Bootstrap Conditions from the Peripheral Model Combined with Duality

Abstract

By requiring that the $\ensuremath{\pi}\ensuremath{\pi}$ absorptive part generated by a peripheral model at moderately high energies be dual (in a semilocal sense) with the Regge behavior in a given process, it is possible to obtain information on the relevant trajectories. In practice we only had to consider singly-peripheral graphs, although at higher energies multiperipheral graphs would also come in. Our general presentation is therefore in terms of the multiperipheral model, although it does not require any of the usual formal apparatus of this approach. The input graphs are constructed in terms of the $\ensuremath{\pi}\ensuremath{\pi}$ amplitude, which is approximated by a Veneziano model. If we then require self-consistency between the input and output trajectories, we can obtain simple algebraic bootstrap conditions on the partial widths of the $\ensuremath{\pi}\ensuremath{\pi}$ resonances in that model. With the help of an additional duality argument it is also possible to obtain the total width of the $g$ meson. Finally, the total cross section at high energies can be included in the bootstrap by making the Freund-Harari hypothesis, and bodily adding on a Pomeranchuk trajectory. The results are in reasonably good agreement with experiment.