Fraction of the gluonium component in $η'$ and $η$

It is interesting to determine the fraction of the gluonium component in $η$ and $η'$ which has been under serious discussion for many years. Measurements on different decay and/or production modes were employed in literatures, thus larger uncertainties were unavoidable. In this paper we suggest to determine the mixing angles of $η-η'-G$ using the data of semileptonic decays of $D$ and $D_s$. We extract the mixing angles $ϕ'$, $ϕ_G$ and the model parameters simultaneously. Thanks to the new measurements carried out by CLEO Collaboration and there are sufficient decay modes to determine both the model parameters and mixing angles. The mixing angles from data are $ϕ'=(41.5\pm2.0)^\circ$ and $\sinϕ_G=0.00\pm0.36$. Even though the central value of $\sinϕ_G$ is still zero, an upper bound is set. Moreover, as suggested in literature, $η(1405)$ is a glueball candidate whereas in our picture, $η(1405)$ may be identified as $G$ with glueonium being its main content. Using all the model-parameters obtained above, we estimate the branching ratios of $D_s^+(D^+)\rightarrow Ge^+ν_e$ where G is identified as $η(1405)$.

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