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Radiative $μ$ and $τ$ leptonic decays at NLO

We present the differential rates and branching ratios of the radiative decays $τ\to l \barν νγ$, with $l=e$ or $μ$, and $μ\to e \barν νγ$ in the Standard Model at next-to-leading order. Radiative corrections are computed taking into account the full depencence on the mass $m_l$ of the final charged leptons, which is necessary for the correct determination of the branching ratios. Only partial agreement is found with previous calculations performed in the $m_l \to 0$ limit. Our results agree with the measurements of the branching ratios $\mathcal{B} (μ\to e \barν νγ)$ and $\mathcal{B} (τ\to μ\barν νγ)$ for a minimum photon energy of 10 MeV in the $μ$ and $τ$ rest frames, respectively. Babar's recent precise measurement of the branching ratio $\mathcal{B} (τ\to e \barν νγ)$, for the same photon energy threshold, differs from our prediction by 3.5 standard deviations.