High-energy bounds from low-energy data

A method for deriving rigorous upper bounds on asymptotic values of total cross sections is discussed. The size of the bound is determined by low-energy data. A functional form for the total cross section at high energies is assumed and bounds on the parameters introduced are obtained. The method is applied to π-π scattering, and numerical estimates, using experimental data, are given for different parametrizations. For an asymptotically constant σ_T(s), we find σ_T(∞)≲40 mb. Better data would improve the accuracy of this bound and could lower it significantly.