Signatures of discrete symmetries in the scalar sector

I discuss methods to identify the presence of dicrete symmetries in the two-Higgs-doublet model by observing the masses and the cubic and quartic interactions of the scalars. The symmetries considered are a $ Z_2 $ symmetry under which $ \phi_2 \rightarrow - \phi_2 $, and a CP symmetry which enforces real coupling constants in the Higgs potential. Those symmetries are spontaneously broken, and the $ Z_2 $ symmetry may also be softly broken. I identify the signatures in the interactions of the scalars that these symmetries leave after their breaking.