Asymptotic Decrease of Scattering Amplitudes

The recent analysis by Orear of the large-angle $p-p$ scattering data at high energies indicates that the scattering amplitude outside the diffraction peak falls off asymptotically like $\exp [-c(\theta \left)s^{\frac{1}{2}}\right]$ for fixed angles, where $c\left(\theta \right)$ is some function of the center-of-mass scattering angle $\theta$ and $s$ is the square of the center-of-mass energy. We show that if the scattering amplitude is $O\left(\exp \right\{-c\left(\theta \right)s^{\frac{1}{2}}\left\}\right)$ as $s\to \infty$ where $c\left(\theta \right)>\mathrm{}0\mathrm{}$ for some fixed $\theta$, then the entire scattering amplitude for this $\theta$ is uniquely determined by the function associated with the left-hand cut in the fixed-angle dispersion relation. The spectral function for the right-hand cut is exhibited as the Fourier transform of an analytic function which is defined by a power series in a certain domain of its regularity. The power series involves only quantities which are determined by the left-hand cut. Unitarity is not explicitly used in the proof, so that this asymptotic requirement plus the nature of the left-hand cut have somehow to imply the unitarity condition and hence the mass spectrum in the $s$ channel at all energies.