D-wave charmonia $$\eta _{c2}(1^1\!D_2)$$ η c 2 ( 1 1 D 2 ) , $$\psi _2(1^3\!D_2)$$ ψ 2 ( 1 3 D 2 ) , and $$\psi _3(1^3\!D_3)$$ ψ 3 ( 1 3 D 3 ) in $$B_c$$ B c decays

Abstract

We study the semi-leptonic and non-leptonic decays of $B_c$ meson to $D$-wave charmonia, namely, $\eta_{c2}(1^1\!D_2)$, $\psi_2(1^3\!D_2)$, and $\psi_3(1^3\!D_3)$. In our calculations, the instantaneous Bethe-Salpeter method is applied to achieve the hadronic matrix elements. This method includes relativistic corrections which are important especially for the higher orbital excited states. For the semi-leptonic decay channels with electron as the final lepton, we get the branching ratios $\mathcal{B}[B_c \rightarrow \eta_{c2}e\bar{\nu}_e] = 5.9^{-0.8}_{+1.0}\times 10^{-4}$, $\mathcal{B}[B_c \rightarrow \psi_2e\bar{\nu}_e]=1.5^{-0.2}_{+0.3}\times 10^{-4}$, and $\mathcal{B}[B_c \rightarrow \psi_3e\bar{\nu}_e]=3.5^{-0.6}_{+0.8}\times 10^{-4}$. The transition form factors, forward-backward asymmetries, and lepton spectra in these processes are also presented. For the non-leptonic decay channels, those with $\rho$ as the lighter meson have the largest branching ratios, $\mathcal{B}[B_c \rightarrow \eta_{c2}\rho] = 8.1^{-1.0}_{+1.0}\times 10^{-4}$, $\mathcal{B}[B_c \rightarrow \psi_2\rho]=9.6^{-1.0}_{+1.0}\times 10^{-5}$, and $\mathcal{B}[B_c \rightarrow \psi_3\rho]=4.1^{-0.7}_{+0.8}\times 10^{-4}$.