Naturally light invisible axion in models with large local discrete symmetries

We show that by introducing appropriate local $Z_N(N>~13)$ symmetries in electroweak models it is possible to implement an automatic Peccei-Quinn symmetry, at the same time keeping the axion protected against gravitational effects. Although we consider here only an extension of the standard model and a particular 3-3-1 model, the strategy can be used in any kind of electroweak model. An interesting feature of this 3-3-1 model is that if we add (i) right-handed neutrinos, (ii) the conservation of the total lepton number, and (iii) a $Z_2$ symmetry, the $Z_{13}$ and the chiral Peccei-Quinn ${U\left(1\right)\mathrm{}}_{\mathrm{PQ}}$ symmetries are both accidental symmetries in the sense that they are not imposed on the Lagrangian but are just a consequence of the particle content of the model, its gauge invariance, renormalizability, and Lorentz invariance. In addition, this model has no domain wall problem.