Precision measurement of the ratio BR($K_{S} \to \pi^{+}\pi^{-}e^{+}e^{-}$)/BR($K_{L} \to \pi^{+}\pi^{-}\pi^{0}_{D}$)

The $K_{S} \rightarrow \pi^{+}\pi^{-}e^{+}e^{-}$ decay mode was investigated using the data collected in 2002 by the NA48/1 collaboration. With about 23,k $K_{S} \rightarrow \pi^{+}\pi^{-}e^{+}e^{-}$ events and 59,k $K_{L} \rightarrow \pi^{+}\pi^{-}\pi^{0}_{D}$ normalization decays, the $K_{S} \rightarrow \pi^{+}\pi^{-}e^{+}e^{-}$ branching ratio relative to the $K_{L}\rightarrow \pi^{+}\pi^{-}\pi^{0}_{D}$ one was determined to be BR($K_{S} \rightarrow \pi^{+}\pi^{-}e^{+}e^{-}$)/BR($K_{L} \rightarrow \pi^{+}\pi^{-}\pi^{0}_{D}$) = $ (3.28 \pm 0.06_{stat}\pm 0.04_{syst})\times 10^{-2}$. This result was used to set the upper limit $|g_{E1}/g_{BR}| \lt 3.0$ at $90%$ CL on the presence, in the decay amplitude, of an E1 direct emission ($g_{E1}$) term relative to the dominant inner bremsstrahlung ($g_{BR}$) term. The CP-violating asymmetry ${cal A}_{\phi}$ in the sin$\phi$,cos$\phi$ distribution of $K_{S} \rightarrow \pi^{+}\pi^{-}e^{+}e^{-}$ events, where $\phi$ is the angle between the $\pi^{+} \pi^{-}$ and the $e^{+}e^{-}$ decay planes in the kaon centre of mass, was found to be ${cal A}_{\phi} = (-0.4 pm 0.8)%$, consistent with zero. These results are in good agreement with a description of the $K_{S} \rightarrow \pi^{+}\pi^{-}e^{+}e^{-}$ decay amplitude dominated by the CP-even inner bremsstrahlung process.