The data on the reactions K^+Xe --> K^0 gamma X and K^+Xe --> K^+ gamma X, obtained with the bubble chamber DIANA, have been analyzed for possible radiative decays of the Theta^+(1540) baryon: Theta^+ --> K^0 p gamma and Theta^+ --> K^+ n gamma. No signals have been observed, and we derive the upper limits Gamma(Theta^+ --> K^0 p gamma) / Gamma(Theta^+ --> K^0 p) < 0.032 and Gamma(Theta^+ --> K^+ n gamma) / Gamma(Theta^+ --> K^+ n) < 0.041 which, using our previous measurement of Gamma(Theta^+ --> KN) = (0.39+-0.10) MeV, translate to Gamma(Theta^+ --> K^0 p gamma) < 8 keV and Gamma(Theta^+ --> K^+ n gamma) < 11 keV at 90% confidence level. We have also measured the cross sections of K^+ -induced reactions involving emission of a neutral pion: sigma(K^+n --> K^0 p pi^0) = (68+-18) mub and sigma(K^+N --> K^+ N pi^0) = (30+-8) mub for incident K^+ momentum of 640 MeV.
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