Limiting two-Higgs-doublet models

We update the constraints on two-Higgs-doublet models (2HDMs) focusing on the parameter space relevant to explain the present muon $g$$-$$2$ anomaly, $\Delta a_{\mu}$, in four different types of models, type I, II, "lepton specific" (or X) and "flipped" (or Y). We show that the strong constraints provided by the electroweak precision data on the mass of the pseudoscalar Higgs, whose contribution may account for $\Delta a_{\mu}$, are evaded in regions where the charged scalar is degenerate with the heavy neutral one and the mixing angles $\alpha$ and $\beta$ satisfy the Standard Model limit $\beta-\alpha \approx \pi /2$. We combine theoretical constraints from vacuum stability and perturbativity with direct and indirect bounds arising from collider and $B$~physics. Possible future constraints from the electron $g$$-$$2$ are also considered. If the 126 GeV resonance discovered at the LHC is interpreted as the light CP-even Higgs boson of the 2HDM, we find that only models of type X can satisfy all the considered theoretical and experimental constraints.