Self-Consistent Calculation of theρ-Meson Regge Pole

The left-hand discontinuities in the partial-wave amplitudes for $\pi -\pi$ scattering are assumed to be dominated by the exchange of the $\rho$ meson in a form suggested by the Regge representation for a resonance. This Regge behavior provides the necessary high-energy cutoff and allows the $\frac{N}{D}$ equations to be solved. The partial-wave $I=1\mathrm{}$ amplitudes are calculated for noninteger angular momenta $l<\mathrm{}1\mathrm{}$ as well as $l=1\mathrm{}$. The trajectory ${\alpha }_{\rho }\mathrm{}\left(s\right)\mathrm{}$ as well as the residue ${\beta }_{\rho }\mathrm{}\left(s\right)\mathrm{}$ of the $\rho$-meson Regge pole are evaluated. An attempt is made to obtain a self-consistent solution for the relevant parameters, namely the position and width of the $\rho$ resonance and ${\alpha }_{\rho }\mathrm{}\left(0\right)\mathrm{}$. The results of this calculation give ${\alpha }_{\rho }\mathrm{}\left(0\right)\mathrm{}\gtrsim 0.9$. The $I=0\mathrm{}$ vacuum trajectory is also discussed.