Pole Dominance and Low-Energy Theorem in the Radiative Decays of Charged Kaons

A low-energy theorem is derived for the structure-dependent axial-vector form factor in the radiative decay $K\to l+\nu +\gamma$ in the soft-kaon approximation. Corrections of the order of ($\frac{{m_K}^2}{{m_V}^2}$) ($V=\rho ,\omega ,\varphi$) are obtained in the pole-dominance approximation. In each approximation, the model predictions of both (i) asymptotic $\mathrm{SU}\left(3\right)\mathrm{}$ and (ii) current mixing are investigated. The quantity $\left|{\gamma }_K\right|\equiv \left|\frac{a_K\mathrm{}\left(0\right)\mathrm{}}{F\left(0\right)\mathrm{}}\right|$ is calculated in both approximations and in both models. It is found that the soft-kaon result is shifted upward by approximately 20%; the separation between the models in the two approximations is of the same order of magnitude.