We report experimental details and results of a new measurement of the decay $K^{+}\to {\pi }^{+}{\pi }^{-}{e}^{+}{\nu }_e{(K}_{e4\mathrm{}}).\mathrm{}$ A sample of more than $400,000K_{e4\mathrm{}}$ events with low background has been collected by Experiment 865 at the Brookhaven Alternate Gradient Synchrotron. From these data, the branching ratio $(4.11\mathrm{}\pm 0.01\pm 0.11)\times {10}^{-5}$ and the $\pi \pi$ invariant mass dependence of the form factors $F,G,$ and H of the weak hadronic current as well as the phase shift difference ${\delta }_0^0-{\delta }_1^1$ for $\pi \pi$ scattering were extracted. Using constraints based on analyticity and chiral symmetry, a new value with considerably improved accuracy for the s-wave $\pi \pi$ scattering length $a_0^0$ has been obtained also: $a_0^0=0.216\mathrm{}\pm 0.013\left(\mathrm{stat}\right)\pm 0.002\left(\mathrm{syst}\right)\pm 0.002\left(\mathrm{theor}\right).\mathrm{}$

• Received 31 January 2003

DOI:https://doi.org/10.1103/PhysRevD.67.072004