  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}\rightarrow \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} $ .

High Energy Physics - Phenomenology