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Semileptonic decays of $B_c$ meson to S-wave charmonium states in the perturbative QCD approach

Inspired by the recent measurement of the ratio of $B_c$ branching fractions to $J/ψπ^+$ and $J/ψμ^+ν_μ$ final states at the LHCb detector, we study the semileptonic decays of $B_c$ meson to the S-wave ground and radially excited 2S and 3S charmonium states with the perturbative QCD approach. After evaluating the form factors for the transitions $B_c\rightarrow P,V$, where $P$ and $V$ denote pseudoscalar and vector S-wave charmonia, respectively, we calculate the branching ratios for all these semileptonic decays. The theoretical uncertainty of hadronic input parameters are reduced by utilizing the light-cone wave function for $B_c$ meson. It is found that the predicted branching ratios range from $10^{-6}$ up to $10^{-2}$ and could be measured by the future LHCb experiment. Our prediction for the ratio of branching fractions $\frac{\mathcal {BR}(B_c^+\rightarrow J/Ψπ^+)}{\mathcal {BR}(B_c^+\rightarrow J/Ψμ^+ν_μ)}$ is in good agreement with the data. For $B_c\rightarrow V l ν_l$ decays, the relative contributions of the longitudinal and transverse polarization are discussed in different momentum transfer squared regions. These predictions will be tested on the ongoing and forthcoming experiments.