Vacuum Stability of Asymptotically Safe Two Higgs Doublet Models

We study different types of Two Higgs Doublet Models (2HDMs) under the assumption that all quartic couplings' beta functions vanish simultaneously at the Planck scale. The Standard Model seems to display this property almost accidentally, because the Higgs boson mass is close to 125 GeV. This also ties closely into the question of whether the theory is stable or metastable. We investigate if such "fixed points" can exist in various $\mathbb{Z}_2$-symmetric 2HDM subclasses, and if the theories that meet these conditions are phenomenologically viable, as well as vacuum stable. We find that the fixed point condition drastically reduces the parameter space of 2HDM theories, but can be met. Fixed points can only exist in type II and type Y models, in regions of large tan$\beta$, and they are only compatible with all existing experimental bounds if the $\mathbb{Z}_2$-symmetry is at least softly broken, with a soft breaking parameter of at least $M_{12}$ $>$ 70 GeV (380 GeV) for type Y (type II) models. The allowed region falls into the alignment limit, with the mixing angle combination $|\alpha - \beta| \approx\frac{\pi}{2}$. While there are both vacuum-stable and vacuum-unstable solutions, only the vacuum-unstable ones really agree with Standard-Model-like CP-even Higgs boson mass values of 125 GeV. The vacuum-stable solutions favour slightly higher values. While scenarios of asymptotically safe 2HDM exist, they cannot improve over the Standard Model regarding the question of vacuum stability.