Froissart Bound on Total Cross-section without Unknown Constants

We determine the scale of the logarithm in the Froissart bound on total cross sections using absolute bounds on the D-wave below threshold for pion-pion scattering. For example, for ${\ensuremath{\pi}}^{0}{\ensuremath{\pi}}^{0}$ scattering, we show that for c.m. energy $\sqrt{s}\ensuremath{\rightarrow}\ensuremath{\infty}$, ${\overline{\ensuremath{\sigma}}}_{\text{tot}}(s,\ensuremath{\infty})\ensuremath{\equiv}s{\ensuremath{\int}}_{s}^{\ensuremath{\infty}}d{s}^{\ensuremath{'}}{\ensuremath{\sigma}}_{\text{tot}}({s}^{\ensuremath{'}})/{{s}^{\ensuremath{'}}}^{2}\ensuremath{\le}\ensuremath{\pi}{({m}_{\ensuremath{\pi}})}^{\ensuremath{-}2}{[\mathrm{ln}(s/{s}_{0})+\phantom{\rule{0ex}{0ex}}(1/2)\mathrm{ln}\text{ }\mathrm{ln}(s/{s}_{0})+1]}^{2}$, where $1/{s}_{0}=17\ensuremath{\pi}\sqrt{\ensuremath{\pi}/2}\text{ }{m}_{\ensuremath{\pi}}^{\ensuremath{-}2}$.