QED Corrections to $B^-\to\tau^-\bar{\nu}_\tau$
Using a sequence of effective field theories (EFTs), we calculate the rate for the leptonic decay $B^-\to\tau^-\bar{\nu}_\tau(\gamma)$ including real and virtual QED corrections, with a cut $E_{\rm cut}\ll\Lambda_{\rm QCD}$ imposed on electromagnetic radiation in the $B$-meson rest frame. We establish a factorization theorem for the rate and evaluate it at $\mathcal{O}(\alpha)$, resumming the leading logarithmic corrections to all orders in perturbation theory. The large mass $m_\tau$ allows us to treat the tau lepton as a heavy fermion below $\mu\sim m_B\sim m_\tau$, leading to an EFT construction that is structurally different and noticeably simpler than that for the muon case. In particular, hadron-structure dependent QED corrections can be described in terms of QED-induced $B^-\to\tau^-$ form factors, which should be calculable on the lattice. Our analysis includes the leading $\Lambda_{\rm QCD}/m_\tau$ corrections as well as the leading corrections in $E_{\rm cut}/\Lambda_{\rm QCD}$. We show that contributions of the form $\Lambda_{\rm QCD}m_B/m_\tau^2$, which are naively sizable, cancel among each other. Contrary to the muon channel, structure-dependent corrections involving $BB^\ast\gamma$ transitions are numerically negligible for the tau case. The remaining logarithmic dependence on $E_\mathrm{cut}$ is mild. We estimate that the present uncertainty in the calculation of the structure-dependent QED corrections is about 0.5\% of the rate. As a byproduct, we present a state-of-the-art prediction for the lepton flavor universality ratio of the tau and muon channels.