Role of vector mesons in rare kaon decays

We study the role of vector mesons in $K\to \pi \pi$, $K\to \pi l^{+}{l}^{-}$, $K_L\to \gamma l^{+}{l}^{-}$, and $K_L\to {\pi }^0\gamma \gamma$ using the chiral Lagrangian in the hidden-symmetry scheme with the Wess-Zumino anomaly. Two main features of our approach are (i) the slight modification of the effective weak chiral Lagrangian and (ii) the existence of the anomalous left-handed current constructed from the Wess-Zumino anomaly and intrinsic parity-violating interactions involving vector mesons. We predict that the decay mode $K_L\to {\pi }^0{e}^{+}{e}^{-}$ is dominated by the indirect $\mathrm{CP}$-violating one-photon-exchange process, which occurs at the level of a few parts in $1010$ in the branching ratio. We can also understand the recent data on $K_L\to \gamma e^{+}{e}^{-}$ and $K_L\to {\pi }^0\gamma \gamma$ in our framework, except for the branching ratio of $K_L\to {\pi }^0\gamma \gamma$.