DirectCPviolation inK→3πdecay

In light of the fact that the presence of the $Z^0$ penguin diagram suppresses strongly $\frac{{\epsilon }_{2\pi }^{\prime }}{\epsilon }$ in $K^0\to 2\pi$ decay for large values of $m_t$, we reanalyze the direct decay-amplitude $\mathrm{CP}$ violation in $K^0\to 3\pi$ decays in the Kobayashi-Maskawa model. The effects due to electroweak penguins, isospin breaking, and higher-order weak chiral Lagrangians are studied in the large-$N_c$ approach. We find that the Li-Wolfenstein relation between ${\epsilon }_{3\pi }^{\prime }$ and ${\epsilon }_{2\pi }^{\prime }$ is modified dramatically: the former receives very large contributions from the higher-derivative chiral terms and sizable contributions from the isospin-breaking correction due to ${\pi }^0-\eta -{\eta }^{\prime }$ mixing and $\eta$, ${\eta }^{\prime }\to 3\pi$ transitions. When the top quark becomes very heavy, effects of the electroweak penguin terms are enhanced. Unlike $\frac{{\epsilon }_{2\pi }^{\prime }}{\epsilon }$, which decreases as $m_t$ increases, the $\mathrm{CP}$-violating parameter $\frac{{\epsilon }_{3\pi }^{\prime }}{\epsilon }$ is of order ${10}^{-2\mathrm{}}$ and increases with the heavy top-quark mass.