$D$-Wave Charmonia ${\eta }_{c2}$($1{}^1{D}_2$), ${\psi }_2$($1{}^3{D}_2$), and ${\psi }_3$($1{}^3{D}_3$) in $B_c$ Decays

We study the semi-leptonic and non-leptonic decays of $B_c$ meson to $D$-wave charmonia, namely, ${\eta }_{c2}(1^1\text{\hspace{0.17em}⁣}{D}_2)$, ${\psi }_2(1^3\text{\hspace{0.17em}⁣}{D}_2)$, and ${\psi }_3(1^3\text{\hspace{0.17em}⁣}{D}_3)$. 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}$[B_c \rightarrow \psi_3\rho]=4.1^{-0.7}_{+0.8}\times 10^{-4}$.