Contribution of $b\to sgg$ through the QCD anomaly in exclusive decays $B^{\pm }\to ({\eta }^{\mathrm{\prime }},\eta )(K^{\pm },K^{\ast \pm })$ and $B^0\to ({\eta }^{\mathrm{\prime }},\eta )(K^0,K^{\ast 0})$

We compute the decay rates for the exclusive decays $B^{\pm} \to (\eta^{\prime},\eta) (K^{\pm}, K^{*\pm})$and$B^{0}\to (\eta^{\prime},\eta) (K^{0}, K^{*0})$ in a QCD-improved factorization framework by including the contribution from the process $b\to sgg\to s({\eta }^{\mathrm{\prime }},\eta )$ through the QCD anomaly. This method provides an alternative estimate of the contribution $b\to sc\bar{c}\to s(\eta ,{\eta }^{\mathrm{\prime }})$ to these decays as compared to the one using the intrinsic charm content of the ${\eta }^{\mathrm{\prime }}$ and $\eta$ mesons determined through the decays $J/\psi \to (\eta,\eta^\prime ,\eta_c) \gamma$. The resulting branching ratios are compared with the CLEO data on $B^{\pm }\to {\eta }^{\mathrm{\prime }}{K}^{\pm }$ and $B^0\to {\eta }^{\mathrm{\prime }}{K}^0$ and predictions are made for the rest.