Experimental Study of the Rare K + Decay Modes: K + → π + π 0 γ , K + → μ + π 0 ν γ , K + → π + γ γ , K + → π + ν ν ¯ , K + → π 0 π 0 e + ν , and K + → e + π 0 ν γ

Six rare stopped-${K}^{+}$ decays have been studied in the 40-inch heavy-liquid bubble chamber at Argonne National Laboratory. The chamber was filled with C${\mathrm{F}}_{3}$Br. Twenty-two radiative ${K}_{\ensuremath{\pi}2}$ events with three converted $\ensuremath{\gamma}$ rays were analyzed. The experiment was sensitive to positive pions with kinetic energy 55 to 102 MeV. The strengths ($\ensuremath{\gamma}$ and $\ensuremath{\beta}$) of the direct processes in the decay were determined to be $\ensuremath{\gamma}=\ensuremath{-}{0.02}_{\ensuremath{-}0.43}^{+0.17}$, $\ensuremath{\beta}=0.0\ifmmode\pm\else\textpm\fi{}0.3$, and the branching ratio is (${1.5}_{\ensuremath{-}0.6}^{+1.1}$)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}4}$ for $55l{T}_{{\ensuremath{\pi}}^{+}}l80$ MeV. These results are consistent with assuming the decay is dominated by internal bremsstrahlung. No events were found in the search for ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\pi}}^{0}\ensuremath{\nu}\ensuremath{\gamma}$, and the upper limit on the branching ratio is reported as 6.1\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ for $\ensuremath{\gamma}$ energies greater than 30 MeV. No examples of the ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\gamma}\ensuremath{\gamma}$ decay mode were found. The experiment was sensitive to pions with kinetic energy 6 to 102 and 114 to 127 MeV. The null result allowed us to discard several theoretical models which made branching-ratio predictions for this decay. Assuming a phase-space model for the pion spectrum, the upper limit is 3.5\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$. The upper limit on the neutral current decay ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}\ensuremath{\nu}\overline{\ensuremath{\nu}}$ is reported as 5.7\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ assuming a vector interaction. The decay ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}{e}^{+}\ensuremath{\nu}$ was observed for the first time. Two events were found in which all four converted $\ensuremath{\gamma}'\mathrm{s}$ were seen. These two events give a branching ratio of (${1.8}_{\ensuremath{-}0.6}^{+2.4}$)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}5}$ for this decay. The form factor ${f}_{1}$ for the decay is $|{f}_{1}|={0.97}_{\ensuremath{-}0.19}^{+0.50}$. These results are in good agreement with $\ensuremath{\Delta}I=\frac{1}{2}$ rule predictions. Seventeen ${K}^{+}\ensuremath{\rightarrow}{e}^{+}{\ensuremath{\pi}}^{0}\ensuremath{\nu}\ensuremath{\gamma}$ events were observed. The branching ratio is reported as $\frac{\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}{e}^{+}{\ensuremath{\pi}}^{0}\ensuremath{\nu}\ensuremath{\gamma})}{\ensuremath{\Gamma}({K}^{+}\ensuremath{\rightarrow}{e}^{+}{\ensuremath{\pi}}^{0}\ensuremath{\nu})}=(0.48\ifmmode\pm\else\textpm\fi{}0.20)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2}$ for ${E}_{\ensuremath{\gamma}}g30$ MeV and ${cos\ensuremath{\theta}}_{e\ensuremath{\gamma}}l0.9$.