On the Higgs spectra of the 3-3-1 model

The minimal scalar sector of the 3-3-1 model is composed by the SU(3)$_L$ triplet scalars $\eta$, $\rho$, $\chi$ and its potential allows the trilinear term $\frac{f}{\sqrt{2}}\chi \eta \rho$. Since $f$ is an energy scale associated to the explicit violation of Peccei-Quinn global symmetry, it is natural to consider in what energy scale such symmetry is broken and its consequences in the spectrum of scalars of the model. here, We show that $f$ determines the spectrum of scalars of the model. Hence, we develop the scalar sector considering $f$ belonging to four energy regimes, namely $f \ll \langle \eta \rangle_0$, $\langle \rho \rangle_0$; $f= \langle \eta \rangle_0$, $\langle \rho \rangle_0$; $f=\langle \chi \rangle_0$ and $f \gg \langle \chi \rangle_0$ and obtain the spectrum of scalars for each case. In the first and second cases the spectrum of scalars presents a set of new scalars belonging to the electroweak scale, while in the third case all new scalars belong to the 3-3-1 scale and the fourth case all the new scalars have masses lying at $f$ scale. All cases have a neutral CP-even scalar mimicking the standard Higgs.