High-EnergyppElastic Diffraction Scattering from Inelastic States

Abstract

Small-angle high-energy elastic proton-proton scattering, i.e., the Pomeranchukon contribution, is determined through unitarity in terms of a model for the inelastic intermediate states. These are represented by two contributions: (i) quasielastic production of free pions, described in terms of a $c$-number source current $\ensuremath{\beta}(k)$ for the pions, assuming factorization of the production amplitude into a part proportional to the elastic $\mathrm{pp}$ amplitude times a part depending on $\ensuremath{\beta}(k)$ and the proton momenta; (ii) peripheral production of a spectrum of ${N}^{*}$ states by the longest range possible force-single-pion exchange. The combination of these two contributions leads to an inhomogeneous integral equation for the elastic amplitude $F(s,\ensuremath{\theta})$ of the form $2\mathrm{Im}F(s,{\ensuremath{\theta}}_{0})=\frac{{M}^{2}}{{(2\ensuremath{\pi})}^{2}}{P}^{2}\ensuremath{\int}d\ensuremath{\Omega}(\ensuremath{\theta},\ensuremath{\varphi}){F}^{*}(s,{\ensuremath{\theta}}^{\ensuremath{'}})F(s,\ensuremath{\theta})Z(\ensuremath{\theta},{\ensuremath{\theta}}^{\ensuremath{'}})+\ensuremath{\Delta}(s,{\ensuremath{\theta}}_{0})$ which becomes the condition for elastic unitarity when $Z(\ensuremath{\theta},{\ensuremath{\theta}}^{\ensuremath{'}})={(P\sqrt{s})}^{\ensuremath{-}1}$ and $\ensuremath{\Delta}(s,{\ensuremath{\theta}}_{0})=0$, i.e., when no pions are produced. The inhomogeneous term is generated by the sum over peripheral ${N}^{*}$ states, while the integral arises from the quasielastic contribution. The assumption of a $c$-number current implies that the pions are produced in a coherent state. The unitarity sum over quasielastic states can be replaced by an integral over coherent states, and their properties, familiar in quantum optics, as well as the factorization assumption, are used to obtain an analytic form for $Z(\ensuremath{\theta},{\ensuremath{\theta}}^{\ensuremath{'}})$ in the high-energy limit. The kernel $Z(\ensuremath{\theta},{\ensuremath{\theta}}^{\ensuremath{'}})$ depends upon the average multiplicity $\overline{n}(P,\ensuremath{\theta})$ of pions in phase space at c.m. proton momentum $P$ and scattering angle $\ensuremath{\theta}$, as well as on more detailed characteristics of the current $\ensuremath{\beta}(k)$. However, $\overline{n}(P,\ensuremath{\theta})$ can be determined from inelastic $\mathrm{pp}$ interactions and is used to establish the general behavior of $\ensuremath{\beta}(k)$, since $\overline{n}(P,\ensuremath{\theta})=\ensuremath{\int}\frac{{d}^{3}k}{{(2\ensuremath{\pi})}^{3}2\ensuremath{\omega}}{|\ensuremath{\beta}(k)|}^{2}$ The above integral equation for $F(s,\ensuremath{\theta})$ is simplified under Fourier-Bessel transformation to its impact-parameter representation when the high-energy, small-angle limit is taken. The resulting nonlinear, inhomogeneous integral equation is solved numerically, and used to determine the elastic differential cross section $\frac{d\ensuremath{\sigma}}{\mathrm{dt}}$. Normalizing the amplitude to the total cross section ${\ensuremath{\sigma}}_{\mathrm{tot}}$ in the forward direction, one finds that the iterative solution can be considered to depend on ${\ensuremath{\sigma}}_{\mathrm{tot}}$, the elastic cross section ${\ensuremath{\sigma}}_{\mathrm{el}}$, a parameter $\ensuremath{\alpha}$ determined by $\overline{n}(P,\ensuremath{\theta})$, and a free parameter $\ensuremath{\lambda}$ that parameterizes the phase of $\ensuremath{\beta}(k)$. With experimentally determined values of ${\ensuremath{\sigma}}_{\mathrm{tot}}$, ${\ensuremath{\sigma}}_{\mathrm{el}}$, and $\ensuremath{\alpha}$, the parameter $\ensuremath{\lambda}$ can be fitted to give $\frac{d\ensuremath{\sigma}}{\mathrm{dt}}$ in agreement with elastic data at 25 GeV/c for $t$ in the range $0<~\ensuremath{-}t<~1.0$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^{2}$.