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Within the frameworks of the light-front quark model (LFQM) and chiralperturbation 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 dependentcontribution. We show the differential decay branching ratio as a function of$x=2E_\gamma/m_K$, 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 $1.54 (1.44)\times10^{-5}$ and $1.57 (1.47)\times 10^{-5}$ in the LFQM and ChPT, respectively.

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