ℤ ₂ breaking effects in 2-loop RG evolution of 2HDM

We investigate the effects of a ℤ ₂ symmetry in the CP-conserving Two-Higgs-Doublet-Model (2HDM); which is often imposed to prevent Flavor-Changing-Neutral-Currents (FCNCs) at tree-level. Specifically, we analyze how a breaking of the ℤ ₂ symmetry spreads during renormalization group evolution; employing general 2-loop renormalization group equations that we have derived. Evolving the model from the electroweak to the Planck scale, we find that while the case of an exact ℤ ₂ symmetric 2HDM is very constrained, a soft breaking of the ℤ ₂ symmetry extends the valid parameter space regions. The effects of a hard ℤ ₂ breaking in the scalar sector as well as the stability of the flavor alignment ansatz are also investigated. We find that while a hard breaking of the ℤ ₂ symmetry in the potential is problematic, since it speeds up the growth of quartic couplings, the generated FCNCs are heavily suppressed. Conversely, we also find that hard ℤ ₂ breaking in the Yukawa sector at most gives moderate ℤ ₂ breaking in the potential; whereas the FCNCs can become quite sizable far away from the ℤ ₂ symmetric regions.