Power-enhanced leading-logarithmic QED corrections to $B_q\to {\mu }^{+}{\mu }^{-}$

We provide a systematic treatment of the previously discovered power-enhanced QED corrections to the leptonic decay $B_q\to {\mu }^{+}{\mu }^{-}$ ($q=d,s$) in the framework of soft-collinear effective theory (SCET). Employing two-step matching on SCET${}_{\text{I}}$ and SCET${}_{\text{II}}$, and the respective renormalization group equations, we sum the leading-logarithmic QED corrections and the mixed QED-QCD corrections to all orders in the couplings for the matrix element of the semileptonic weak effective operator $\sim Q_9$. We propose a treatment of the $B$-meson decay constant and light-cone distribution amplitude in the presence of process-specific QED corrections. Finally we include ultrasoft photon radiation and provide updated values of the non-radiative and radiative branching fractions of $B_q\to {\mu }^{+}{\mu }^{-}$ decay that include the double-logarithmic QED and QCD corrections.