Numerical Solution of the Pion-Pion Strip-ApproximationNDEquation

Chew's strip-approximation $\frac{N}{D}$ equations have been solved numerically with a generalized potential of the form corresponding to elementary-particle $\rho$ exchange plus the contribution from Pomeranchon exchange required by the fact that the phase shift at the strip boundary is generally nonzero. Trajectories and reduced residue functions are found. The trajectories have reasonable shapes, slopes in agreement with experiment [${\alpha }^{\prime }\approx 0.3/{(\mathrm{BeV}/c)}^2$] for reasonably chosen strip widths, and end points in the region of $l\gtrsim 0$. Increasing the phase shift at the strip boundary displaces trajectories upward, while increasing the strip width tends to flatten trajectories. The behavior of the reduced residues is found to be representable by a simple approximate formula in terms of the input potential. The potential investigated, with neglect of inelastic scattering, is incapable of generating a $J=2\mathrm{}$, $I=0\mathrm{}$ resonance and yields a $\rho$ width several times too large. Increasing the phase shift at the strip boundary tends to improve the situation.