Sums of Direct-Channel Regge-Pole Contributions and Crossing Symmetry

Starting with a sequence of parallel rising trajectories in the $s$ channel, we give examples of residue functions for which the Regge-Mandelstam pole contributions can be explicitly summed. The sum has Regge behavior in the $s$ channel as well as the $t$ channel, and satisfies fixed-$t$ dispersion relations and finite-energy sum rules. The residue functions we start with do satisfy the usual analyticity properties and threshold properties, have the Mandelstam symmetry factor, and show the expected exponential behavior for large $\mathrm{}\left|s\right|\mathrm{}$. These results are achieved without fixing either the slope or the intercept of the leading trajectory and without specifying $\mathrm{Im}\alpha$ in detail in the low- and intermediate-energy region. We use these examples to clarify some of the problems related to the use of finite-energy sum rules, both as phenomenological relations and as dynamical equations. The way a finite but large number of $s$-channel resonances can be summed to give Regge behavior in $s$ is explicitly demonstrated. We also indicate how the observation by Schmid on the relation of the $\rho$-exchange contribution in pion-nucleon charge-exchange scattering to direct-channel resonances can be understood in terms of a direct-channel Regge-Mandelstam analysis. Finally, we point out a general method for generating more examples of other residue functions with the desired properties.