Bootstrap of Pion-Pion Scattering in the Unitarized Strip Approximation

We give the result of a bootstrap of the $\rho$ and Pomeranchuk trajectories in pion-pion scattering, using the unitarized strip approximation. In this scheme, we construct an analytic, crossing-symmetric, unitary amplitude with Regge asymptotic behavior. This is achieved by starting with strips in which the double spectral function is parametrized by Regge poles, in the way devised by Chew and Jones, and adding the elastic double spectral functions in each channel calculated by the Mandelstam iteration method. Unitarity is imposed using the $\frac{N}{D}$ method with inelasticity. We find self-consistent trajectories which bear a fair resemblance to those found experimentally, at least for small $\left|t\right|$, and obtain a $\rho$ of width 155 MeV, which is very much better than earlier calculations. This and related improvements are due to the proper inclusion of Pomeranchon exchange. The requirement of self-consistency does not fix the trajectories uniquely, but the range of solutions is comparatively narrow, with $0.32\le {\alpha }_{\rho }\mathrm{}\left(0\right)\mathrm{}\le 0.69$. There are still some unsatisfactory features in comparison with the phenomenological trajectories: We always find that ${\alpha }_{\rho }\left(\infty \right)>-1\mathrm{}$, there is a too rapid increase of $\mathrm{Im}\alpha$ just above threshold, and the trajectories do not rise much above $\mathrm{Re}\alpha =1\mathrm{}$. We draw some conclusions about the prospects for this sort of $S$-matrix dynamics.