Minkowski space structure of the Higgs potential in the two-Higgs-doublet model

The Higgs potential of 2HDM keeps its generic form under the group of transformation $GL(2,C)$, which is larger than the usually considered reparametrization group $SU(2)$. This reparametrization symmetry induces the Minkowski space structure in the orbit space of 2HDM. Exploiting this property, we present a geometric analysis of the number and properties of stationary points of the most general 2HDM potential. In particular, we prove that charge-breaking and neutral vacua never coexist in 2HDM and establish conditions when the most general explicitly $CP$-conserving Higgs potential has spontaneously $CP$-violating minima. We also define the prototypical model of a given 2HDM, which has six free parameters less than the original one but still contains all the essential physics. Our analysis avoids manipulation with high-order algebraic equations.