Direct and indirect $\mathrm{CP}$ violation in the decay $K_L\to {\pi }^{+}{\pi }^{-}{e}^{+}{e}^{-}$

The decay $K_L\to {\pi }^{+}{\pi }^{-}{e}^{+}{e}^{-}$ is analyzed in a model containing (i) a $\mathrm{CP}$-conserving amplitude associated with the $M1\mathrm{}$ transition in $K_L\to {\pi }^{+}{\pi }^{-}\gamma$, (ii) an indirect $\mathrm{CP}$-violating amplitude related to the bremsstrahlung part of $K_L\to {\pi }^{+}{\pi }^{-}\gamma$, and (iii) a direct $\mathrm{CP}$-violating term associated with the short-distance interaction $s\overline{d}\to e^{+}{e}^{-}$. Interference of the first two components produces a large $\mathrm{CP}$-violating asymmetry (∼14%) in the distribution of the angle $\Phi$ between the $e^{+}{e}^{-}$ and ${\pi }^{+}{\pi }^{-}$ planes. The full angular distribution contains two further $\mathrm{CP}$-violating observables. Effects of direct $\mathrm{CP}$ violation are found to be numerically small.