Radiative $\mu$ and $\tau$ leptonic decays at NLO

We present the differential rates and branching ratios of the radiative decays $\tau \to l \bar{\nu} \nu \gamma$, with $l=e$ or $\mu$, and $\mu \to e \bar{\nu} \nu \gamma$ 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 $\mathcal{B} (\mu \to e \bar{\nu} \nu \gamma)$ and $\mathcal{B} (\tau \to \mu \bar{\nu} \nu \gamma)$ for a minimum photon energy of 10 MeV in the $\mu$ and $\tau$ rest frames, respectively. Babar's recent precise measurement of the branching ratio $\mathcal{B} (\tau \to e \bar{\nu} \nu \gamma)$, for the same photon energy threshold, differs from our prediction by 3.5 standard deviations.