Veneziano Model forππ→πS, WhereSHas Arbitrary Spin and Parity

The Veneziano model is applied to the reaction $\pi \pi \to \pi S$, where $S$ is any particle of spin $S$, isospin 0 or 1. For particles with natural parity, $P={\mathrm{}(-1\mathrm{})\mathrm{}}^S$, we require that our amplitude not be coupled to any odd daughters. This fixes the relative magnitudes of the helicity amplitudes and constrains the particles $S(\omega ,A_2,\cdots )$ to lie on a trajectory which is parallel to the degenerate $\rho$ and $f$ trajectories. For particles with unnatural parity, $P=-{\mathrm{}(-1\mathrm{})\mathrm{}}^S$, we demand that the amplitude satisfy the Adler consistency condition. This gives mass restrictions on the sequence of particles $\pi ,A_1,\cdots$.