Question of Peccei-Quinn symmetry and quark masses in the economical 3-3-1 model

We show that there is an infinite number of U(1) symmetries like Peccei-Quinn symmetry in the 3-3-1 model with minimal scalar sector-two scalar triplets. Moreover, all of them are completely broken due to the model's scalars by themselves (notice that these scalars as known to have been often used to break the gauge symmetry and to generate the masses for the model's particles). There is not any residual Peccei-Quinn symmetry. Because of the minimal scalar content, there are some quarks that are massless at tree level, but they can get consistent mass contributions at one loop due to this fact. Interestingly, axions as associated with the mentioned U(1)s breaking (including Majoron due to lepton-charge breaking) are all gauged away because they are also the Goldstone bosons responsible for the gauge symmetry breaking as usual.