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 $\pi \pi$ scattering. E.g. for ${\pi }^0{\pi }^0$ scattering we show that for c.m. energy $\sqrt{s}\to \mathrm{\infty }$, $\bar{\sigma}{tot}(s,\infty)\equiv s\int{s} ^{\infty} ds’\sigma_{tot}(s’)/s’^2 \leq \pi (m_{\pi})^{-2} [\ln (s/s_0)+(1/2)\ln \ln (s/s_0) +1]^2$where$m_\pi^2/s_0= 17\pi \sqrt{\pi/2} $ .