We present a non-perturbative lattice calculation of the form factors which contribute to the amplitudes for the radiative decays $Pto ell bar u_ell gamma$, where $P$ is a pseudoscalar meson and $ell$ is a charged lepton. Together with the non-perturbative determination of the corrections to the processes $Pto ell bar u_ell$ due to the exchange of a virtual photon, this allows accurate predictions at $O(alpha_{em})$ to be made for leptonic decay rates for pseudoscalar mesons ranging from the pion to the $D_s$ meson. We are able to separate unambiguously and non-pertubatively the point-like contribution, from the structure-dependent, infrared-safe, terms in the amplitude. The fully non-perturbative $O(a)$ improved calculation of the inclusive leptonic decay rates will lead to the determination of the corresponding Cabibbo-Kobayashi-Maskawa (CKM) matrix elements also at $O(alpha_{em})$. Prospects for a precise evaluation of leptonic decay rates with emission of a hard photon are also very interesting, especially for the decays of heavy $D$ and $B$ mesons for which currently only model-dependent predictions are available to compare with existing experimental data.
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