Interpretation of High-Energy Large-Angle Scattering

It is shown that if the analytically continued partial-wave amplitude is assumed to have $l$ dependence $a_{\pm }(s, l)=\Sigma \stackrel{n}{l=0\mathrm{}}{C_m}^{\pm }\mathrm{}\left(s\right)\mathrm{}{l}^m{\mathrm{}(l+1\mathrm{})\mathrm{}}^m$ for $l<\mathrm{}{l}_0\mathrm{}\left(s\right)\mathrm{}$ and finite $n$, the scattering amplitude is bounded by $\exp \{-\mathrm{const}{\left[l_0\mathrm{}\right(s\left)\mathrm{}sin\theta \mathrm{}\right(s\left)\right]}^{\frac{1}{2}}\}$ at high energies. Here $a_{+}(s, l)\left[a_{-}\right(s, l\left)\right]$ is equal to $a_l\mathrm{}\left(s\right)\mathrm{}$ for even (odd) integer $l$. The most physical example of this dependence is that in which a central area of the scatterer becomes maximally absorptive.