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$η_c$ mixing effects on charmonium and $B$ meson decays

We include the $η_c$ meson into the $η$-$η'$-$G$ 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 $η_c$ and $G$ is found to be about $11^\circ$ from the measured decay widths of the $η_c$ 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 $η$-$η'$-$G$ mixing is confirmed, while $m_{qq}$ grows to above the pion mass, and thus increases perturbative QCD predictions for the branching ratios $Br(B\toη'K)$. We then analyze the $η_c$-mixing effects on charmonium magnetic dipole transitions, and on the $B\toη^{(\prime)}K_S$ branching ratios and CP asymmetries, which further improve the consistency between theoretical predictions and data. A predominant observation is that the $η_c$ mixing enhances the perturbative QCD predictions for $Br(B\toη'K)$ by 18%, but does not alter those for $Br(B\toηK)$. The puzzle due to the large $Br(B\toη'K)$ data is then resolved.