ReactionsAπ→AπandAπ→ππin the Veneziano Model

In order to find the effect of spin and isospin on the construction of a Veneziano amplitude, we have analyzed the reaction $A\pi \to A\pi$. This reaction involves the (degenerate) $\rho -f$ trajectory in both channels. In order to construct an amplitude possessing crossing symmetry, proper asymptotic behavior, and pole structure, as well as signature and factorization for the leading trajectory, it is necessary to include three independent beta functions in an expansion of the form (polynomials) × (beta functions). It is found that, to avoid parity doubling on the leading trajectory and to keep the $A\rho \pi$ coupling from vanishing, it is necessary to have a conspiracy among three of the invariant amplitudes for all values of $s$. The conspiracy, signature, and factorization constraints provide a large number of constraints, but do not fully determine the amplitude. The $A\rho \pi$ couplings are undetermined by the foregoing constraints. The amplitude is sufficiently flexible to satisfy the Adler condition and to agree with current-algebra determination of the threshold amplitude. (These conditions only lead to two constraints.) The simpler reaction $A\pi \to \pi \pi$ is analyzed in detail. For this reaction, it is possible to find a minimal amplitude satisfying asymptotic behavior and soft-pion constraints, and only one over-all undetermined constant.