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Experimental measurements of decay branching fractions of semitaunic andsemimuonic $B_c$ into $J/\psi$ has challenged the lepton flavor universality instandard model with about two standard deviations. In this paper, we firstinvestigate the unitary constraint on form factors of $B_c$ meson into $S$-waveand $P$-wave charmonium. Such constraint leads to the exploration of the$R_{J/\psi}$ and other ratios $R_{\eta_c}$, $R_{h_c}$, and $R_{\chi_{cJ}}$ in amodel-independent way. These results together with future experimentalmeasurements can be used to explore the lepton flavor universality in a moresystematic way. In addition, we point out that the helicity-dependent ratios$R^L_{J/\psi}$ and $R^\perp_{J/\psi}$ can also provide complementaryinformation.

Model independent investigation of the $R_{J/\psi,\eta_c}$ and ratios of decay widths of semileptonic $B_c$ decays into a P-wave charmonium