Experimental Study of the Rare $K^{+}$ Decay Modes: $K^{+}\to {\pi }^{+}{\pi }^0\gamma$, $K^{+}\to {\mu }^{+}{\pi }^0\nu \gamma$, $K^{+}\to {\pi }^{+}\gamma \gamma$, $K^{+}\to {\pi }^{+}\nu \overline{\nu }$, $K^{+}\to {\pi }^0{\pi }^0{e}^{+}\nu$, and $K^{+}\to e^{+}{\pi }^0\nu \gamma$

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 $K_{\pi 2}$ events with three converted $\gamma$ rays were analyzed. The experiment was sensitive to positive pions with kinetic energy 55 to 102 MeV. The strengths ($\gamma$ and $\beta$) of the direct processes in the decay were determined to be $\gamma =-{0.02}_{-0.43\mathrm{}}^{+0.17\mathrm{}}$, $\beta =0.0\mathrm{}\pm 0.3$, and the branching ratio is (${1.5}_{-0.6\mathrm{}}^{+1.1\mathrm{}}$)×${10}^{-4\mathrm{}}$ for $55<\mathrm{}{T}_{{\pi }^{+}}<80\mathrm{}$ MeV. These results are consistent with assuming the decay is dominated by internal bremsstrahlung. No events were found in the search for $K^{+}\to {\mu }^{+}{\pi }^0\nu \gamma$, and the upper limit on the branching ratio is reported as 6.1×${10}^{-5\mathrm{}}$ for $\gamma$ energies greater than 30 MeV. No examples of the $K^{+}\to {\pi }^{+}\gamma \gamma$ 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×${10}^{-5\mathrm{}}$. The upper limit on the neutral current decay $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ is reported as 5.7×${10}^{-5\mathrm{}}$ assuming a vector interaction. The decay $K^{+}\to {\pi }^0{\pi }^0{e}^{+}\nu$ was observed for the first time. Two events were found in which all four converted $\gamma \text{'}\mathrm{s}$ were seen. These two events give a branching ratio of (${1.8}_{-0.6\mathrm{}}^{+2.4\mathrm{}}$)×${10}^{-5\mathrm{}}$ for this decay. The form factor $f_1$ for the decay is $\left|f_1\right|={0.97}_{-0.19\mathrm{}}^{+0.50\mathrm{}}$. These results are in good agreement...