Supersymmetric Models with Approximate CP

Supersymmetric models with an approximate CP, $10^{-3} \lsim \phi_{CP} \ll 1$, are a viable framework for the description of nature. The full high energy theory has exact CP and horizontal symmetries that are spontaneously broken with a naturally induced hierarchy of scales, ${\mathrm{\Lambda }}_{CP}\ll {\mathrm{\Lambda }}_H$. Consequently, the effective low energy theory, that is the supersymmetric Standard Model, has CP broken explicitly but by a small parameter. The ${ϵ}_K$ parameter is accounted for by supersymmetric contributions. The predictions for other CP violating observables are very different from the Standard Model. In particular, CP violating effects in neutral B decays into final CP eigenstates such as $B\to \psi K_S$ and in $K\to \pi \nu \bar{\nu }$ decays are very small. This framework, though, is strongly disfavored by the recent measurements of ${ϵ}_K^{\mathrm{\prime }}\mathrm{/}{ϵ}_K$.