The K^+ to pi^0 mu^+ u gamma ($K_{mu 3 gamma}$) decay has been measured with stopped positive kaons at the KEK 12 GeV proton synchrotron. A $K_{mu 3 gamma}$ sample containing 125 events was obtained. The partial branching ratio $Br(K_{mu 3 gamma}, E_{gamma}>30 {rm MeV}, theta_{mu^+ gamma}>20^{circ})$ was found to be $[2.4 pm 0.5(stat) pm 0.6(syst)]times 10^{-5}$, which is in good agreement with theoretical predictions.
The $K^+ to pi^0 pi^0 e^+ u$ ($K_{e4}^{00}$) decay has been measured with stopped positive kaons for a data sample of 216 events. A comparison of the observed spectra with a Monte Carlo simulation determined the $K_{e4}^{00}$ form factor. The results are compatible with the $K^+ to pi^+ pi^- e^+ u$ data, as estimated from the $Delta I=1/2$ rule. We also established that the $K_{e4}^{00}$ channel can be used to determine the $pi$-$pi$ scattering lengths.
The structure dependent radiative $K^+ rightarrow e^+ u gamma$ ($K_{e2gamma}^{rm SD}$) decay was investigated with stopped positive kaons. The $e^+$ momentum spectra were measured with and without a photon in coincidence and analyzed with Monte Carlo simulations for acceptance and detector response to extract the ratio of the $K_{e2gamma}^{rm SD}$ and $K^+ rightarrow e^+ u$ ($K_{e2}$) branching ratios. A value of $Br(K_{e2gamma}^{rm SD}) / Br(K_{e2}) = 1.22pm0.07_{rm stat} pm 0.04_{rm syst}$ was obtained, which is significantly larger than the value inferred from a previous experimental result for $Br(K_{e2gamma}^{rm SD})/Br(K^+rightarrow mu^+ u)$.
Using data collected by the fixed target Fermilab experiment FOCUS, we present several first measurements for the semileptonic decay $D^0 to bar{K}^0pi^-mu^+ u$. Using a model that includes a $bar{K}^0 pi^-$ S-wave component, we measure the form factor ratios to be r_v= 1.706+-0.677+-0.342 and r_2= 0.912+-0.370+-0.104 and the S-wave amplitude to be A=0.347+-0.222+-0.053 GeV^-1. Finally, we measure the vector semileptonic branching ratio $frac{Gamma(D^0 to K^{*}(892){-}mu^+ u)}{Gamma(D^0 to bar{K}^0pi^-pi^+)}= 0.337+-0.034+-0.013.
A search for T-violating muon polarization in the $K^+ to pi^0 mu^+ u$ decay has been performed at KEK using stopped kaons. A new improved limit was obtained, $P_T= -0.0017 pm 0.0023 (stat) pm 0.0011 (syst)$, corresponding to a 90 % confidence limit of $| P_T| < 0.0050$. The T-violating parameter was also determined to be Im$xi = -0.0053 pm 0.0071 (stat) pm 0.0036 (syst)$, and $|{rm Im}xi| < 0.016$ (90 % CL).
We have performed a measurement of the K^+ -> pi^+ pi^0 gamma decay and have observed 2 X 10^4 events. The best fit to the decay spectrum gives a branching ratio for direct photon emission of (4.7pm0.8pm0.3) X 10^{-6} in the pi^+ kinetic energy region of 55 to 90 MeV and requires no component due to interference with inner bremsstrahlung.