Symmetries of two Higgs doublet model and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>C</mml:mi><mml:mi>P</mml:mi></mml:math>violation

We use the invariance of a physical picture under a change of Lagrangian, the reparameterization invariance in the space of Lagrangians and its particular case---the rephrasing invariance---for analysis of the two-Higgs-doublet extension of the standard model. We found that some parameters of theory like $\mathrm{tan}\ensuremath{\beta}$ are reparameterization dependent and therefore cannot be fundamental. We use the ${Z}_{2}$ symmetry of the Lagrangian, which prevents a ${\ensuremath{\phi}}_{1}\ensuremath{\leftrightarrow}{\ensuremath{\phi}}_{2}$ transition, and the different levels of its violation, soft and hard, to describe the physical content of the model. In general, the broken ${Z}_{2}$ symmetry allows for a $CP$ violation in the physical Higgs sector. We argue that the two-Higgs-doublet model with a soft breaking of ${Z}_{2}$ symmetry is a natural model in the description of electroweak symmetry breaking. To simplify the analysis, we choose among different forms of Lagrangian describing the same physical reality a specific one, in which the vacuum expectation values of both Higgs fields are real. A possible $CP$ violation in the Higgs sector is described by using a two-step procedure with the first step identical to a diagonalization of the mass matrix for $CP$-even fields in the $CP$-conserving case. We find a very simple, necessary, and sufficient condition for a $CP$ violation in the Higgs sector. We determine the range of parameters for which $CP$ violation and flavor-changing neutral current effects are naturally small---it corresponds to a small dimensionless mass parameter $\ensuremath{\nu}=\mathrm{Re}{m}_{12}^{2}/(2{v}_{1}{v}_{2})$. We show that for small $\ensuremath{\nu}$ some Higgs bosons can be heavy---with mass up to about 0.6 TeV---without violating of the unitarity constraints. If $\ensuremath{\nu}$ is large, all Higgs bosons except one can be arbitrarily heavy. We discuss, in particular, main features of this case, which corresponds for $\ensuremath{\nu}\ensuremath{\rightarrow}\ensuremath{\infty}$ to a decoupling of heavy Higgs bosons. In the model II for Yukawa interactions we obtain the set of relations among the couplings to gauge bosons and to fermions which allows us to analyze different physical situations (including $CP$ violation) in terms of these very couplings, instead of the parameters of Lagrangian.