Pomeranchuk-type theorems for total and elastic cross sections

In the framework of local field theory the asymptotic properties of the crossing-odd forward scattering amplitude $F\left(E\right)\mathrm{}$ are investigated, $E$ being the laboratory energy. Confronting the analyticity properties of the logarithm of the amplitude with the Froissart-Martin bound, we obtain a series of sufficient conditions for a fast asymptotic vanishing of $\mathrm{Im}\frac{F\left(E\right)\mathrm{}}{F}$ and $\mathrm{Re}\frac{F\left(E\right)\mathrm{}}{E}$ in the mean. Analogous conditions for the vanishing of the total-cross-section difference $\Delta \sigma \mathrm{}\left(E\right)\mathrm{}$ follow via the optical theorem. We also find conditions for the existence of Meiman's generalized high-energy limit of $\Delta \sigma \mathrm{}\left(E\right)\mathrm{}$. Finally, two independent asymptotic bounds on $\Delta \sigma \mathrm{}\left(E\right)\mathrm{}$ are derived. The method is extended to nonforward scattering to give asymptotic bounds on differential cross sections.