Limit on High-Energy Cross Section from Analyticity in Lehmann Ellipses

High-energy limits on the scattering amplitude and the total cross section follow from that analyticity of the scattering amplitude in ellipses in ${cos\theta }_{\mathrm{c}.\mathrm{m}.\mathrm{}}$ which has been proved rigorously by Lehmann, together with unitarity. The limits for the scattering amplitude are $\mathrm{}\left|T\right|<\left(\mathrm{const}\right)W^4{\ln }^{2}W$, for the forward and backward directions, and $\mathrm{}\left|T\right|<\left(\mathrm{const}\right)W^3{\ln }^{\frac{3}{2}}W$ for other fixed directions, where $W$ is the total center-of-mass (c.m.) energy. For the total cross section the limit is $\sigma <\left(\mathrm{const}\right)W^2{\ln }^{2}W$.