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

Abstract

Motivated by the first LHCb searches for the rare $\bar{B}_{s,d}^{0} \to J/ψμ^{+}μ^{-}$ decays, we perform a detailed study of these processes within the QCD factorization formalism. Since the transverse size of the $J/ψ$ 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 the 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 $\bar{B}_{s}^{0}\to J/ψμ^{+}μ^{-}$ and $\bar{B}_{d}^{0} \to J/ψμ^{+}μ^{-}$, integrated over the dimuon invariant mass squared $q^2$ from $1\,\mathrm{GeV}^2$ to $(m_{B_{s,d}}-m_{J/ψ})^2$, can reach, respectively, up to $2.21\times10^{-9}$ and $7.69\times10^{-11}$ at the leading order in $α_s$. After incorporating the non-factorizable one-loop vertex corrections, these branching ratios are further reduced by about one order of magnitude, with $\mathcal{B}(\bar{B}_{s}^{0} \to J/ψμ^{+}μ^{-})|_{q^2 \geq 1\,\mathrm{GeV}^2}=2.88\times10^{-10}$ and $\mathcal{B}(\bar{B}_{d}^{0} \to J/ψμ^{+}μ^{-})|_{q^2 \geq 1\,\mathrm{GeV}^2}=1.07\times10^{-11}$. 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 $J/ψ$ meson, which could be probed by the future LHCb and Belle II experiments with more accumulated data.

Figures (8)

• Figure 1: Leading tree-level Feynman diagram contributing to the rare $\bar{B}_{q}^{0} \to J/\psi \mu^{+} \mu^{-}$ decays within the SM, where the circled cross marks possible insertions of the four-quark operators present in eq. \ref{['eq:Heff']}. Other diagrams with the virtual photon emitted from the remaining three quark lines are further suppressed by $\Lambda_{\rm QCD}/m_b$, and will be neglected throughout this paper.
• Figure 2: Non-factorizable one-loop vertex corrections to the rare $\bar{B}_{q}^{0} \to J/\psi \mu^{+} \mu^{-}$ decays within the SM. The other captions are the same as in figure \ref{['fig:LO_diags']}.
• Figure 3: The $q^2$ dependence of the differential decay branching ratio for $\bar{B}_{s}^{0} \to J/\psi \mu^{+}\mu^{-}$ at the LO (upper panel) and NLO (lower panel) in $\alpha_s$, for the three distinct models listed in table \ref{['tab:model_parameters']} for the leading-twist $B$-meson LCDA. The bands result from the variation of the parameter $\hat{\sigma}_{1}$ within the ranges specified by the last column in table \ref{['tab:model_parameters']}.
• Figure 4: The $q^2$ dependence of the differential decay branching ratio for $\bar{B}_{d}^{0} \to J/\psi \mu^{+}\mu^{-}$ at the LO (upper panel) and NLO (lower panel) in $\alpha_s$. The other captions are the same as in figure \ref{['benekemodelsBsBR']}.
• Figure 5: The differential branching ratios $d\mathcal{B}_{|H_{0,-,+}|^{2}}/dq^{2}$ from three individual helicity contributions as a function of $q^2$ for $\bar{B}_{s}^{0} \to J/\psi \mu^{+}\mu^{-}$ (left) and $\bar{B}_{d}^{0}\to J/\psi \mu^{+}\mu^{-}$ (right) decays at the NLO in $\alpha_s$, taking the simple exponential model for the leading-twist $B$-meson LCDA as a representative example.