Determining the $\eta-\eta'$ mixing by the newly measured $BR(D(D_s)\to\eta(\eta')+\bar l+\nu_l$

The mixing of $\eta-\eta'$ or $\eta-\eta'-G$ is of a great theoretical interest, because it concerns many aspects of the underlying dynamics and hadronic structure of pseudoscalar mesons and glueball. Determining the mixing parameters by fitting data is by no means trivial. In order to extract the mixing parameters from the available processes where hadrons are involved, theoretical evaluation of hadronic matrix elements is necessary. Therefore model-dependence is somehow unavoidable. In fact, it is impossible to extract the mixing angle from a unique experiment because the model parameters must be obtained by fitting other experiments. Recently $BR(D\to\eta+\bar l+\nu_l)$ and $BR(D_s\to\eta(\eta')+\bar l+\nu_l)$ have been measured, thus we are able to determine the $\eta-\eta'$ mixing solely from the semileptonic decays of D-mesons where contamination from the final state interactions is absent. Thus we hope that the model-dependence of the extraction can be somehow alleviated. Once $BR(D\to\eta'+\bar l+\nu_l)$ is measured, we can further determine all the mixing parameters for $\eta-\eta'-G$. As more data are accumulated, the determination will be more accurate. In this work, we obtain the transition matrix elements of $D_{(s)}\to \eta^{(\prime)}$ using the light-front quark model whose feasibility and reasonability for such processes have been tested.