Solving the strong CP problem with horizontal gauge symmetry

We present a solution to the strong CP problem that relies on the horizontal gauge symmetry and CP invariance in a full theory. Similar to other Nelson-Barr-type solutions, CP violation in both the strong and weak sectors of the Standard Model (SM) is attributed to the condensation of complex scalars Phi in the model. The model is differentiated from others in that it explains the hierarchy in the quark-Higgs Yukawa coupling in the SM based on a series of sequential breakings of the horizontal SU(3)(f) gauge symmetry. The experimental constraint (theta) over bar less than or similar to 10(-10) less than or similar to 10(13)-10(14) GeV (the vacuum expectation value of complex scalars) and lambda less than or similar to 10(-6) (the scalar quartic coupling). We show that this small coupling is natural in the sense of 't Hooft naturalness. Compared to other Nelson-Barr-type models with a CP-breaking scale A(cp)less than or similar to 10(8) GeV, our model is more advantageous in terms of consistency with thermal leptogenesis.