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 $\mathrm{\Phi }$ 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 $\overline{\theta }\lesssim {10}^{-10}$ requires $⟨\mathrm{\Phi }⟩\lesssim {10}^{13}–{10}^{14}\mathrm{GeV}$ (the vacuum expectation value of complex scalars) and $\lambda \lesssim {10}^{-6}$ (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 ${\mathrm{\Lambda }}_{CP}\lesssim {10}^8\mathrm{GeV}$, our model is more advantageous in terms of consistency with thermal leptogenesis.

Figure 1

One-loop correction to the dimension-seven operator with coefficients $c_{2,ij}^{(u,7)}$. The right panel corresponds to a correction of the dimension-seven operator with coefficients $c_{2,ij}^{(u,7)}$. The left panel corresponds to a correction of the dimension-seven operator with coefficients $c_{2,ji}^{(u,7)}$. The blobs correspond to the dimension-five operators with the coefficient $c_2^{(u,5)}$. The arrow on the internal line is directed from ${\mathrm{\Phi }}^{†}$ to $\mathrm{\Phi }$.

Figure 2

The diagram for the one-loop radiative correction to the ${\lambda }_{-}$ quartic interaction of the complex scalar fields defined in Eq. (21). The blobs correspond to the dimension-five operator with the coefficient $c_{2,i}^{(u,5)}$ defined in Eq. (10).

Figure 3

Diagram that produces a low-energy effective SM quark-Higgs Yukawa coupling after spontaneous breaking of $SU(3{)}_f$ and $CP$. The sum over the $Z_2$ index $i$ is assumed. The blob corresponds to the higher-dimensional operator in Eq. (10). The cross represents the mass insertion of the heavy fermion.