High-Energy Bounds on Scattering Amplitudes and Oscillations in the Diffraction-Peak Region

Rigorous bounds are established on the absorptive part of the scattering amplitude, $A(s,t)\mathrm{}$, for $t$ real and within the Lehman-Martin ellipse. This result is used to prove an upper bound on ${\left[\frac{d\ln A(s,t)\mathrm{}}{\mathrm{dt}}\right]}_{t=0\mathrm{}}$, and to show that $A(s,t)\mathrm{}$ cannot have a zero in the region $0>t>\mathrm{}\frac{-4\mathrm{}{t}_0}{{\left(\ln s\right)}^2}$ for $s\to \infty$ (${\sqrt{t}}_0=2\mathrm{}{m}_{\pi }$ for $\pi \pi$, $\pi N$, $\mathrm{KN}$, and $\mathrm{NN}$ scattering). No assumption is made about the high-energy behavior of the total cross sections.