  In all scalar extensions of the standard model of particle interactions, the
one Higgs boson responsible for electroweak symmetry breaking always mixes with
other neutral scalars at tree level unless a symmetry prevents it. An
unexplored important option is that the mixing may be radiative, and thus
guaranteed to be small. Two first such examples are discussed. One is based on
the soft breaking of the discrete symmetry $Z_3$. The other starts with the
non-Abelian discrete symmetry $A_4$ which is then softly broken to $Z_3$, and
results in the emergence of an interesting dark-matter candidate together with
a light mediator for the dark matter to have its own long-range interaction.
