The complete gauge-invariant matrix element for the decays ${\overline{K}}^0\to {\pi }^{+}{l}^{-}\overline{\nu }\gamma$ ($l=e$ or $\mu$) is derived using the soft-photon theorems of Low and of Adler and Dothan. These theorems, along with several corollaries of them, are reviewed in detail and their application demonstrated by reference to the radiative $\overline{K}_{l3\mathrm{}}^{}{}_{}{}^0$ decay mode. The square of the matrix element is calculated using the theorem of Burnett and Kroll, and is compared with the result of a direct computer evaluation of the appropriate traces. Structure-dependent terms are discussed, and the dominant terms among those linear in the photon energy are estimated in the softpion and kaon limits. Results for the radiative photon spectra are given, together with the decay rates for a specific value of the minimum photon energy $E_0$.

• Received 17 March 1970

DOI:https://doi.org/10.1103/PhysRevD.2.542