Isospin breaking, coupled-channel effects, and X(3872)

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 $3π$ and $2π$ come from a virtual omega and rho meson respectively, we obtain the ratio of these two hidden-charm decay modes: $\mathcal{B}(X(3872)\rightarrow π^+π^-π^0 J/ψ)/\mathcal{B}(X(3872)\rightarrow π^+π^- J/ψ)=0.42$ for the binding energy being 0.3 MeV, which is consistent with the experimental value.

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