Precise tests of low energy QCD from K(e4)decay properties

We report results from the analysis of the \mbox {\)\mathrm {K}^{\pm}\(}\rightarrow \pi^{+} \pi^{-} \mathrm{e}^{\pm} \nu ( \mbox {\)\mathrm {K}_{\mathrm {e}4}\(} ) decay by the NA48/2 collaboration at the CERN SPS, based on the total statistics of 1.13 million decays collected in 2003–2004. The hadronic form factors in the S- and P-wave and their variation with energy are obtained. The phase difference between the S- and P-wave states of the ππ system is accurately measured and allows a precise determination of \mbox {\)a_{0}^{0}\(} and \mbox {\)a_{0}^{2}\(} , the I = 0 and I = 2 S-wave ππ scattering lengths: \mbox {\)a_{0}^{0}\(}= 0.2220 \pm 0.0128 \mbox {\)\mathrm {_{stat}}\(}\pm 0.0050 \mbox {\)\mathrm {_{syst}}\(}\pm 0.0037\mbox {\)\mathrm {_{th}}\(},\mbox {\)a_{0}^{2}\(}= -0.0432 \pm 0.0086 \mbox {\)\mathrm {_{stat}}\(}\pm 0.0034 \mbox {\)\mathrm {_{syst}}\(}\pm 0.0028\mbox {\)\mathrm {_{th}}\(} . Combination of this result with the other NA48/2 measurement obtained in the study of \mbox {\)\mbox {\(\mathrm {K}^{\pm}\)}\rightarrow \mbox {\(\pi ^{0}\)}\mbox {\(\pi ^{0}\)}\mbox {\(\scriptstyle \pi ^{\pm }\)}\(} decays brings an improved determination of \mbox {\)a_{0}^{0}\(} and the first precise experimental measurement of \mbox {\)a_{0}^{2}\(} , providing a stringent test of Chiral Perturbation Theory predictions and lattice QCD calculations.