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We report results from the analysis of the K-+/- -> pi(+)pi(-) C-+/-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 pp 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 pi pi scattering lengths: a(0) (0) = 0.2220 +/- 0.0128 (stat) +/- 0.0050(syst) +/- 0.0037th, a(2) (0) = - 0.0432 +/- 0.0086(stat) +/- 0.0034(syst) +/- 0.0028th. Combination of this result with the other NA48/ 2 measurement obtained in the study of K +/- -> pi(0)pi(0) pi(+/-) 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)

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