Refactorization of endpoint divergencies for the ${\cal O}_7$ contribution to $\bar B_s \to \mu^+\mu^-$

Abstract

We report on the construction of a factorization theorem for the contribution of the electromagnetic dipole operator ${\cal O}_7$ to the $\bar B_s \to \mu^+\mu^-$ decay amplitude. The leading-order contribution from a QED box diagram 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 analyse 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, the hard-collinear scattering kernel, and the $\bar B_s$-meson distribution amplitude.