Two Higgs Doublet Solutions to the Strong CP Problem

We solve the strong CP problem in a broad class of two Higgs doublet theories that will be probed at the Large Hadron Collider and at future colliders. These theories feature CP and Abelian flavor symmetries, both broken softly in the scalar potential, that yield realistic quark masses and mixings. The flavor symmetry charges are chosen so that $\bar{\theta}=0$ at tree level for all values of the Yukawa and quartic couplings of the theory. We prove that in all such theories the 1-loop contribution to $\bar{\theta}$ also vanishes, independently of the mass scale of the second Higgs doublet. We study two illustrative models with flavor group $\mathbb{Z}_3$. The direct contributions to the neutron electric dipole moment are negligible in both models. While the 2-loop contribution to $\bar{\theta}$ is less than $10^{-12}$ in one model, it can be as large as $10^{-10}$ in the other, yielding the prospect of a signal in planned experiments. Even with Abelian flavor symmetries, CP violation in neutral kaon mixing is generally expected to yield naturalness bounds on the masses of additional Higgs doublets of order 20 TeV. We prove that for all models in our class, where the flavor symmetry forces $\bar{\theta}$ to vanish at tree-level, the flavor-changing neutral currents are CP-conserving, yielding model-dependent bounds from neutral meson mixing near 1 TeV. Mixing of the two CP-even scalars gives corrections to the couplings of the 125 GeV Higgs state to $\bar{t}t, \bar{b}b, \bar{c}c, \bar{\tau}\tau$ and $\bar{\mu} \mu$, giving possible signals at high luminosity runs at LHC and at future colliders. Furthermore, distinctive correlations between corrections in the various channels can probe the underlying flavor symmetry.