New evidence for the saturation of the Froissart bound

Fits to high energy data alone cannot cleanly discriminate between asymptotic $\ln s$ and ${ln}^{2}s$ behavior of total hadronic cross sections. We demonstrate that this is no longer true when we require that these amplitudes also describe, on average, low energy data dominated by resonances. We simultaneously fit real analytic amplitudes to high energy measurements of: (i) the ${\pi }^{+}p$ and ${\pi }^{-}p$ total cross sections and $\rho$-values (ratio of the real to the imaginary portion of the forward scattering amplitude), for $\sqrt{s}\ge 6$ GeV, while requiring that the asymptotic fits smoothly join the ${\sigma }_{{\pi }^{+}p}$ and ${\sigma }_{{\pi }^{-}p}$ total cross sections at $\sqrt{s}=2.6$ GeV—both in magnitude and slope , and (ii) separately simultaneously fit the $\overline{p}p$ and $pp$ total cross sections and $\rho$-values for $\sqrt{s}\ge 6$ GeV, while requiring that their asymptotic fits smoothly join the the ${\sigma }_{\overline{p}p}$ and ${\sigma }_{pp}$ total cross sections at $\sqrt{s}=4.0$ GeV—again both in magnitude and slope. In both cases, we have used all of the extensive data of the Particle Data Group [K. Hagiwara et al. (Particle Data Group), Phys. Rev. D 66, 010001 (2002).]. However, we then subject these data to a screening process, the Sieve algorithm [M. M. Block, physics/0506010.], in order to eliminate outliers that can skew a ${\chi }^2$ fit. With the Sieve algorithm, a robust fit using a Lorentzian distribution is first made to all of the data to sieve out abnormally high $\Delta {\chi }_i^2$, the individual i$\mathrm{th}$ point’s contribution to the total ${\chi }^2$. The ${\chi }^2$ fits are then made to the sieved data. Both the $\pi p$ and nucleon-nucleon systems strongly favor a high energy ${ln}^{2}s$ fit of the form: ${\sigma }^{\pm }=c_0+c_1\ln (\frac{\nu }{m})+c_2{ln}^2(\frac{\nu }{m})+{\beta }_{{\mathcal{P}}^{\prime }}(\frac{\nu }{m}{)}^{\mu -1}\pm \delta (\frac{\nu }{m}{)}^{\alpha -1}$, basically excluding a $\ln s$ fit of the form: ${\sigma }^{\pm }=c_0+c_1\ln (\frac{\nu }{m})+{\beta }_{{\mathcal{P}}^{\prime }}(\frac{\nu }{m}{)}^{\mu -1}\pm \delta (\frac{\nu }{m}{)}^{\alpha -1}$. The upper sign is for ${\pi }^{+}p$ ($pp$) and the lower sign is for ${\pi }^{-}p$ ($\overline{p}p$) scattering, where $\nu$ is the laboratory pion (proton) energy, and $m$ is the pion (proton) mass.