On the Dynamical Determination of High Energy Scattering Cross Sections

Citation

Sharp, David Howland (1964) On the Dynamical Determination of High Energy Scattering Cross Sections. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/4D8K-BP97. https://resolver.caltech.edu/CaltechETD:etd-10072002-084711

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. This thesis deals with the problem of obtaining a quantitative understanding of high energy cross sections and angular distributions and their connection with low energy resonances exchanged in crossed channels. The work presented here takes as its starting point the conjecture that scattering amplitudes may be expressed as a sum of Regge poles. In a general introduction we summarize the basic ideas and current status of the Regge conjecture. Here we also review briefly the main results of this work and try to cast them into perspective before plunging into details. In Part II of the thesis a detailed analysis of NN and [...] scattering on the basis of the Regge hypothesis is carried out. The Regge expansions of a set of ten invariant amplitudes describing NN-scattering are presented, with residues expressed in factorized form. Expressions involving both the full Legendre functions and their asymptotic forms are given. Spin sums are carried out to obtain simple and convenient expressions for the contributions of the P, [rho], [omega] and P' trajectories to the differential cross sections. The optical theorem has been applied to find the contribution of the P, P', [rho] and [omega] trajectories to the spin-averaged total cross sections. Finally, the available data on the total and differential cross sections for NN-scattering has been analysed to extract information about the Regge pole parameters. The possible effect of the spin structure of the amplitudes and the variation with energy of the Legendre functions has been taken into account. In Part III, we show that the analytic properties of the Regge parameters plus the unitarity condition satisfied by the partial wave amplitude lead to a set of coupled non-linear integral equations for the Regge pole parameters. We then show that these equations can be written in a very simple form which makes many of their mathematical properties transparent and permits their numerical solution by iteration. These equations have been solved numerically in several interesting cases. In the potential theory case, where our results could be compared with those obtained from the Schrodinger equation, the agreement was good in most cases. In the relativistic case, we calculated the position of the Pomeranchuk trajectory, the [rho]-meson trajectory and the second vacuum trajectory P'. Inelastic contributions were neglected. One notable result of this set of calculations is that the function [...] for the Pomeranchuk trajectory as determined by our equations agrees well with the results obtained by Foley et al. from an analysis of the [...] angular distributions in the -0.8 (BeV/c)[superscript 2] < t < t -0.2 (BeV/c)[superscript 2]. No spin 2 resonance is found to lie on this trajectory. As for the [rho]-trajectory, we find that [...] (t), -0.8 (BeV/c)[superscript 2] < t [...] 0 is larger than 0.9 for a wide range of input parameters. The width of the [rho] resonance, as determined by our equations, is several times larger than the experimental width. This probably means that inelastic contributions must be included to obtain a correct value for the width.

Thesis Files