  We present a non-perturbative calculation of the form factors which
contribute to the amplitudes for the radiative decays $P\to \ell \bar \nu_\ell
\gamma$, where $P$ is a pseudoscalar meson and $\ell$ is a charged lepton.
Together with the non-perturbative determination of the virtual photon
corrections to the processes $P\to \ell \bar \nu_\ell$, this will allow
accurate predictions to be made at $O(\alpha_{em})$ for leptonic decay rates
for pseudoscalar mesons ranging from the pion to the $B$ meson. We are able to
separate unambiguously the point-like contribution, the square of which leads
to the infrared divergence in the decay rate, 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
significantly improved precision in the determination of the corresponding
Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. Precise predictions for the
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.

High Energy Physics - Lattice;High Energy Physics - Phenomenology