Theory of Reggeized Bootstraps

Using a representation for the partial-wave S matrix developed recently by the authors, a representation for the total scattering amplitude is constructed with Regge trajectories as the sole input. The resulting representation has the following properties regardless of the number of trajectories included: (a) It has the correct analytic structure; namely, the double spectral function is nonvanishing in a region bounded by the curved boundaries implied by the Mandelstam representation. (b) It is unitary. (c) It gives the correct threshold behavior for both the real and the imaginary parts of the total amplitude. (d) It converges rapidly. A potential theory example is worked out explicitly and with one trajectory input, numerically an excellent agreement with the exact result for the total amplitude is obtained. The representation is then generalized to the relativistic case. Invoking crossing, a set of relatively simple bootstrap equations is obtained for a self-consistent determination of the Regge trajectories. Also, an alternative procedure is proposed which imposes approximate crossing symmetry in the unphysical region to determine the Regge parameters. The latter method is operationally more attractive since it obviates solving any dispersion integrals, and also has the added advantage of avoiding inelasticity. Finally, in this method, no statements (or approximations) regarding the $I=2$ channel need be made, since the latter can be eliminated from the appropriate crossing relation.