Problem for theories with spontaneousCPviolation and natural flavor conservation

Abstract

Using a vacuum-saturation approximation, Vainshtein, Zakharov, and Shifman have shown that $L=L_{\mathrm{QCD}}+L_{\mathrm{EW}}$ can explain the $\Delta I=\frac{1}{2}$ rule of strange-particle decays. Requiring $L_{\mathrm{EW}}$ to possess spontaneous $\mathrm{CP}$ violation and natural flavor conservation, we estimate $\frac{{\epsilon }^{\prime }}{\epsilon }$ using a similar approximation. We show that a very crude computation results in a very stringent limit $0.050>\left|\frac{{\epsilon }^{\prime }}{\epsilon }\right|>\mathrm{}0.048\mathrm{}$. This estimate is in conflict with the experimental measurement $\left|\frac{{\epsilon }^{\prime }}{\epsilon }\right|=0.003\mathrm{}\pm 0.015$. This is a problem for theories with spontaneous $\mathrm{CP}$ violation and natural flavor conservation if the above understanding of the $\Delta I=\frac{1}{2}$ rule is correct.