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Exclusive rare $B_s\to (K,η,η')\ell^+\ell^-$ decays in the light-front quark model

Using the light-front quark model, we calculate the transition form factors, decay rates, and longitudinal lepton polarization asymmetries for the exclusive rare $B_s\to (K,η^{(\prime)})(\ell^+\ell^-,ν_{\ell}\bar{ν_{\ell}}$ ($\ell=e,μ,τ$) decays within the standard model, taking into account the $η-η'$ mixing angle. For the mixing angle $θ=-20^{\circ}$ ($-10^{\circ}$) in the octet-singlet basis, we obtain ${\rm BR}(B_s\to η\sumν_{\ell}\barν_{\ell})=1.1 (1.7)\times 10^{-6}$, ${\rm BR}(B_s\to ημ^+μ^-)=1.5 (2.4)\times 10^{-7}$, ${\rm BR}(B_s\to ητ^+τ^-)=3.8 (5.8)\times 10^{-8}$, ${\rm BR}(B_s\to η'\sumν_{\ell}\barν_{\ell})=1.8 (1.3)\times 10^{-6}$, ${\rm BR}(B_s\to η'μ^+μ^-)=2.4 (1.8)\times 10^{-7}$, and ${\rm BR}(B_s\to η'τ^+τ^-)=3.4 (2.6)\times 10^{-8}$, respectively. The branching ratios for the $B_s\to K(ν_{\ell}\bar{ν_{\ell}},\ell^+\ell^-)$ decays are at least an order of magnitude smaller than those for the $B_s\to η^{(\prime)}(ν_{\ell}\bar{ν_{\ell}},\ell^+\ell^-)$ decays. The averaged values of the lepton polarization asymmetries for $B_s\to (K,η^{(\prime)})\ell^+\ell^-$ are obtained as $\la P^K_L\ra_μ=\la P^η_L\ra_μ=\la P^{η'}_L\ra_μ=-0.98$, $\la P^K_L\ra_τ=-0.24$, $\la P^η_L\ra_τ=-0.20$ and $\la P^{η'}_L\ra_τ=-0.14$, respectively.