Cross Sections at High Energies

If the difference σ₊(E)-σ_-(E) of particle and antiparticle total cross sections changes sign at most at a finite number of energies, then the odd part f₊(E)-f_-(E) 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 σ₊(E)-σ_-(E) and f₊(E)-f_-(E). For example, if f₊(E)-f_-(E)ElnᵐE is bounded, and m≤1/2 then we have the Pomeranchuk result that σ₊(E)-σ_-(E)→0. Even if m>1/2 it seems likely that although the difference of σ₊(E) and σ_-(E) may not tend to zero, their ratio does tend to one.