Measurement of the CP violation parameter $|\eta_{+−}|$ and the charge asymmetry in $K^{\pm} \rightarrow 3\pi$ decays by NA48 and NA48/2

Throughout all periods of data taking, the main goal of the NA48 experiment has been the search for CP violation (CPV) in kaon decays. The observable $\eta_{+−}$ links the parameters of indirect and direct CPV $(\eta_{+−} = \epsilon + \epsilon^{\prime})$ and is defined as the CP violating amplitude ratio of the neutral kaon decaying into two charged pions: $\eta_{+−} = A(K_{L} \rightarrow \pi^{+}\pi^{-})/A(K_{S} \rightarrow \pi^{+}\pi^{-})$. NA48 has determined $|\eta_{+−}|$ via the measurement of the ratio of decay rates $\Gamma (K_{L} \rightarrow \pi^{+}\pi^{-})/\Gamma(K_{L} \rightarrow \pi e\nu)$, denoted as $\Gamma_{K2\pi}/\Gamma_{Ke3}$, using a pure $K_{L}$ beam in 1999. From a sample of 47000 $K_{2\pi}$ and five million $K_{e3}$ decays, we obtained results for $\Gamma K_{2\pi}/\Gamma_{Ke3}$, BR$(K_{L} \rightarrow \pi^{+}\pi^{-})$ and $|\eta_{+−}|$. Complementary with $\epsilon^{0}/\epsilon$, the observable for direct CPV in the charged kaons sector is the asymmetry Ag = $(g^{+} − g^{−})/(g^{+} + g^{−})$ of the linear slope parameter $g$ in the Dalitz plot of $K^{\pm} \rightarrow 3\pi$ decays. The NA48/2 experiment used simultaneous $K^{+}/K^{−}$ beams, and from the data samples taken in 2003 and 2004, $3 \times 10^{9} K^{\pm} \rightarrow \pi^{\pm}\pi^{+}\pi^{-}$ and $9 \times 10^{7} K^{\pm} \rightarrow \pi^{\pm}\pi^{0}\pi^{0}$ were selected. The charge asymmetry parameter Ag was determined with a total uncertainty of $∼ 2 \times 10^{−4}$ for each mode, ten times more accurate than previous measurements.