Further Analysis of Di-gluon Fusion Mechanism for the decays of $B \to K \eta^{\prime}$

Di-gluon fusion mechanism might account for the large branching ratios of $B\to K \eta^{\prime}$. But because we know little about the effective $\eta^{\prime}gg$ vertex, there are large uncertainties in perturbative QCD estimations. In this paper, we try several kinds of $\eta^{\prime}gg$ form factors and compare the numerical results with the experiment. We find that, though we know little about $\eta^{\prime}gg$ form factor, if di-gluon fusion mechanism is important in $B\to K \eta^{\prime}$, the branching ratios of the decays $B\to K^{\ast} \eta^{\prime}$ would be around $10^{-5}$ which can be tested by future experiments.