Dispersive calculation of complex Regge trajectories for the lightest f2 resonances and the K⁎(892)

We apply a recently developed dispersive formalism to calculate the Regge trajectories of the f2(1270) , f2′(1525) and K⁎(892) mesons. Trajectories are calculated, not fitted to a family of resonances. Assuming that these resonances can be treated in the elastic approximation, the only input are the pole position and residue of a resonance. In all three cases, the predicted Regge trajectories are almost real and linear, with slopes in agreement with the universal value of order 1 GeV −2 . We also show how these results barely change when considering more than two subtractions in the dispersive formalism

Identification of non-ordinary mesons from the dispersive connection between their poles and their Regge trajectories: The <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math> 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 <math altimg="si2.gif" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi><mo stretchy="false">(</mo><mn>770</mn><mo stretchy="false">)</mo></math> resonance pole in ππ scattering, we obtain its linear Regge trajectory, characteristic of ordinary quark–antiquark states. In contrast, the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math> pole—the sigma meson—which dominates scalar isoscalar ππ scattering, yields a nonlinear trajectory with a much smaller slope at the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math> mass. Conversely, imposing a linear Regge trajectory for the <math altimg="si1.gif" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mn>500</mn><mo stretchy="false">)</mo></math>, 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