Radiative corrections to $K_{\ell 3}$ decays

Abstract. We present a complete calculation of the $K_{\ell 3}$ decays $K^+ \to \pi^0 \ell^+ \nu_{\ell}$ and $K^0 \to \pi^- \ell^+ \nu_\ell$ to $\mathcal O(p^4, (m_d-m_u) p^2, e^2 p^2)$ in chiral perturbation theory with virtual photons and leptons. We introduce the concept of generalized form factors and kinematical densities in the presence of electromagnetism, and propose a possible treatment of the real photon emission in $K^+_{\ell 3}$ decays. We illustrate our results by applying them to the extraction of the Kobayashi–Maskawa matrix element $|V_{us}|$ from the experimental $K^+_{e3}$ decay parameters.