Current Algebra, K l 3 + Form Factors, and Radiative K l 3 + Decay

TL;DR: In this article, the authors derived the complete gauge-invariant matrix element for the decays of the photon energy, up to and including terms linear in photon energy $k. The contributions of order (k}^{\ensuremath{-}1}$ 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.

Abstract: The complete gauge-invariant matrix element for the decays ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{l}^{+}\ensuremath{\nu}\ensuremath{\gamma}$ is derived up to and including terms linear in the photon energy $k$. The contributions of order ${k}^{\ensuremath{-}1}$ 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 ${K}^{+}\ensuremath{\rightarrow}{l}^{+}\ensuremath{\nu}\ensuremath{\gamma}$ 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 ${K}_{l3}$ parameters ${f}_{+}(0)$, $\ensuremath{\xi}$, and ${\ensuremath{\lambda}}_{\ifmmode\pm\else\textpm\fi{}}$. Results are given for the rates and photon spectra in both ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{e}^{+}\ensuremath{\nu}\ensuremath{\gamma}$ and ${K}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}{\ensuremath{\mu}}^{+}\ensuremath{\nu}\ensuremath{\gamma}$ decays for representative values of the ${K}_{l3}$ parameters. With improved experimental data, it is then possible to check the predictions of various theoretical models both for the $t$ dependence of the ${K}_{l3}$ form factors and for the size of the structure-dependent matrix elements.