Unified study of $B_s^0 \to X(3872) \pi^+\pi^- (K^+ K^-)$ and $B_s^0 \to \psi(2S) \pi^+\pi^- (K^+ K^-)$ processes

Abstract

We perform a unified description of the experimental data of the $\pi^+\pi^-$ invariant mass spectra of $B_s^0 \to \psi(2S) \pi^+\pi^-$, the $\pi^+\pi^-$ and $K^+ K^-$ invariant mass spectra of $B_s^0 \to X(3872) \pi^+\pi^- (K^+ K^-)$, and the ratio of branch fractions $\mathcal{B}[B_s^0 \to X(3872)(K^+K^-)_{{\rm non-}\phi}]/ \mathcal{B}[B_s^0 \to X(3872)\pi^+ \pi^-)$. The strong final state interactions between the two pseudoscalars are taken into account using a parametrization fulfilling unitarity and analyticity. We find that there is universality in the coupling constants for $B_s^0 \to \psi(2S) \pi^+\pi^-$ and $B_s^0 \to J/\psi \pi^+\pi^-$ processes. While the couplings of $B_s^0 \to X(3872) \pi^+\pi^-$ are about half of magnitude smaller than the couplings of $B_s^0 \to \psi(2S) \pi^+\pi^-$, which indicates that the $X(3872)$ is different from a pure charmonium state. Furthermore, we find that the $f_0(1500)$ plays an important role in the $B_s^0 \to \psi(2S) \pi^+\pi^-$ and the $B_s^0 \to X(3872) \pi^+\pi^- (K^+ K^-)$ processes, though the phase space of $B_s^0 \to X(3872) f_0(1500)$ is small. Also we predict the ratio of branch fractions $\mathcal{B}[B_s^0 \to \psi(2S)(K^+K^-)_{{\rm non-}\phi}]/ \mathcal{B}[B_s^0 \to \psi(2S)\pi^+ \pi^-]$ and the $K^+ K^-$ invariant mass distribution of $B_s^0 \to \psi(2S)K^+K^-$.