The Production of $X(3940)$ and $X(4160)$ in $B_c$ decays

Abstract

Considering $X(3940)$ and $X(4160)$ as ${\eta }_c(3S)$ and ${\eta }_c(4S)$, we study the productions of $X(3940)$ and $X(4160)$ in exclusive weak decays of $B_c$ meson by the improved Bethe-Salpeter(B-S) Method. Using the relativistic B-S equation and Mandelstam formalism, we calculate the corresponding decay form factors. The predictions of the corresponding branching ratios are: $Br(B_c^{+}\to X(3940)e^{+}{\nu }_e)$$=1.0\times 10^{-4}$ and $Br(B_c^{+}\to X(4160)e^{+}{\nu }_e)=2.4\times 10^{-5}$. That will provide us a new way to observe the $X(3940)$ and $X(4160)$ in the future, as well as to improve the knowledge of $B_c$ meson decay.

15 pages, 7 figures