Analysis of $K^{+}\to e^{+}{\nu }_e\gamma$ in light-front quark model and chiral perturbation theory of order $p^6$

Within the frameworks of the light-front quark model (LFQM) and chiral perturbation theory (ChPT) of $O(p^6)$, we reevaluate the form factors of the $K^{+}\to \gamma$ transition. We use these form factors to study the decay of $K^{+}\to e^{+}{\nu }_e\gamma$, which is dominated by the structure dependent contribution. We show the differential decay branching ratio as a function of $x=2E_{\gamma }\mathrm{/}{m}_K$, where $E_{\gamma }$ ($m_K$) is the photon energy (kaon mass). Explicitly, we find that, in the standard model with the cut of $x=0.01$ (0.1), the decay branching ratio of $K^{+}\to e^{+}{\nu }_e\gamma$ is $1.54 (1.44)\times 10^{-5}$and$1.57 (1.47)\times 10^{-5}$ in the LFQM and ChPT, respectively.