$ quarks; and at least a CP-even, $H_1$, plus non-hermitian neutral, $\phi^0$, $\phi^{0\dagger}$, scalar bosons. The model may be tested by looking for the possible production of such particles at the LHC.