[en] We construct line shapes for the X(3872), which 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*⁰D⁰ threshold. If the resonance is identified with the P-wave charmonium state χ_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*⁰D⁰ 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 χ_c1^' to the D*⁰D⁰ threshold. In particular, the data do not exclude a separate χ_c1^' resonance between the D*⁺D^- threshold and 4000 MeV.