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

Experimental measurements of decay branching fractions of semitaunic and semimuonic $B_c$ into $J\mathrm{/}\psi$ has challenged the lepton flavor universality in standard model with about two standard deviations. In this paper, we first investigate the unitary constraint on form factors of $B_c$ meson into $S$-wave and $P$-wave charmonium. Such constraint leads to the exploration of the $R_{J\mathrm{/}\psi }$ and other ratios $R_{{\eta }_c}$, $R_{h_c}$, and $R_{{\chi }_{cJ}}$ in a model-independent way. These results together with future experimental measurements can be used to explore the lepton flavor universality in a more systematic way. In addition, we point out that the helicity-dependent ratios $R_{J\mathrm{/}\psi }^L$ and $R_{J\mathrm{/}\psi }^{\perp }$ can also provide complementary information.