New Measurement of ${\mathit{K}}_{\mathit{e}4}^{+}$ Decay and the $\mathit{s}$-Wave $\pi \pi$-Scattering Length ${\mathit{a}}_0^0$

A sample of $4\times {10}^5$ events from the decay $K^{+}\to {\pi }^{+}{\pi }^{-}{e}^{+}{\nu }_e\left(K_{e4\mathrm{}}\right)$ has been collected in experiment E865 at the Brookhaven Alternating Gradient Synchrotron. The analysis of these data yields new measurements of the $K_{e4\mathrm{}}$ branching ratio $\left[\right(4.11\pm 0.01\pm 0.11)\times {10}^{-5\mathrm{}}]$, the $s$-wave $\pi \pi$ scattering length $[a_0^0=0.216\mathrm{}\pm 0.013(\mathrm{stat})\pm 0.004(\mathrm{syst})\pm 0.005(\mathrm{theor}\left)\right]$, and the form factors $F$, $G$, and $H$ of the hadronic current and their dependence on the invariant $\pi \pi$ mass.