Current Algebra,Kl3+Form Factors, and RadiativeKl3+Decay

The complete gauge-invariant matrix element for the decays $K^{+}\to {\pi }^0{l}^{+}\nu \gamma$ is derived up to and including terms linear in the photon energy $k$. The contributions of order $k^{-1\mathrm{}}$ and $k^0$ are evaluated through Low's theorem, while the contributions linear in $k$, which represent the most important structure-dependent terms, are related to the vector and axial-vector couplings in $K^{+}\to l^{+}\nu \gamma$ via the hypothesis of partially conserved axial-vector current (PCAC). It is shown that the spectra and rates from the radiative decay provide an independent means of measuring the $K_{l3\mathrm{}}$ parameters $f_{+}\mathrm{}\left(0\right)\mathrm{}$, $\xi$, and ${\lambda }_{\pm }$. Results are given for the rates and photon spectra in both $K^{+}\to {\pi }^0{e}^{+}\nu \gamma$ and $K^{+}\to {\pi }^0{\mu }^{+}\nu \gamma$ decays for representative values of the $K_{l3\mathrm{}}$ parameters. With improved experimental data, it is then possible to check the predictions of various theoretical models both for the $t$ dependence of the $K_{l3\mathrm{}}$ form factors and for the size of the structure-dependent matrix elements.