Measurement of the ratio $\Gamma(K_{L} \rightarrow \pi^{+}\pi^{-}) / \Gamma(K_{L} \rightarrow \pi^{\pm}e^{\mp}\nu)$ and extraction of the CP violation parameter $|\eta_{+-}|$

We present a measurement of the ratio of the decay rates $\Gamma (K_L \rightarrow\pi^+\pi^-) / \Gamma(K_L \rightarrow\pi^{\pm}e^{\mp}\nu)$, denoted as $\Gamma_{K2\pi}/\Gamma_{Ke3}$. The analysis is based on data taken during a dedicated run in 1999 by the NA48 experiment at the CERN SPS. Using a sample of 47000 $K_{2\pi}$ and five million $K_{e3}$ decays, we find $\Gamma_{K2\pi}/\Gamma_{Ke3} = (4.835 \pm 0.022_{stat.} \pm 0.016_{syst.}) \times 10^{-3}$. From this we derive the branching ratio of the CP violating decay $K_L \rightarrow\pi^+\pi^-$ and the CP violation parameter $|\eta_{+-}|$. Excluding the CP conserving direct photon emission component $K_L \rightarrow \pi^+\pi^-\gamma$, we obtain the results BR$(K_L \rightarrow \pi^+\pi^-) = (1.941 \pm 0.019) \times 10^{-3}$ and $|\eta_{+-}| = (2.223 \pm 0.012) \times 10^{-3}$.