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The exclusive leptonic decay $B \to l \nu$ is arguably the simplest $B$-meson decay, but it becomes surprisingly intricate once QED corrections are included. The latter bring in several distinct scales, and separating point-like from structure-dependent effects requires a sequence of effective theories, ranging from HQET and SCET down to a hadronic description at low energies. In this talk I will sketch the derivation of a QCD$\times$QED factorization theorem for $B \to \mu \nu (\gamma)$, explain the EFT setup used to disentangle virtual and real radiation, and present our phenomenological predictions.