We investigate the role played by the three-body DDˉπ dynamics on
the near-threshold resonance X(3872) charmonium state, which is assumed to be
formed by nonperturbative DDˉ∗ 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 DDˉπ intermediate state. In particular, for a resonance
peaked at 0.5 MeV below the D0Dˉ∗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
D0D0ˉπ0 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 DDˉπ channel below DDˉ∗ 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 DDˉ∗ interactions only, consistent with earlier claims.
Coupled-channel effects (inclusion of the charged DDˉ∗ channel) turn
out to have a moderate impact on the results.Comment: 34 pages, 6 figures, version to appear in Phys.Rev.