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

TL;DR

This paper predicts the production rates of the mesons $X(3940)$ and $X(4160)$ in $B_c$ decays using an improved Bethe-Salpeter approach, aiding future experimental observations and understanding of $B_c$ decay mechanisms.

Contribution

It introduces a relativistic Bethe-Salpeter method to calculate decay form factors and branching ratios for $B_c$ decays into $X(3940)$ and $X(4160)$, assuming they are $ ta_c(3S)$ and $ ta_c(4S)$ states.

Findings

Predicted branching ratio for $B_c^+ o X(3940)e^+ u_e$ is $1.0\times10^{-4}$.

Predicted branching ratio for $B_c^+ o X(4160)e^+ u_e$ is $2.4\times10^{-5}$.

Provides a new method for observing $X(3940)$ and $X(4160)$ in future experiments.

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\times10^{-4}$ and $Br(B_c^+\to X(4160)e^+\nu_e)=2.4\times10^{-5}$. 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.