Fermion Trajectory Parametrization and Superconvergence

A parametrization of the nucleon and $\Delta$ trajectory is given, using both the position and the width of the resonances in a dispersion relation. The nucleon trajectory requires and accommodates a $D$-wave $\pi N$ resonance in the neighborhood of 1670 MeV. Another significant outcome of the scheme is that the $\Delta$ trajectory has an intercept, at $W=0\mathrm{}$, of -0.69 and hence leads to a superconvergence relation for ${\pi }^{-}p$ scattering invariant amplitudes in the nearly backward direction ($u=0\mathrm{}$). The resulting sum rules are satisfied only if there exists a strong attractive $s$-wave interaction (for example, a $\sigma$ meson) or other equivalent effects (such as an $f$ meson) in the $\pi \pi \to N\overline{N}$ channel.