Investigation of the Pion Trajectory in a Regge-Pole Model

Reactions of $\pi$, $K$, $\overline{p}$, and $p$ resulting in $\Delta \mathrm{}\left(1238\right)\mathrm{}$ production are examined. It is shown that whenever an exchange with $P={\mathrm{}(-)\mathrm{}}^{J+1\mathrm{}}$ is allowed by the known selection rules, this exchange dominates the production mechanism at small momentum transfers. This phenomenon is described in terms of a Regge-pole model for pion exchange. Arguments are given to show that although the pion trajectory is lower than the $\rho$ and $A_2$ trajectories, $\pi$ exchange can dominate at small $\mathrm{}\left|t\right|\mathrm{}$ values. The pion trajectory is determined from the experimental data, and fits for the differential cross sections at various energies are given for $\mathrm{}\left|t\right|\mathrm{}\le 0.25$ ${(\mathrm{G}\mathrm{e}\mathrm{V}/\mathit{c})}^2$.