Scalar and gauge sectors in the 3-Higgs Doublet Model under the S3-symmetry

We analyse the Higgs sector of an $S_3$ model with three Higgs doublets and no explicit CP violation. After electroweak breaking there are nine physical Higgs bosons, one of which corresponds to the Standard Model one. We study the scalar and gauge sectors of this model, taking into account the conditions set by the minimisation and stability of the potential. We calculate the masses, trilinear and quartic Higgs self-couplings, and Higgs-gauge couplings. We consider two possible alignment scenarios, where only one of the three neutral scalars has couplings to the gauge bosons and corresponds to the SM Higgs, and whose trilinear and quartic couplings reduce exactly to the SM ones. We also obtain numerically the parameter space allowed for the scalar masses in each of the alignment scenarios. With this information we are able to analyse the structure of the one-loop scalar and gauge corrections to the neutral scalar mass matrix. We show that a residual $\mathcal{Z}2$ symmetry is preserved at NLO, keeping the decoupling of one of the neutral scalars, and a possible mixing between the other two. Small values for the off-diagonal element favour a large mixing between the $S_3$ singlet and doublet.