Phenomenology of extended multiHiggs doublet models with $$S_4$$ family symmetry

Abstract We propose extended 3HDM and 4HDM models where the SM gauge symmetry is enlarged by the spontaneously broken $$S_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> group, the preserved $$Z_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and broken $$Z_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> cyclic symmetries. Each model has three active SU (2) scalar doublets, in addition, the first one has an extra inert scalar singlet, whereas the second model has an inert scalar doublet. Furthermore, each model has several extra gauge singlet scalars, which are triplets under $$S_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> . Both models yield the same structure of the mass matrices for the fermion sector, where a radiative seesaw generates the tiny light active neutrino masses at one-loop level, through the $$S_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> triplets. The presence of flavor changing neutral currents mediated by heavy scalars allowed us to study the $$(K^0-\overline{K}^0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>K</mml:mi> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>-</mml:mo> <mml:msup> <mml:mover> <mml:mi>K</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mn>0</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$(B_{d,s}^0-\overline{B}_{d,s}^0)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mo>-</mml:mo> <mml:msubsup> <mml:mover> <mml:mi>B</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>,</mml:mo> <mml:mi>s</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> meson mixings, in the parameter space that currently satisfies the experimental constraints. On the other hand, due to the preserved $$Z_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> symmetry, our proposed models have stable scalar and fermionic dark matter candidates. Furthermore, these models are consistent with the current pattern of SM fermion masses and mixings, with the measured dark matter relic abundance and successfully accommodate the constraints arising from meson oscillations and oblique parameters. The extra scalars in our models provide radiative corrections to the oblique parameters, where due to the presence of the scalar inert doublet, renders the 4HDM less restrictive than the 3HDM one.