Remarks on the Forward Peak at High Energies

In the framework of the axiomatic field theory, it is proved that $s-u$ crossing-symmetric elastic-scattering amplitude of any two stable particles has forward peak at high energies, unless $\left|\frac{\mathrm{Re}f(s,1\mathrm{})\mathrm{}}{\mathrm{Im}f(s,1\mathrm{})\mathrm{}}\right|\to \infty$ or $\mathrm{}\left|f\right(s,1\mathrm{}\left)\right|<C{\left(\ln s\right)}^{-1\mathrm{}}$. If one exchange amplitude dominates the others, this result also holds for any elastic-scattering amplitude without $s-u$ symmetry. It turns out that the dispersion relation is not essential and the analytic property recently found by Bros, Epstein, and Glaser is sufficient for the proof. In the proof, the properties of the Herglotz function are extensively used.