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 $K^{\pm} \pi^{+}\pi^{-}e^{\pm}\nu$ by the NA48/2 collaboration at the CERN SPS, based on a partial sample of more than 670 000 $K_{e4}$ 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 their 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 a00 and a20 . Under the assumption of isospin symmetry and using numerical solutions of the Roy equations, the following values are obtained in the plane a00 , a20 : a00 = 0.233± 0.016stat ±0.007syst, a20 =0.0471±0.011stat ±0.004syst. The presence of potentially large isospin effects is also considered and will allow comparison with precise predictions from Chiral Perturbation Theory