New Results on Rare and Forbidden Semileptonic $K^+$ Decays

Experiment E865 at the Brookhaven AGS was set up primarily to search for the lepton flavour violating decay $K^+\to \pi^+\mu^+e^-$ ($K_{\pi\mu e}$) with high sensitivity. The flexibility of the apparatus allowed also to obtain more than an order of magnitude larger than previously available event samples on the following decay modes: $ \pi^+e^+e^- (K_{\pi ee})$, $\pi^+\mu^+\mu^- (K_{\pi\mu\mu})$, $\pi^+\pi^-e^+\nu_e (K_{e4})$, $\mu^+e^+e^-\nu_\mu$, and $e^+e^+e^-\nu_e$. The report focusses on the $K_{\pi\mu e}$ results and those on other lepton flavour violating decays as well as the $K_{e4}$ data, from which a new, quite precise value for the s-wave $\pi\pi$ scattering length can be deduced.