Stability and symmetry breaking in a three-Higgs-doublet model with lepton family symmetry $O(2)\otimes {\mathbb{Z}}_2$

Motivated by the neutrino data, an extension of the Standard Model with three Higgs-boson doublets has been proposed. Imposing an $O(2)\otimes {\mathbb{Z}}_2$ family symmetry, a neutrino mixing matrix with ${\theta }_{23}=\pi /4$ and ${\theta }_{13}=0$ appears in a natural way. Even though these values for the mixing matrix do not follow the recent experimental constraints, they are nevertheless a good approximation. We study the Higgs potential of this model in detail. We apply recent methods which allow for the study of any three-Higgs-boson doublet model. It turns out that for a variety of parameters the potential is stable, has the correct electroweak symmetry breaking, and gives the correct vacuum expectation value.

Figure 1

Stability and stationarity solutions of the 3HDM Higgs potential, varying the two quartic parameters $a_1$ and $a_2$ of the potential (1.6). The other quartic parameters are set to $a_3=-5$, $a_4=-0.0474$, $a_5=1.5$, $a_6=2$, $a_7=3$. The quadratic parameters are fixed by the equations (1.13), the vacuum expectation value $v_0$, and the ratio $v_1/v_2=m_{\mu }/m_{\tau }$ as well as $v_3=v_0/\sqrt{2}$. The larger full circles (blue) show points where the potential is unstable. The open circles (red) show parameters where no correct electroweak symmetry breaking appears or the vacuum expectation value $v_0$ is unequal to 246 GeV. Finally, the small dots (green) have a viable global minimum corresponding to the correct vacuum expectation value.