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

Abstract

We fill up the missing piece in our own re-analysis of the long-distance electromagnetic radiative corrections to semileptonic kaon decays by performing a rigorous study in the $K\to \pi {\mu }^{+}{\nu }_{\mu }(\gamma )$ channels. With appropriate experimental and lattice inputs, we achieve a precision level of $10^{-4}$ in these channels. This is comparable to our previous analysis in the $K\to \pi e^{+}{\nu }_e(\gamma )$ channels. With this new result, we present an 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 $K^0\to {\pi }^{-}$ transition form factor at $N_f=2+1+1$ and $N_f=2+1$, respectively.

Version accepted by JHEP, including a slight change of the title

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