Analysis of Bₛ⁰ → X(3872) [ψ(2S)] π⁺π⁻ (K⁺ K⁻) decays

We have phenomenologically investigated the decays Bₛ⁰ → X(3872) π⁺π⁻ (K⁺ K⁻) and Bₛ⁰ → ψ(2S) π⁺ π⁻ (K⁺K⁻). In our analysis, the scalar meson f₀(980) is formed through the final state interactions of coupled channels π π and KK̄. Our findings indicate that the π⁺π⁻ invariant mass distribution of the Bₛ⁰ → ψ(2S)π⁺π⁻ decay can be accurately reproduced. Furthermore, we have explored the π⁺π⁻ (K⁺ K⁻) invariant mass distribution of the Bₛ⁰ → X(3872) π⁺π⁻ (K⁺ K⁻) decay, accounting for the different production mechanisms between $X(3872)$ and ψ(2S), up to a global factor. It is found that the production rates for $X(3872)$ and ψ(2S) are much different, which indicates that the structure of $X(3872)$ is more complicated than the ψ(2S), which is a conventional cc̄ state. Additionally, we have considered the contributions from f₀(1500) to π⁺π⁻ and the φ meson to K⁺ K⁻ in our analysis. Utilizing the model parameters, we have calculated the branching fraction of Bₛ⁰ → X(3872) K⁺ K⁻, and anticipate that the findings of our study can be experimentally tested in the future.