Precise tests of low energy QCD from $K_{e4}$ decay properties

We report results from the analysis of the $K^{\pm} \rightarrow \pi^{+} \pi^{-} e^{\pm} \nu$ (Ke4) decay by the NA48/2 collaboration at the CERN SPS, based on the total statistics of 1.13 million decays collected in 2003-2004. The hadronic form factors in the S- and P-wave and their variation with energy are obtained. The phase difference between the S- and P-wave states of the $\pi \pi$ system is accurately measured and allows a precise determination of $a_{0}^{0}$ and $a_{0}^{2}$, the I=0 and I=2 S-wave $\pi \pi$ scattering lengths: $a_{0}^{0} = 0.2220 \pm 0.0128_{stat} \pm 0.0050_{syst} \pm 0.0037_{th}, a_{0}^{2}$ = $-0.0432 \pm 0.0086_{stat} \pm 0.0034_{syst} \pm 0.0028_{th}.$ Combination of this result with the other NA48/2 measurement obtained in the study of $K^{\pm} \rightarrow \pi^{0} \pi^{0} \pi^{\pm}$ decays brings an improved determination of $a_{0}^{0}$ and the first precise experimental measurement of $a_{0}^{2}$, providing a stringent test of Chiral Perturbation Theory predictions and lattice QCD calculations. Using constraints based on analyticity and chiral symmetry, even more precise values are obtained: $a_{0}^{0} = 0.2196 \pm 0.0028_{stat} \pm 0.0020_{syst}$ and $a_{0}^{2} = -0.0444 \pm 0.0007_{stat} \pm 0.0005_{syst} \pm 0.0008_{ChPT}$.