Radiative Kaon Decays $K^{\pm }\to {\pi }^{\pm }{\pi }^0\gamma$ and Direct CP Violation

It is stressed that a measurement of the electric dipole amplitude for direct photon emission in $\text{kpm}$ decays through its interference with inner bremsstrahlung is important for differentiating among various models. Effects of amplitude CP violation in the radiative decays of the charged kaon are analyzed in the Standard Model in conjunction with the large $N_c$ approach. We point out that gluon and electromagnetic penguin contributions to the CP-violating asymmetry between the Dalitz plots of $\text{kpm}$ are of equal weight. The magnitude of CP asymmetry ranges from $2\times {10}^{-6}$ to $1\times 10^{-5}$ when the photon energy in the kaon rest frame varies from 50 MeV to 170 MeV.