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

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 $\exp [-c_{\gamma }(z_s\left)s^{\gamma }\ln s\right]$, where $c_{\gamma }\left(z_s\right)$ and $\gamma$ are positive. The precise value of $\gamma$ 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 $\gamma =\frac{1}{2}$, and for the case of a linearly rising trajectory, $\gamma =1\mathrm{}$. Furthermore, we obtain a nonzero lower bound at $z_s=0\mathrm{}$, which heretofore was given as zero.