Strong and weakCPin a model with a new gauged U(1) symmetry

We discuss a natural solution to the strong $\mathrm{CP}$ problem with spontaneous $\mathrm{CP}$ breaking in the context of $\mathrm{S}{\mathrm{U}}_C\mathrm{}\left(3\right)\mathrm{}\times \mathrm{S}{\mathrm{U}}_L\mathrm{}\left(2\right)\mathrm{}\times {\mathrm{U}}_Y\mathrm{}\left(1\right)\mathrm{}\times {\mathrm{U}}_{\mathrm{new}}\mathrm{}\left(1\right)\mathrm{}$ with two vectorlike quarks. The gauged ${\mathrm{U}}_{\mathrm{new}}$(1) symmetry is imposed such that $\overline{\theta }$ remains zero at the tree level when $\mathrm{CP}$ and ${\mathrm{U}}_{\mathrm{new}}$(1) are broken by singlet Higgs scalars. Flavor-changing neutral currents at the tree level are naturally suppressed because only singlet Higgs scalars are introduced. In order to agree with the experimental measurement of $\epsilon$ and $\frac{{\epsilon }^{\prime }}{\epsilon }$, we find that the $\mathrm{CP}$-breaking scale and the masses of the new quarks are less than 2.0 TeV and 530 GeV, respectively.