We include the $\eta_c$ meson into the $\eta$-$\eta'$-$G$ mixing formalism constructed in our previous work, where $G$ represents the pseudoscalar gluball. The mixing angles in this tetramixing matrix are constrained by theoretical and experimental implications from relevant hadronic processes. Especially, the angle between $\eta_c$ and $G$ is found to be about $11^\circ$ from the measured decay widths of the $\eta_c$ meson. The pseudoscalar glueball mass $m_G$, the pseudoscalar densities $m_{ qq,ss,cc}$ and the U(1) anomaly matrix elements associated with the mixed states are solved from the anomalous Ward identities. The solution $m_G\approx 1.4$ GeV obtained from the $\eta$-$\eta'$-$G$ mixing is confirmed, while $m_{ qq}$ grows to above the pion mass, and thus increases perturbative QCD predictions for the branching ratios $Br(B\to\eta'K)$. We then analyze the $\eta_c$-mixing effects on charmonium magnetic dipole transitions, and on the $B\to\eta^{ (\prime)}K_S$ branching ratios and CP asymmetries, which further improve the consistency between theoretical predictions and data. A predominant observation is that the $\eta_c$ mixing enhances the perturbative QCD predictions for $Br(B\to\eta'K)$ by 18%, but does not alter those for $Br(B\to\eta K)$. The puzzle due to the large $Br(B\to\eta'K)$ data is then resolved.
Categories: hep-ph hep-ex