Model-independent radiative corrections toMl3±decays

The radiative corrections to first order in $\alpha$ to the three-body semileptonic decays of pseudoscalar mesons, $M_{l3}$, are treated in a general fashion, which is suitable for model-independent experimental analysis. Special attention is paid to the energy dependence of the form factors involved. No approximation is made about the smallness of the four-momentum transfer. As an application an expression for the energy spectrum of the charged lepton is derived. The procedure developed in this paper is valid for ${\pi }_{e3}$, $K_{l3}^{\pm }$, $D_{l3}^{\pm }$, or any charged $M_{l3}$ decay, $l$ being an electron or a muon, provided hard bremsstrahlung photons are experimentally discriminated.