Precise tests of low energy QCD from $\mbox {$\mathrm {K}_{\mathrm {e}4}$}$ decay properties

We report results from the analysis of the K + -e (K e4 ) 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 system is accurately measured and allows a precise determination of a 0 0 and a 2 0 , the I = 0 and I = 2 S-wave scattering lengths: a 0 0 = 0.2220 0.0128 stat 0.0050 syst 0.0037 th , a 2 0 = -0.0432 0.0086 stat 0.0034 syst 0.0028 th . Combination of this result with the other NA48/2 measurement obtained in the study of K 0 0 decays brings an improved determination of a 0 0 and the first precise experimental measurement of a 2 0 , 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 0.0028 stat 0.0020 syst and a 2 0 = -0.0444 0.0007 stat 0.0005 syst 0.0008 ChPT .