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.