Radiative corrections to \(K_{\ell 3}\) decays

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.