High-Energy Elastic Scattering at Low Momentum Transfers

The $K^{\pm }p$, ${\pi }^{\pm }p$, $\mathrm{pp}$, and $\overline{p}p$ data in the laboratory-energy region between 7 and 20 BeV and momentum transfer squared, $-t$, less than 0.5 ${\left(\frac{\mathrm{BeV}}{c}\right)}^2$ are analyzed in terms of the $P$, $P^{\prime }$, and $\omega$ Regge poles. A linear approximation to the trajectories is made with slopes ${\alpha }^{\prime }$ assumed to be equal. The reduced residues of $P$ and $P^{\prime }$ are taken to be of the form ${(1-b_{i}t)}^{-{\epsilon }_i}$, $i=P$, $P^{\prime }(b_i>0)$. In order to explain the difference between the antiparticle ($K^{-}p$ and $\overline{p}p$) and particle ($K^{+}p$ and $\mathrm{pp}$) differential cross sections, the $\omega$ residue should have a zero at a negative value of $t$. Hence, the reduced residue for $\omega$ is taken to be of the form $(1+\frac{t}{t_0}{(1-b_{\omega }t)}^{-{\epsilon }_{\omega }}$, where $t_0$ is the position of the zero. We choose ${\epsilon }_P={\epsilon }_{P^{\prime }}=2.5\mathrm{}$ and ${\epsilon }_{\omega }=3.5\mathrm{}$ in order to conform to the high-momentum-transfer behavior ($\frac{d\sigma }{\mathrm{dt}}\sim t^{-5\mathrm{}}$) observed in $\mathrm{pp}$ scattering. The $t=0\mathrm{}$ values of the residues and the trajectory intercepts are known from other considerations. Covering the above range of energy and momentum transfer, we thus have five parameters for each of the antiparticle-particle sets, $K^{\pm }p$ and $pp-\overline{p}p$, and three parameters for ${\pi }^{\pm }p$, of which ${\alpha }^{\prime }$ and (from factorization) the $t_0\text{'}\mathrm{s}$ should be the same between the different sets. The ${\alpha }^{\prime }$ values turn out to be the same (=0.41 ${\left(\frac{\mathrm{BeV}}{c}\right)}^{-2\mathrm{}}$) for each set, while the $t_0$ values are reasonably close: 0.061 ${\left(\frac{\mathrm{BeV}}{c}\right)}^2$ for $K^{\pm }p$ and 0.074 ${\left(\frac{\mathrm{BeV}}{c}\right)}^2$ for $\overline{p}p-pp$. It is found that the residues of $P$ contribute substantially to the diffraction widths. A crude estimate of the contribution of branch cuts indicates that they will not be important compared to $P$ in the above region of energy and momentum transfer.