The inert two-Higgs-doublet model (i2HDM) is a theoretically well-motivated example of a minimal consistent dark matter (DM) model which provides monojet, mono-$Z$, mono-Higgs, and vector-boson-$\mathrm{fusion}+{\mathit{E}}_{\mathrm{T}}^{\mathrm{miss}}$ signatures at the LHC, complemented by signals in direct and indirect DM search experiments. In this paper we have performed a detailed analysis of the constraints in the full five-dimensional parameter space of the i2HDM, coming from perturbativity, unitarity, electroweak precision data, Higgs data from the LHC, DM relic density, direct/indirect DM detection, and LHC monojet analysis, as well as implications of experimental LHC studies on disappearing charged tracks relevant to a high DM mass region. We demonstrate the complementarity of the above constraints and present projections for future LHC data and direct DM detection experiments to probe further i2HDM parameter space. The model is implemented into the CalcHEP and micrOMEGAs packages, which are publicly available at the HEPMDB database, and it is ready for a further exploration in the context of the LHC, relic density, and DM direct detection.

We obtain tree-level unitarity constraints for the most general Two-Higgs-Doublet Model (2HDM) with explicit $CP$-violation. We briefly discuss correspondence between possible violation of tree-level unitarity limitation and physical content of the theory.

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.