Large-Nchigher-order weak chiral Lagrangians coupled to external electromagnetic fields: Applications to radiative kaon decays

Using factorization and bosonization of quark currents derived from the anomalous Wess-Zumino-Witten terms and from the electroweak perturbations to a QCD-motivated model for nonanomalous $p^4$ chiral Lagrangians, we obtain leading $\frac{1}{N_c}$ fourth-order $\Delta S=1\mathrm{}$ weak chiral Lagrangians coupled to external electromagnetic fields, valid to the zeroth order of gluonic corrections. Applications to radiative $K$ decays, e.g., $K\to \pi {\gamma }^{*}\to \pi l^{+}{l}^{-}$, $K\to \pi \gamma \gamma$, $K\to \pi \pi \gamma$, and $K\to \pi l\nu \gamma$, are studied. The twofold ambiguity for the predictions of $K\to \pi l^{+}{l}^{-}$ is suggestively resolved. The branching ratio of $K^{\pm }\to {\pi }^{\pm }\gamma \gamma$ is predicted to be 5.1 × ${10}^{-7\mathrm{}}$, while the current upper bound based on a pion phase-space spectrum is ${10}^{-6\mathrm{}}$. The direct $K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$ decays offer an excellent test on both anomalous and nonanomalous weak chiral Lagrangians; the agreement between theory and experiment is marvelous. The branching ratio of the interference between innerbremsstrahlung and direct-emission amplitudes in the process $K_L\to \pi e\nu \gamma$ is found to be of order ${10}^{-5\mathrm{}}$.