Experimental Study ofKe4+Decay

The rare $K_{e4\mathrm{}}^{}{}_{}{}^{+}$ decay modes ${\pi }^{+}{\pi }^{-}{e}^{+}\nu$, ${\pi }^{+}{\pi }^{+}{e}^{-}\overline{\nu }$, and ${\pi }^{+}{\pi }^{+}{\mu }^{-}\overline{\nu }$ were searched for in a sample of approximately 3×${10}^6$ $K^{+}$ decays. A total of 69 events of the mode ${\pi }^{+}{\pi }^{-}{e}^{+}\nu$ were observed. The other two modes are forbidden by the $\Delta S=\Delta Q$ rule and no examples of them were found. This experiment gives an upper limit for the amplitude ratio $X=\left|\frac{\left[A\right(\Delta S=-\Delta Q\left)\right]}{\left[A\right(\Delta S=\Delta Q\left)\right]}\right|$, for the axial-vector current of 0.25, with the assumption of no final-state $\pi -\pi$ interaction. The effects of final-state interactions in the $\pi -\pi$ system modify the interpretation of this result and may be large. The rate for the ${\pi }^{+}{\pi }^{-}{e}^{+}\nu$ mode was determined to be (2.9±0.6)×${10}^3$ ${\sec }^{-1\mathrm{}}$. The angular correlations between the planes formed by the lepton pair and the pion pair, and the angular distribution of ${\pi }^{+}$ with respect to the dipion line of flight in the dipion rest system for the 69 events, were studied. The dipion mass spectrum is presented for the 69 ${\pi }^{+}{\pi }^{-}{e}^{+}\nu$ events. The value of the ratio of the two form factors in $K_{e4\mathrm{}}$ decay ($p$ wave to $s$ wave) was found to be $\eta =0.8\mathrm{}\pm 0.30$ from an analysis of these correlations. An estimate of the low-energy $\pi -\pi$ phase shift was determined to be $"{\delta }_0-{\delta }_1"=35\mathrm{}°\pm 30°$ from the angular correlations. The difference "${\delta }_0-{\delta }_1$" between the