Off-Mass-Shell Theory of the Bootstrap and Composite-Particle Scattering

Previous work on a crossing-symmetric bootstrap model for the asymptotic vertex function is generalized to an analysis of the asymptotic behavior of composite-particle scattering amplitudes. Using off-mass-shell integral equations, but making no reference to field theory, a complete, unitary, crossing-symmetric bootstrap theory is constructed and a consistent asymptotic solution for all of the $n$-particle amplitudes is obtained. The asymptotic prediction for the two-body scattering amplitude, valid in the limit in which $\mathrm{}\left|s\right|\mathrm{}$, $\mathrm{}\left|t\right|\mathrm{}$, and $\mathrm{}\left|u\right|\mathrm{}$ approach infinity, is compared with the qualitative features of the wide-angle proton-proton scattering data, and a severe restriction is thereby placed on the asymptotic rate of decrease of the hadronic vertex functions. This restriction is consistent with a new lower bound, $F\left(t\right)\mathrm{}>~c{e}^{-b{\mathrm{}(-t)\mathrm{}}^{\frac{1}{4}}}$ for $t\to \infty$, on the hadronic form factors, which is derived here under the assumption of the Wick rotatibility of the Bethe-Salpeter equation. While previous nonrelativistic "bootstrap" models predict $F\left(t\right)=c{e}^{-b{\mathrm{}(-t)\mathrm{}}^{\frac{1}{2}}}$, the fourth-root behavior is shown to be consistent with the relativistic bootstrap theory.