Semileptonic decays $B_c^+\to D^{(*)}_{(s)}(l^+\nu,l^+l^-,\nu\bar\nu)$ in the perturbative QCD approach

In this paper we study the semileptonic decays of $B_c^+\to D^{(*)}_{(s)}(l^+\nu_l,l^+l^-,\nu\bar\nu)$ (here $l$ stands for $e$, $\mu$, or $\tau$). After evaluating the $B_c^+ \to (D_{(s)},D^*_{(s)})$ transition form factors $F_{0,+,T}(q^2)$ and $V(q^2), A_{0,1,2}(q^2), T_{1,2,3}(q^2)$ by employing the perturbative QCD factorization approach, we calculate the branching ratios for all these semileptonic decays. Our predictions for the values of the $B_c^+ \to D_{(s)}$ and $B_c^+ \to D^*_{(s)}$ transition form factors are consistent with those obtained by using other methods. The branching ratios of the decay modes with $\bar\nu\nu$ are almost an order of magnitude larger than the corresponding decays with $l^+l^-$ after the summation over the three neutrino generations. The branching ratios for the decays with $b\to d$ transitions are much smaller than those decays with the $b\to s$ transitions, due to the Cabibbo-Kobayashi-Maskawa suppression. We define ratios $R_D$ and $R_{D^*}$ for the branching ratios with the $\tau$ lepton versus $\mu$, $e$ lepton final states to cancel the uncertainties of the form factors, which could possibly be tested in the near future.