$\bar{B}_{s,d}^{0} \to J/\psi \mu^{+}\mu^{-}$ Decays in QCD Factorization

Motivated by the first LHCb searches for the rare $\bar{B}_{s,d}^{0} \to J/\psi\mu^{+}\mu^{-}$ decays, we perform a detailed study of these processes within the QCD factorization formalism. Since the transverse size of the $J/\psi$ meson is small in the heavy quark mass limit, this formalism is generally expected to hold for these decays. We include both the leading- and next-to-leading-order QCD corrections to the hard-scattering kernels, which are convoluted with the light-cone distribution amplitudes (LCDAs) of the initial- and final-state hadrons. It is numerically found that, depending on the model parameters for the leading-twist $B$-meson LCDA, the maximum branching ratios of $\bar{B}_{s}^{0}\to J/\psi\mu^{+}\mu^{-}$ and $\bar{B}_{d}^{0} \to J/\psi\mu^{+}\mu^{-}$, integrated over the dimuon invariant mass squared $q^2$ from $1\,\mathrm{GeV}^2$ to $(m_{B_{s,d}}-m_{J/\psi})^2$, can reach up to $2.21\times10^{-9}$ and $7.69\times10^{-11}$ at the leading order in $\alpha_s$, respectively. After incorporating the non-factorizable one-loop vertex corrections, these branching ratios are further reduced by about one order of magnitude, with $\mathcal{B}(\bar{B}_{s}^{0} \to J/\psi\mu^{+}\mu^{-})|_{q^2 \geq 1\,\mathrm{GeV}^2}=2.88\times10^{-10}$ and $\mathcal{B}(\bar{B}_{d}^{0} \to J/\psi\mu^{+}\mu^{-})|_{q^2 \geq 1\,\mathrm{GeV}^2}=1.07\times10^{-11}$. In addition, we have presented the dimuon invariant mass distributions of the individual and total helicity amplitudes squared, as well as the differential and integrated longitudinal polarization fractions of the $J/\psi$ meson, which could be probed by the future LHCb and Belle II experiments with more accumulated data.