Systematic analysis of the transition form factors of $B_c$ to $D$-wave charmonia and corresponding semileptonic decays

In this article, the vector, axial vector and tensor form factors of $B_c$ to $D$-wave charmonia ${\psi }_J$ ($J=1,2,3$) and ${\eta }_{c2}$ are analyzed within the framework of three-point QCD sum rules. With the estimated vector and axial vector form factors, we study the decay widths and branching ratios of semileptonic decays $B_c\to {\psi }_{J}l\overline{{\nu }_l}$ and $B_c\to {\eta }_{c2}l\overline{{\nu }_l}$. Our results indicate that the semileptonic decay branching ratios of $B_c$ to $D$-wave charmonia decreases as the total angler momentum of the final state $D$-wave charmonia increases. For decay processes $B_c\to {\psi }_{1}l{\bar{\nu }}_l$, $B_c\to {\psi }_2[{\eta }_{c2}]l{\bar{\nu }}_l$ and $B_c\to {\psi }_{3}l{\bar{\nu }}_l$, the branching ratios can reach the order ${10}^{-3}$, ${10}^{-4}$ and ${10}^{-5}$, respectively, which can provide a valuable reference for future experimental measurements for $B_c$ meson by LHCb collaboration. Furthermore, these results also can provide useful information to study the properties of $B_c$ meson and $D$-wave charmonia.