On the contribution of the electromagnetic dipole operator ${\cal O}_7$ to the $\bar B_s \to \mu^+\mu^-$ decay amplitude

Abstract

We construct a factorization theorem that allows to systematically include QCD corrections to the contribution of the electromagnetic dipole operator in the effective weak Hamiltonian to the $\bar B_s \to \mu^+\mu^-$ decay amplitude. We first rederive the known result for the leading-order QED box diagram, which features a double-logarithmic enhancement associated to the different rapidities of the light quark in the $\bar B_s$ meson and the energetic muons in the final state. We provide a detailed analysis of the cancellation of the related endpoint divergences appearing in individual momentum regions, and show how the rapidity logarithms can be isolated by suitable subtractions applied to the corresponding bare factorization theorem. This allows us to include in a straightforward manner the QCD corrections arising from the renormalization-group running of the hard matching coefficient of the electromagnetic dipole operator in soft-collinear effective theory, the hard-collinear scattering kernel, and the $B_s$-meson distribution amplitude. Focusing on the contribution from the double endpoint logarithms, we derive a compact formula that resums the leading-logarithmic QCD corrections.