Power Behavior of the Scattering Amplitude at Fixed Momentum Transfer and a Reconsideration of the Cerulus-Martin Lower Bound

Abstract

We have shown that the assumption of maximal analyticity of first degree and fixed-$t$ power behavior of the scattering amplitudes in general imply a lower bound at a fixed angle. The fixed-angle lower bound takes the form $\mathrm{exp}[\ensuremath{-}{c}_{\ensuremath{\gamma}}({z}_{s}){s}^{\ensuremath{\gamma}}\mathrm{ln}s]$, where ${c}_{\ensuremath{\gamma}}({z}_{s})$ and $\ensuremath{\gamma}$ are positive. The precise value of $\ensuremath{\gamma}$ depends on the specific assumptions on the fixed-$t$ bound of the scattering amplitude. In particular, the assumptions made by Cerulus and Martin correspond to $\ensuremath{\gamma}=\frac{1}{2}$, and for the case of a linearly rising trajectory, $\ensuremath{\gamma}=1$. Furthermore, we obtain a nonzero lower bound at ${z}_{s}=0$, which heretofore was given as zero.