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Using Line Shapes to Discriminate between Binding Mechanisms for the X(3872)

We construct line shapes for the X(3872) that generalize the Flatte and zero-range line shapes that have been considered previously. These line shapes are associated with scattering amplitudes that are exactly unitary for real values of the interaction parameters and can be derived from a renormalizable quantum field theory. The new line shapes can be used to discriminate between the alternative binding mechanisms in which the X(3872) is generated either dynamically by charm meson interactions or by a resonance near the D^{*0} Dbar^0 threshold. If the resonance is identified with the P-wave charmonium state chi'_{c1}, the interaction parameters can be constrained by using charmonium phenomenology. We analyze data on the X(3872) and also data from the Belle and Babar Collaborations on the invariant mass distribution of the charm mesons from the decay B -> K + D^{*0} Dbar^0 up to 4000 MeV. Our analysis is compatible with the mechanism for the X(3872) being either a fine-tuning of charm meson interactions or the fine-tuning of the chi'_{c1} to the D^{*0} Dbar^0 threshold. In particular, the data do not exclude a separate chi'_{c1} resonance between the D^{*+} D^- threshold and 4000 MeV.