Some Simple Results on the Duality of Non-Parity-Doubled Systems

We show that in some cases the crossing relations and kinematic behavior of helicity amplitudes alone suffice to rule out combinations of mutually bootstrapping trajectories. In particular, no set of $g=-$ natural-parity trajectories can bootstrap itself in the reaction $\pi \rho \to \rho \pi$; and no set of meson trajectories of definite (natural or unnatural) parity can be self-dual in baryon-antibaryon or quark-antiquark scattering. The techniques described are most useful when the number of independent helicity amplitudes involved in the reaction is small, and several of these can be associated with definite parity for exchanged trajectories.