Evidence for the saturation of the Froissart bound

It is well known that fits to high energy data cannot sharply discriminate between asymptotic $\ln s$ and ${ln}^{2}s$ behavior of total cross section. We show that this is no longer the case when we impose the condition that the amplitudes also describe, on average, low energy data dominated by resonances. We demonstrate this by fitting real analytic amplitudes to high energy measurements of the $\gamma p$ total cross section, for $\sqrt{s}\ge 4\mathrm{G}\mathrm{e}\mathrm{V}$. We subsequently require that the asymptotic fit smoothly join the $\sqrt{s}=2.01\mathrm{G}\mathrm{e}\mathrm{V}$ cross section described by Damashek and Gilman [M. Damashek and F. J. Gilman, Phys. Rev. D 1, 1319 (1970).] as a sum of Breit-Wigner resonances. The results strongly favor the high energy ${ln}^{2}s$ fit of the form ${\sigma }_{\gamma p}=c_0+c_1\ln (\nu /m)+c_2{ln}^2(\nu /m)+{\beta }_{{\mathcal{P}}^{\prime }}/\sqrt{\nu /m}$, basically excluding a $\ln s$ fit of the form ${\sigma }_{\gamma p}=c_0+c_1\ln (\nu /m)+{\beta }_{{\mathcal{P}}^{\prime }}/\sqrt{\nu /m}$, where $\nu$ is the laboratory photon energy. This evidence for saturation of the Froissart [M. Froissart, Phys. Rev. 123, 1053 (1961).] bound for $\gamma p$ interactions is confirmed by applying the same analysis to $\pi p$ data using vector meson dominance.