Unitarity Bounds of the Scattering Amplitude and the Diffraction Peak

From unitarity alone a lower bound for the derivative of the absorptive part of the forward scattering amplitude with respect to the momentum transfer is obtained, in terms of the elastic and total cross sections. Comparison with high-energy scattering experiments shows that the actual value of this derivative is rather close to the lower bound, which provides some information on the partial-wave distribution. Our result can also be used to obtain consistency requirements on theoretical models. If Regge behavior is assumed for high-energy scattering, namely, $F(s, t)\simeq f\left(t\right)\mathrm{}{s}^{\alpha \mathrm{}\left(t\right)\mathrm{}}$, then one can show that either ${\alpha }^{\prime }\mathrm{}\left(0\right)\mathrm{}>~\epsilon >0\mathrm{}$ or $\alpha \mathrm{}\left(t\right)\mathrm{}\equiv \mathrm{const}.\mathrm{}$