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We fill up the missing piece in our own re-analysis of the long-distanceelectromagnetic radiative corrections to semileptonic kaon decays by performinga rigorous study in the $K\rightarrow\pi\mu^+\nu_\mu(\gamma)$ channels. Withappropriate experimental and lattice inputs, we achieve a precision level of$10^{-4}$ in these channels. This is comparable to our previous analysis in the$K\rightarrow\pi e^+\nu_e(\gamma)$ channels. With this new result, we presentan updated global analysis to extract the Standard Model parameter $|V_{us}|$from semileptonic kaon decays. We obtain $|V_{us}|=0.22308(55)$ and$0.22356(73)$, using the lattice average of the $K^0\rightarrow\pi^-$transition form factor at $N_f=2+1+1$ and $N_f=2+1$, respectively.

Complete theory of radiative corrections to $K_{\ell 3}$ decays and the $V_{us}$ update