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Considering $Z(3930)$ and $X(4160)$ as $\chi_{c2}(2P)$ and $\chi_{c2}(3P)$states, the semileptonic and nonleptonic of $B_c$ decays to $Z(3930)$ and$X(4160)$ are studied by the improved Bethe-Salpeter(B-S) Method. The formfactors of decay are calculated through the overlap integrals of the meson wavefunctions in the whole accessible kinematical range. The influence ofrelativistic corrections are considered in the exclusive decays. Branchingratios of $B_c$ weak decays to $Z(3930)$ and $X(4160)$ are predicted. Some ofthe branching ratios are: $Br(B_c^+\toZ(3930)e^+\nu_e)$$=(3.03^{+0.09}_{-0.16})\times 10^{-4}$ and $Br(B_c^+\toX(4160)e^+\nu_e)$$=(3.55^{+0.83}_{-0.35})\times 10^{-6}$. These results mayprovide useful information to discover $Z(3930)$ and $X(4160)$ and thenecessary information for the phenomenological study of $B_c$ physics.

The weak decay $B_c$ to $Z(3930)$ and $X(4160)$ by Bethe-Salpeter method