Contribution of the electromagnetic dipole operator ${\mathcal{O}}_7$ to the ${\overline{B}}_s\to {\mu }^{+}{\mu }^{-}$ decay amplitude

We construct a factorization theorem that allows one to systematically include QCD corrections to the contribution of the electromagnetic dipole operator in the effective weak Hamiltonian to the ${\overline{B}}_s\to {\mu }^{+}{\mu }^{-}$ decay amplitude. We first rederive the known result for the leading-order QED box diagram, which features a double-logarithmic enhancement associated with the different rapidities of the light quark in the ${\overline{B}}_s$ 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.

Figure 1

QED box diagram describing the leading contribution of the operator ${\mathcal{O}}_7$ (indicated by the black square) to the ${\overline{B}}_s\to {\mu }^{+}{\mu }^{-}$ decay amplitude. The analogous diagram with the role of ${\mu }^{+}$ and ${\mu }^{-}$ interchanged gives the same result and is not shown.

Figure 2

Illustration of short-distance QCD corrections to the anti-soft-collinear region. Left: hard correction to $b\to s{\gamma }^{*}$ vertex. Right: anti-hard-collinear corrections to spectator scattering.

Figure 3

For illustration of the RG effect, we plot the prefactors in front of the expansion coefficients $a_k$ defined in Eq. (70) as a function of the auxiliary scale ${\omega }_0$ and normalized to the case $g=V=0$, which we call ${\widehat{f}}_k({\omega }_0)\equiv \frac{f_k({\omega }_0)}{f_k({\omega }_0){|}_{g=V=0}}$, for $k=0$, 1, 2.