Analyticity Properties of Production Amplitudes

We study the analytic properties of production amplitudes as functions of the momentum transfer ${\Delta }^2$ between one of the incoming particles and one of the outgoing particles, when the total energy and the three further parameters determining the relative motion of the three outgoing particles in the center-of-mass system are held fixed. We find that suitable combinations of the amplitudes are analytic functions of ${\Delta }^2$ regular within an ellipse in the ${\Delta }^2$ plane. It is also shown that in the same domain the cross section $\frac{{\partial }^2\sigma }{\partial {\Delta }^2\partial w^2}$ is an analytic and regular function of ${\Delta }^2$, $w^2$ being the total mass of two of the outgoing particles. The poles in ${\Delta }^2$ conjectured by Chew and Low never lie inside the domain of regularity.