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 ${\pi }^0{\pi }^0$ scattering, we show that for c.m. energy $\sqrt{s}\to \infty$, ${\overline{\sigma }}_{\text{tot}}(s,\infty )\equiv s{\int }_s^{\infty }d{s}^{\prime }{\sigma }_{\text{tot}}(s^{\prime })/{s^{\prime }}^2\le \pi {(m_{\pi })}^{-2}{[\ln (s/s_0)+(1/2)\ln \ln (s/s_0)+1]}^2$, where $1/s_0=17\pi \sqrt{\pi /2}{m}_{\pi }^{-2}$.

• Received 24 June 2013

DOI:https://doi.org/10.1103/PhysRevD.89.045015