Identification of non-ordinary mesons from the dispersive connection between their poles and their Regge trajectories: The f_0(500) resonance

We show how the Regge trajectory of a resonance can be obtained from its pole in a scattering process and analytic constraints in the complex angular momentum plane. The method is suited for resonances that dominate an elastic scattering amplitude. In particular, from the ρ(770) resonance pole in ππ scattering, we obtain its linear Regge trajectory, characteristic of ordinary quark–antiquark states. In contrast, the f_0(500) pole—the sigma meson—which dominates scalar isoscalar ππ scattering, yields a nonlinear trajectory with a much smaller slope at the f0(500) mass. Conversely, imposing a linear Regge trajectory for the f0(500), with a slope of typical size, yields an elastic amplitude at odds with the data. This provides strong support for the non-ordinary nature of the sigma meson