Three-body DDπ dynamics for the X(3872)

We investigate the role played by the three-body DDπ dynamics on the near-threshold resonance X(3872) charmonium state, which is assumed to be formed by nonperturbative DD* dynamics. It is demonstrated that, as compared to the naive static-pions approximation, the imaginary parts that originate from the inclusion of dynamical pions reduce substantially the width from the DDπ intermediate state. In particular, for a resonance peaked at 0.5 MeV below the D0D*0 threshold, this contribution to the width is reduced by about a factor of 2, and the effect of the pion dynamics on the width grows as long as the resonance is shifted towards the D0D0π0 threshold. Although the physical width of the X is dominated by inelastic channels, our finding should still be of importance for the X line shapes in the DDπ channel below DD* threshold. For example, in the scattering length approximation, the imaginary part of the scattering length includes effects of all the pion dynamics and does not only stem from the D* width. Meanwhile, we find that another important quantity for the X phenomenology, the residue at the X pole, is weakly sensitive to dynamical pions. In particular, we find that the binding energy dependence of this quantity from the full calculation is close to that found from a model with pointlike DD* interactions only, consistent with earlier claims. Coupled-channel effects (inclusion of the charged DD* channel) turn out to have a moderate impact on the results.