Formulation and Numerical Solution of a Set of Dynamical Equations for the Regge Pole Parameters

Abstract

In a previous paper, the principles of analyticity and unitarity were shown to lead to a set of coupled nonlinear integral equations for the Regge pole parameters. In this paper, we demonstrate, for both boson and fermion trajectories, 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. We then proceed to carry out their numerical solution in a number of interesting cases. Because our equations are approximate, we first solved the equations in the potential-theory case, where our results could be compared with those obtained from the Schrödinger equation. The agreement in most cases is good. Then we turn to the determination of the Regge pole parameters which describe relativistic $\pi \pi$ scattering at high energies. Neglecting the inelastic contributions, we calculate the Pomeranchuk trajectory, the $\rho$-meson trajectory, and the second vacuum trajectory $P^{\prime }$. One notable result of this set of calculations is that the function Re $\alpha \mathrm{}\left(t\right)\mathrm{}$ for the Pomeranchuk trajectory, as determined by our equations, agrees well with the results obtained by Foley et al. from an analysis of the ${\pi }^{-}p$ angular distributions in the range $-0.8\mathrm{}{(\mathrm{BeV}/c)}^2<t<-0.2\mathrm{}{(\mathrm{BeV}/c)}^2$. No spin-2 resonance is found to lie on this trajectory. As for the $\rho$ trajectory, we find that ${\alpha }_{\rho }\mathrm{}\left(t\right)\mathrm{}$, $-0.8\mathrm{}{(\mathrm{BeV}/c)}^2<t\le 0$, is larger than 0.9 for a wide range of input parameters. The width of the $\rho$ resonance, as determined from our equation, 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. Finally, we outline various problems which remain to be investigated.