3HDM with $\Delta(27)$ symmetry and its phenomenological consequences

We perform a comprehensive analysis of a version of the 3-Higgs doublet model whose scalar potential is invariant under a global $\Delta(27)$ discrete symmetry and where the three scalar doublets are chosen to transform as a triplet under this discrete group. For each of the known tree-level minima we study the mass spectra and use the oblique parameters $STU$ as well as perturbative unitarity to constrain the parameter space of the model. We then discuss phenomenological consequences of some leading order flavour mixing quark Yukawa couplings by considering the flavour violation process $b \to s \gamma$. We show that perturbative unitarity significantly constrains parameters of the model while, conversely, the beyond the Standard Model contributions to the $b \to s \gamma$ decay are automatically tamed by the symmetry.