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 ${\mathbb{Z}}_9$ 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 vacuum expectation value 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, $U^{\pm }$, $V^0$, $V^{0†}$, 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, ${\varphi }^0$, ${\varphi }^{0†}$, scalar bosons. The model may be tested by looking for the possible production of such particles at the LHC.