New high statistics measurement of $K_{e4}$ decay form factors and $\pi\pi$ scattering phase shifts

We report results from a new measurement of the $K_{e4}$ decay $K^{\pm} \rightarrow \pi^{+}\pi^{-}e^{\pm}\nu$ by the NA48/2 collaboration at the CERN SPS, based on a partial sample of more than 670000 $K_{e4}$ decays in both charged modes collected in 2003. The form factors of the hadronic current (F, G, H) and $\pi\pi$ phase difference ($\delta=\delta_{s}-\delta_{p}$) have been measured in ten independent bins of the $\pi\pi$ mass spectrum to investigate the variation. A sizeable acceptance at large $\pi\pi$ mass, a low background and a very good resolution contribute to an improved experimental accuracy, a factor two better than in the previous measurement, when extracting the $\pi\pi$ scattering lengths $a^{0}_{0}$ and $a^{2}_{0}$. Under the assumption of isospin symmetry and using numerical solutions of the Roy equations, the following values are obtained in the plane ($a^{0}_{0}, a^{2}_{0}): a^{0}_{0}=0.233 \pm 0.016_{stat} \pm 0.007_{syst}, a^{2}_{0}=-0.0471 \pm 0.011_{stat} \pm 0.004_{syst}$. The presence of potentially large isospin effects is also considered and will allow comparison with precise predictions from Chiral Perturbation Theory.