Abstract
For the radiative leptonic kaon decays of $K^{+} \to \mu^{+}\nu_{\mu}\gamma$, we calculate the form factors within Chiral perturbaion theory and Light-front model. For Chiral perturbaion theory, we describe the form factors of the pseudoscalar mesons to first nonleading order in the low-energy expansion, and the branching ratio of $O(p^{4})$ is $2.94\times 10^{-3}$ with cut $214.5\frac{MeV}{C}\leq p_{\mu }\leq 231.5\frac{MeV}{C}$. For Light-front model, we study the form factors for the $K^{+} \to \gamma$ transition in the entire physical range of momentum transference, and find the branching ratio is $2.926\times 10^{-3}$ with the same cut. Instead of tending to it, we find the results of $O(p^{6})$ in CHPT, which consider terms generated by vector and axial vector resonance exchange with strangeness, get far away from the results of LF than $O(p^{4})$ in CHPT.