We introduce a class of multi-Higgs doublet extensions of the Standard Model that solve the strong $CP$ problem with profound consequences for the flavor sector. The Yukawa matrices are constrained to have many zero entries by a “Higgs-flavor” symmetry, $G_{\mathrm{HF}}$, that acts on Higgs and quark fields. The violation of both $CP$ and $G_{\mathrm{HF}}$ occurs in the Higgs mass matrix so that, for certain choices of $G_{\mathrm{HF}}$ charges, the strong $CP$ parameter $\overline{\theta }$ is zero at tree level. Radiative corrections to $\overline{\theta }$ are computed in this class of theories. They vanish in realistic two-Higgs doublet models with $G_{\mathrm{HF}}={\mathbb{Z}}_3$. We also construct realistic three-Higgs models with $G_{\mathrm{HF}}=\mathrm{U}(1)$, where the one-loop results for $\overline{\theta }$ are model-dependent. Requiring $\overline{\theta }<{10}^{-10}$ has important implications for the flavor problem by constraining the Yukawa coupling and Higgs mass matrices. Contributions to $\overline{\theta }$ from higher-dimension operators are computed at one loop and can also be sufficiently small, although the hierarchy problem of this class of theories is worse than in the Standard Model.

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• Cabibbo–Kobayashi–Maskawa matrix
• Extensions of Higgs sector
• Symmetries
• CP symmetry
• CP violation