New measurement of the $K^{\pm} \to \pi^+\pi^-e^{\pm}\nu(K_{e4} )$ decay branching ratio and hadronic form factors

A sample of more than one million $K^{\pm} \rightarrow \pi^{+} \pi^{-} e^{\pm} \nu$ ($K_{e4}$) decay candidates with less than one percent background contamination has been collected by the NA48/2 experiment at the CERN SPS in 2003--2004, allowing a detailed study of the decay properties. The branching ratio, inclusive of $K_{e4\gamma}$ decays, is measured to be BR$(K_{e4}) = (4.257 \pm 0.016_{exp} \pm0.031_{ext}) \times 10^{-5}$ with a total relative error of 0.8%. This measurement complements the study of S- and P-wave hadronic form factors by assigning absolute values to the relative hadronic form factors obtained earlier in a simultaneous analysis of the $\pi\pi$ scattering lengths conducted on the same data sample. The overall form factor normalization $f_s = 5.705 \pm 0.017_{exp} \pm 0.031_{ext}$ is obtained with a total relative precision of 0.6%.