[Paper Review] Isospin breaking, Coupled-channel effects and Diagnosis of X(3872)

Ning Li, Shi-Lin Zhu|arXiv (Cornell University)|Jul 17, 2012

Quantum Chromodynamics and Particle Interactions4 citations

TL;DR

This paper proposes that the X(3872) state arises as a loosely bound $D\bar{D}^*$ molecule with $J^{PC}=1^{++}$, incorporating isospin breaking via $D$-$D^*$ mass splitting and $D\bar{D}^*$ coupled-channel effects. Using both one-pion-exchange (OPE) and one-boson-exchange (OBE) models with S-D wave mixing, it predicts a 26.24% isovector component at 0.30 MeV binding energy and a $\pi^+\pi^-\pi^0 J/\psi / \pi^+\pi^- J/\psi$ decay ratio of 0.42, matching experimental data.

ABSTRACT

We re-investigate the possibility of X(3872) as a $D\bar{D}^*$ molecule with $J^{PC}=1^{++}$ within the framework of both the one-pion-exchange (OPE) model and the one-boson-exchange (OBE) model. After careful treatment of the S-D wave mixing, the mass difference between the neutral and charged $D(D^*)$ mesons and the coupling of the $D(D^*)$ pair to $D^*\bar{D}^*$, a loosely bound molecular state X(3872) emerges quite naturally with large isospin violation in its flavor wave function. For example, the isovector component is 26.24% if the binding energy is 0.30 MeV, where the isospin breaking effect is amplified by the tiny binding energy. After taking into account the phase space difference and assuming the $3\pi$ and $2\pi$ come from a virtual omega and rho meson respectively, we obtain the ratio of these two hidden-charm decay modes: $\mathcal{B}(X(3872) ightarrow \pi^+\pi^-\pi^0 J/\psi)/\mathcal{B}(X(3872) ightarrow \pi^+\pi^- J/\psi)=0.42$ for the binding energy being 0.3 MeV, which is consistent with the experimental value.

Motivation & Objective

• To investigate whether X(3872) can be explained as a $D\bar{D}^*$ molecular state with $J^{PC}=1^{++}$.
• To examine the role of isospin breaking due to the mass difference between neutral and charged $D(D^*)$ mesons.
• To assess the impact of $D\bar{D}^*$ coupled-channel effects on the molecular state formation.
• To predict the branching ratio of hidden-charm decay modes $\pi^+\pi^-\pi^0 J/\psi$ and $\pi^+\pi^- J/\psi$ and compare with experiment.

Proposed method

• Employing the one-pion-exchange (OPE) model to describe the $D\bar{D}^*$ interaction with isospin violation.
• Applying the one-boson-exchange (OBE) model as an alternative framework for the strong interaction dynamics.
• Including S-D wave mixing effects in the $D\bar{D}^*$ system to account for tensor forces.
• Incorporating the mass splitting between neutral and charged $D(D^*)$ mesons to model isospin breaking.
• Including the $D^*\bar{D}^*$ channel coupling to simulate in-medium effects and enhance molecular binding.
• Calculating decay branching ratios by modeling $3\pi$ and $2\pi$ final states via virtual $\omega$ and $\rho$ mesons, respectively.

Experimental results

Research questions

• RQ1 Can the X(3872) state be consistently described as a $D\bar{D}^*$ molecule with $J^{PC}=1^{++}$ under isospin-breaking effects?
• RQ2 How does the $D$-$D^*$ mass splitting influence the flavor structure of the X(3872) wave function?
• RQ3 To what extent do $D\bar{D}^*$ coupled-channel effects contribute to the formation of a loosely bound molecular state?
• RQ4 What is the predicted ratio of $\mathcal{B}(X(3872)\to \pi^+\pi^-\pi^0 J/\psi)$ to $\mathcal{B}(X(3872)\to \pi^+\pi^- J/\psi)$, and how does it compare to experiment?

Key findings

• A loosely bound $D\bar{D}^*$ molecular state with $J^{PC}=1^{++}$ emerges naturally when isospin breaking and coupled-channel effects are included.
• The isovector component of the X(3872) wave function reaches 26.24% at a binding energy of 0.30 MeV, indicating significant isospin violation.
• The isospin-breaking effect is amplified by the small binding energy, enhancing the mixing of isospin components.
• The predicted ratio of hidden-charm decay branching fractions is $\mathcal{B}(X(3872)\to \pi^+\pi^-\pi^0 J/\psi)/\mathcal{B}(X(3872)\to \pi^+\pi^- J/\psi) = 0.42$ for a 0.30 MeV binding energy.
• This predicted ratio is in good agreement with the experimental value, supporting the molecular interpretation of X(3872).