Diffraction Scattering and Singularities in the Angular Momentum Plane

The singularities in the angular momentum plane, which describe the vacuum intermediate state dominating high-energy scattering, are assumed to consist of a single Regge pole and a fixed branch cut. If the energy variable ($\surd s$) only enters the equations in the normalized form $w=\frac{s}{2M_1{M}_2}$, where $M_1$ and $M_2$ are masses of the particles, the value of $w$ for lower mass interactions (e.g., $\pi -N$) will be much larger than for high mass interactions (e.g., $N-N$), for the same value of the lab energy. In the combined pole and cut model the vacuum pole dominates at lower values of $w$, and the cut becomes dominant as $w\to \infty$, so that the $p+p$ scattering between 7 and 20 BeV/c will be mainly described by the single vacuum pole, causing the width of the diffraction peak to shrink, whilst the ${\pi }^{-}+p$ scattering between 7 and 17 BeV/c will be described by the fixed cut, which leads to no shrinkage. This permits the $p+p$ and ${\pi }^{-}+p$ data to be combined into one. A fit is obtained to the combined $p+p$ and ${\pi }^{-}+p$ data for the elastic differential cross section obtained recently by Foley et al. Another consequence of the vacuum-pole-cut model is that, in the energy range under consideration, the total cross section is not yet constant, but has the form ${\sigma }_T=a+\frac{b}{\ln w}$, which agrees well with ${\pi }^{\pm }+p$ scattering data. This behavior of ${\sigma }_T$ should also describe the average of the $p+p$ and $\overline{p}+p$ total cross sections. Estimates of $a$ and $b$ using $p+p$, $\overline{p}+p$, and ${\pi }^{\pm }+p$ data agree with those obtained for ${\pi }^{\pm }+p$ alone to within 6%. This strongly supports our method of combining $N-N$ and $\pi -N$ scattering data, and is in good agreement with the pole and cut model. A test of the theory can be obtained from experiments on $K-N$ scattering.