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

Yun-Hua Chen

TL;DR

The paper addresses how to coherently describe the decays $B_s^0 \to X(3872)\pi\pi$ and $B_s^0 \to \psi(2S)\pi\pi$ (and related $K^+K^-$ modes) within a single framework. It combines a chiral-heavy-quark effective description of the $B_s^0\psi(2S)\!\!\!$ and $B_s^0 X(3872)$ couplings with a dispersive, unitary three-channel final-state-interaction formalism to account for strong $S$-wave rescattering among $\pi\pi$, $K\bar K$, and an effective $4\pi$ channel, ensuring compatibility with low-energy chiral constraints and high-energy analyticity. The analysis extracts couplings for the $B_s^0\psi(2S) PP$ and $B_s^0 X(3872) PP$ vertices, finds universality of the charmonium-production couplings across $B_s^0$ decays, and reveals that the $X(3872)$ couplings are substantially smaller than those for $\psi(2S)$, supporting a non-pure charmonium interpretation. It also identifies a non-negligible role for $f_0(1500)$ in these processes and makes predictions for the ratio and line-shape of the $\psi(2S)(K^+K^-)_{\mathrm{non-}\phi}$ channel, providing experimentally testable consequences for the internal structure of $X(3872)$ and the dynamics of heavy-hadron decays.

Abstract

We perform a unified description of the experimental data of the $π^+π^-$ invariant mass spectra of $B_s^0 \to ψ(2S) π^+π^-$, the $π^+π^-$ and $K^+ K^-$ invariant mass spectra of $B_s^0 \to X(3872) π^+π^- (K^+ K^-)$, and the ratio of branch fractions $\mathcal{B}[B_s^0 \to X(3872)(K^+K^-)_{{\rm non-}φ}]/ \mathcal{B}[B_s^0 \to X(3872)π^+ π^-)$. 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 $B_s^0 \to ψ(2S) π^+π^-$ and $B_s^0 \to J/ψπ^+π^-$ processes. While the couplings of $B_s^0 \to X(3872) π^+π^-$ are about half of magnitude smaller than the couplings of $B_s^0 \to ψ(2S) π^+π^-$, 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 $B_s^0 \to ψ(2S) π^+π^-$ and the $B_s^0 \to X(3872) π^+π^- (K^+ K^-)$ 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 $\mathcal{B}[B_s^0 \to ψ(2S)(K^+K^-)_{{\rm non-}φ}]/ \mathcal{B}[B_s^0 \to ψ(2S)π^+ π^-]$ and the $K^+ K^-$ invariant mass distribution of $B_s^0 \to ψ(2S)(K^+K^-)_{{\rm non-}φ}$.

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

TL;DR

The paper addresses how to coherently describe the decays and (and related modes) within a single framework. It combines a chiral-heavy-quark effective description of the and couplings with a dispersive, unitary three-channel final-state-interaction formalism to account for strong -wave rescattering among , , and an effective channel, ensuring compatibility with low-energy chiral constraints and high-energy analyticity. The analysis extracts couplings for the and vertices, finds universality of the charmonium-production couplings across decays, and reveals that the couplings are substantially smaller than those for , supporting a non-pure charmonium interpretation. It also identifies a non-negligible role for in these processes and makes predictions for the ratio and line-shape of the channel, providing experimentally testable consequences for the internal structure of and the dynamics of heavy-hadron decays.

Abstract

We perform a unified description of the experimental data of the invariant mass spectra of , the and invariant mass spectra of , and the ratio of branch fractions . 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 and processes. While the couplings of are about half of magnitude smaller than the couplings of , which indicates that the is different from a pure charmonium state. Furthermore, we find that the plays an important role in the and the processes, though the phase space of is small. Also we predict the ratio of branch fractions and the invariant mass distribution of .

Paper Structure

This paper contains 8 sections, 29 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical framework
  3. Lagrangians of contact $B_s^0 \psi(2S) \pi^+\pi^- (K^+ K^-)$ couplings
  4. Lagrangians of contact $B_s^0 X(3872) \pi^+\pi^- (K^+ K^-)$ couplings
  5. Amplitudes of $B_s^0 \to \psi(2S)[X(3872)] \pi^+\pi^- (K^+ K^-)$ processes
  6. Treatment of final-state interactions
  7. Phenomenological discussion
  8. Conclusions

Figures (3)

  • Figure 1: The $B_s^0 \to \psi(2S) \pi^+\pi^- (K^+ K^-)$ diagram to leading order via $W^+$ exchange.
  • Figure 2: Fit results of the $\pi\pi$ invariant mass distributions of $B_s^0 \to \psi(2S) \pi^+\pi^-$ (left) and $B_s^0 \to X(3872) \pi^+\pi^-$ processes (middle), and the $K\bar{K}$ invariant mass distributions of $B_s^0 \to X(3872) K^+ K^-$ process (right) for Fits I (top) and II (bottom). The red solid lines represent the best fit results, while the blue dotted lines correspond to the contributions from the $f_0(980)$ only.
  • Figure 3: Theoretical prediction of the $K\bar{K}$ invariant mass distributions of $B_s^0 \to \psi(2S)(K^+K^-)_{{\rm non-}\phi}$ process for Fits I (left) and II (right). The red solid lines represent the best fit results, while the blue dotted lines correspond to the contributions from the $f_0(980)$ only.