Universal $\rho$ Coupling and the Adler-Weisberger Theorem for ${\pi }^{-}{\rho }^{+}$ and ${\pi }^{-}{A_1}^{+}$ in the Veneziano Model

We write minimal Veneziano representations, normalized to the $t$-channel $\rho$ poles, for the ${\pi }^{-}{\rho }^{+}\to {\pi }^{-}{\rho }^{+}$ and ${\pi }^{-}{A_1}^{+}\to {\pi }^{-}{A_1}^{+}$ invariant amplitudes relevant to the Adler-Weisberger theorem. Using these amplitudes and the results of other authors for ${\pi }^{-}{\pi }^{+}$ and ${\pi }^{-}{K}^{+}$, we find that the Adler-Weisberger theorem is satisfied for ${\pi }^{-}{\pi }^{+}$, ${\pi }^{-}{K}^{+}$, ${\pi }^{-}{\rho }^{+}$, and ${\pi }^{-}{A_1}^{+}$ elastic scattering if the $\rho \pi \pi$, $\rho \mathrm{KK}$, $\rho \rho \rho$, and $\rho \mathrm{AA}$ charge couplings all have the same (universal) value.