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Motivated by a dynamical enhancement of the electromagnetic corrections by apower of $\Lambda_{\mathrm{QCD}}/m_b$ in $B_{d,s}\, \to\, \mu^+\, \mu^-$ atnext-to-leading order (NLO), we extend the QED factorization effects on theleptonic $B$ meson decays with light muon leptons to tauonic final states,$B_{d,s} \, \to\, \tau^+\, \tau^-$, using soft-collinear effective theory(SCET). This extension is necessary owing to the appearance of the large $\tau$mass, which will lead to different power counting in SCET and also differentresults. We provide a complete NLO electromagnetic corrections to $B_{d,s} \,\to\, \tau^+\, \tau^-$, which include hard functions and hard-collinearfunctions below the bottom quark mass scale $\mu_b$. The power enhancedelectromagnetic effects from hard-collinear contributions on $B_{d,s}\, \to\,\mu^+\, \mu^-$ discussed before also exist in $B_{d,s} \, \to\, \tau^+\,\tau^-$. However the logarithm term arising from contributions ofhard-collinear photon and lepton virtualities for $B_{d,s}\, \to\, \tau^+\,\tau^-$ is not large as it is in muon case due to the hard-collinear scale of$\tau$ mass, which lead to only approximately $0.04\%$ QED corrections to thebranching fraction of $B_{d,s} \, \to\, \tau^+\, \tau^-$ compared with overallreduction about $0.5\%$ in $B_{d,s}\, \to\, \mu^+\, \mu^-$.

Complete analysis on QED corrections to $B_{q}\, \to\, \tau^+\, \tau^-$