Abstract
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. Using constraints based on analyticity and chiral symmetry, even more precise values are obtained: \(\mbox {$a_{0}^{0}$}=0.2196 \pm 0.0028\mbox {$\mathrm {_{stat}}$}\pm 0.0020 \mbox {$\mathrm {_{syst}}$}\) and \(\mbox {$a_{0}^{2}$}= -0.0444 \pm 0.0007 \mbox {$\mathrm {_{stat}}$}\pm 0.0005 \mbox {$\mathrm {_{syst}}$}\pm 0.0008 \mbox {$\mathrm {_{ChPT}}$}\).
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The NA48/2 Collaboration., Batley, J.R., Kalmus, G. et al. Precise tests of low energy QCD from \(\mbox {$\mathrm {K}_{\mathrm {e}4}$}\) decay properties. Eur. Phys. J. C 70, 635–657 (2010). https://doi.org/10.1140/epjc/s10052-010-1480-6
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DOI: https://doi.org/10.1140/epjc/s10052-010-1480-6