Cross Sections at High Energies

If the difference ${\sigma }_{+}\mathrm{}\left(E\right)-{\sigma }_{-}\mathrm{}\left(E\right)\mathrm{}$ of particle and antiparticle total cross sections changes sign at most at a finite number of energies, then the odd part $f_{+}\mathrm{}\left(E\right)-f_{-}\mathrm{}\left(E\right)\mathrm{}$ of the forward scattering amplitude has a very useful representation, as a rational times a "Herglotz" function. The representation implies a correlation between the high-energy asymptotic behavior of ${\sigma }_{+}\mathrm{}\left(E\right)-{\sigma }_{-}\mathrm{}\left(E\right)\mathrm{}$ and $f_{+}\mathrm{}\left(E\right)-f_{-}\mathrm{}\left(E\right)\mathrm{}$. For example, if $\frac{\left\{f_{+}\mathrm{}\right(E)-f_{-}\mathrm{}(E\left)\right\}}{E{\ln }^{m}E}$ is bounded, and $m\le \frac{1}{2}$ then we have the Pomeranchuk result that ${\sigma }_{+}\mathrm{}\left(E\right)-{\sigma }_{-}\mathrm{}\left(E\right)\mathrm{}\to 0$. Even if $m>\mathrm{}\frac{1}{2}$ it seems likely that although the difference of ${\sigma }_{+}\mathrm{}\left(E\right)\mathrm{}$ and ${\sigma }_{-}\mathrm{}\left(E\right)\mathrm{}$ may not tend to zero, their ratio does tend to one.