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(θ)s1/2] for fixed angles, where c(θ) is some function of the center-of-mass scattering angle θ and s is the square of the center-of-mass energy. We show that if the scattering amplitude is O(exp-c(θ)s1/2) as s→∞ where c(θ)>0 for some fixed θ, then the entire scattering amplitude for this θ 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.If the scattering amplitude is O[exp-c(θ)s1/2] where c(θ)>0 for a nonzero interval in θ, then the amplitude is uniquely determined by its left-hand cut for all θ by analytic continuation. In this case, the solutions of the partial-wave dispersion relations are also unique.