Bootstrap Calculations ofππScattering Using the Mandelstam Iteration

Abstract

The results of some strip-model calculations of $\ensuremath{\pi}\ensuremath{\pi}$ scattering are presented. In these calculations, unitarity is imposed by means of the Mandelstam iteration. This procedure has the advantage that the output trajectories and residues (including their imaginary parts) may be computed above as well as below threshold; this is at present not feasible in calculations using the $\frac{N}{D}$ technique. First, a bootstrap calculation of the $\ensuremath{\rho}$ trajectory is carried out, neglecting Pomeranchuk exchange. The extra requirement of self-consistency above threshold is very strict, but a solution with satisfactory consistency between -1 and +2 Ge${\mathrm{V}}^{2}$ is obtained. The scale of energy is established by giving the $\ensuremath{\rho}$ resonance the physical mass; the self-consistent $\ensuremath{\rho}$ width is then about 400 MeV. Various dynamical approximations are investigated, and it is shown explicitly that the pion mass is not a significant parameter of the dynamics. Finally, a bootstrap calculation of both the $\ensuremath{\rho}$ and Pomeranchuk trajectories is presented. Except for the Pomeranchuk residue, the results show satisfactory self-consistency throughout the range -1 to +2 Ge${\mathrm{V}}^{2}$. The self-consistent Pomeranchuk trajectory has an intercept ${\ensuremath{\alpha}}_{P}(0)\ensuremath{\approx}1$ and a slope $\ensuremath{\alpha}_{P}^{}{}_{}{}^{\ensuremath{'}}(0)\ensuremath{\approx}0.5$ Ge${\mathrm{V}}^{\ensuremath{-}2}$. The inclusion of Pomeranchuk exchange increases the slope of the $\ensuremath{\rho}$ trajectory, but the effect of this on the $\ensuremath{\rho}$ resonance width is offset by a more rapid increase in the imaginary part of the trajectory, and the self-consistent $\ensuremath{\rho}$ width is now about 600 MeV. In contrast to the more encouraging consequences of including Pomeranchuk exchange in recent $\frac{N}{D}$ calculations, these results suggest that the physical $\ensuremath{\rho}$ is in fact primarily a bound state of some channel other than $\ensuremath{\pi}\ensuremath{\pi}$.