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Testing lepton flavor universality in terms of data of BES III and charm-tau factory

The recent measurements on $R_K$ and $R_π$ imply that there exists a possible violation of the leptonic flavor universality which is one of the cornerstones of the standard model. It is suggested that a mixing between sterile and active neutrinos might induce such a violation. In this work we consider the scenarios with one or two sterile neutrinos to explicitly realize the data while the constraints from the available experiments have been taken into account. Moreover, as indicated in literature, the deviation of the real PMNS matrix from the symmetric patterns may be due to a $μ-τ$ asymmetry, therefore the measurements on $R_{D(D_s)eμ} = Γ(D(D_s)\rightarrow e^+ν_e)/Γ(D(D_s)\rightarrow μ^+ν_μ)$ and $R_{D(D_s)μτ} = Γ(D(D_s)\rightarrow μ^+ν_μ)/Γ(D(D_s)\rightarrow μ^+τ_τ)$ (and for some other heavy mesons $B^{\pm}$ and $B_c$ etc.) may shed more light on physics responsible for the violation of the leptonic flavor universality. The data of BES III are available to test the universality and that of the future charm-tau factory will provide more accurate information towards the aspect, in this work, we will discuss $R_{D(D_s)eμ}$ and $R_{D(D_s)μτ}$ in all details and also briefly consider the cases for $B^{\pm}$ and $B_c$.