On the construction of scattering amplitudes from experimental data and analyticity

Abstract

The modulus (phase) representation of analytic functions, which expresses them in terms of their complex zeroes and a dispersion integral over their modulus (phase) on the real axis is studied in general. Under certain conditions it is shown that the scattering amplitude in nonrelativistic potential scattering has a finite number of zeroes in the complex energy plane ( s) at fixed momentum transfer ( t), and the fixed- t modulus representation and the t = 0 phase representation in the s-plane are given for it. The modulus representation for the relativistic π± p → π± p and pp → π +π − transversity amplitudes in the laboratory frame are presented, after discussing their complex zeroes. The t= 0 phase representation for π − p → π − p is also ggive. All of them give the corresponding scattering amplitude if differential cross sections and polarizations are known. and if the complex zeroes and certain unphysical region contributions are known. Several methods are proposed to find out the latter two: superconvergent modulus representations and superconvergent sum rules coming from them, combination of t = 0 phase and modulus representations, sum rules coming from analyticity and nonoscillation at infinity, approximate modulus representations for high-energy scattering and optimalized polymials to perform analytic extrapolations. The generalization to N-N scattering is treated briefly, the main consquence being to stress the interest of measuring the polarization parameters C nn , D nn and K nn .