Veneziano-Like Model for the Isoscalar-Current Three-Pion Amplitude

Abstract

Using mutilated five-point functions, we construct an amplitude for ${S}^{\ensuremath{\mu}}+\ensuremath{\pi}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$, where ${S}^{\ensuremath{\mu}}$ is an isoscalar current. The amplitude satisfies crossing symmetry and the Adler condition, and contains an infinite sequence of poles in the current momentum squared. We find that it reduces to the Veneziano amplitude on the $\ensuremath{\omega}$ mass shell. For fixed ${q}^{2}$ and $t$, as $s\ensuremath{\rightarrow}\ensuremath{\infty}$ the amplitude behaves as $\ensuremath{\Gamma}(1\ensuremath{-}\ensuremath{\alpha}(t))B(1\ensuremath{-}\ensuremath{\alpha}({q}^{2}),{\ensuremath{\alpha}}_{\ensuremath{\rho}}(t)\ensuremath{-}{\ensuremath{\gamma}}_{1}){s}^{\ensuremath{\alpha}(t)}$. We find by normalizing the amplitude that ${\ensuremath{\gamma}}_{1}=\ensuremath{-}1$, and thus the amplitude has fixed multiplicative poles at ${\ensuremath{\alpha}}_{\ensuremath{\rho}}(t)=\ensuremath{-}1,\ensuremath{-}2,\dots{}$, when we are not at an $\ensuremath{\omega}$ daughter pole. We then compare this model with vector dominance.