Reactionωπ→ωπin the Veneziano Model

A set of invariant amplitudes for the reaction $\omega \pi \to \omega \pi$ is constructed within the framework of the Veneziano model by expansions of the form $\Sigma \left(\mathrm{polynomials}\right)\times \left(\mathrm{beta}\mathrm{function}\right)$. The dominant asymptotic behavior corresponds to the $f$ trajectory in the $t$ channel and the $\rho$ and $B$ trajectories in the $s$ and $u$ channels. Separate treatments are given for the $\rho$ and $B$ contributions. In every case the leading trajectory poles have correct spin-parity as a consequence of enforcing proper asymptotic behavior on the invariant amplitudes. These relations are nontrivial and differ for the $\rho$ and $B$ contributions. The detailed construction of the amplitudes for $\omega \pi$ scattering differs from most previous applications because of the absence of an exotic channel. $\omega \pi$ scattering is notable in the number of terms required to represent the amplitude; a 72-parameter amplitude built using $B_{11}$, $B_{12}$, and $B_{21}$ cannot satisfy simple physical requirements for the $\rho$ trajectory. One must add further terms lacking the $\rho$ pole (such as $B_{22}$) in order to avoid decoupling the $\rho$ trajectory. The $B$ contribution is easily treated in direct analogy to the $\rho$ trajectory in $A\pi$ scattering. The implication of the hypothesis of partially conserved axial-vector current (PCAC) that the invariant amplitude $T_1$ (coefficient of $e^{\prime }·e$) should vanish at the Adler point is somewhat delicate because of the near degeneracy of the $\omega$ and $\rho$ mesons. The signature and amplitude conspiracy relations ensure the suppression of the apparent pole.