Abstract
We present a solution to the strong CP problem, which 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 in the Standard model (SM) is attributed to the condensation of complex scalars $\Phi$ in the model. The model is differentiated by others in that it explains the hierarchy in quark-Higgs Yukawa coupling in the SM based on a series of sequential breaking of the horizontal $SU(3)_{f}$ gauge symmetry. The experimental constraint $\overline{\theta}\lesssim10^{-10}$ requires $<\!\!\Phi\!\!>\,\lesssim\,10^{13}-10^{14}{\rm GeV}$ (vacuum expectation value of complex scalars) and $\lambda\,\lesssim\,10^{-6}$ (scalar quartic coupling). We show that this small coupling is natural in the sense of 'tHooft naturalness. Compared to other models of Nelson-Barr type with CP breaking scale $\Lambda_{CP}\lesssim10^{8}{\rm GeV}$, our model is more advantageous in terms of consistency with the thermal leptogenesis.
Highlights
The smallness of a parameter in a theory can be considered natural provided certain additional symmetries are restored in the limit where the parameter is sent to zero [1]
Beginning as a gauged CP-invariant theory, spontaneous CP violation is triggered by the condensation of the complex scalar field Φi
The horizontal SUð3Þf is spontaneously broken around the energy scale ∼1013–1014 GeV. This is in contrast with other NelsonBarr-type models where the CP breaking occurs for ΛCP ≲ 108 GeV by dimension-five operators [26]
Summary
The smallness of a parameter in a theory can be considered natural provided certain additional symmetries are restored in the limit where the parameter is sent to zero [1]. The model assumes a complex scalar sector, and the vacuum thereof breaks CP. It is the VEV of this complex scalar that makes the next-to-leadingorder contribution to the fermion mass matrix complex, thereby inducing the CP-violating KM phase in the weak sector of the SM. Strong CP problem by relying on the physics of spontaneous CP violation, it would be natural to suspect that the underlying origins of θ, the KM phase, and the hierarchy of fermion masses may be very closely related to one another. The particle content of the model is extended by introducing into the SM particles three complex scalars Φαi and six heavy Dirac fermions Ψuα and Ψdα, where α and i are indices for SUð3Þf and Zð2iÞ, respectively (α, i 1⁄4 1, 2, 3). We examine additional contributions to the fermion mass matrices which lead to δArg1⁄2det MuMd ≠ 0
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