Froissart Bound on Total Cross-section without Unknown Constants

TL;DR

This paper establishes a precise logarithmic bound on total cross-sections in pion scattering, removing unknown constants by leveraging bounds on D-wave scattering amplitudes below threshold.

Contribution

It derives an explicit form of the Froissart bound on total cross-sections without unknown constants using bounds on the D-wave amplitude in pion scattering.

Findings

Bound on total cross-section expressed with explicit logarithmic terms.

Quantitative relation between scattering energy and cross-section limit.

No unknown constants needed in the Froissart bound derivation.

Abstract

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} $ .