Self-ConsistentρTrajectory in the New Form of the Strip Approximation

The new form of the strip approximation, devised by Chew, is applied to the problem of "bootstrapping" a $\rho$ trajectory in the $\pi -\pi$ system. Even in the absence of other trajectories it is possible to obtain a self-consistent $\rho$ trajectory and residue functions $\alpha \mathrm{}\left(t\right)\mathrm{}$ and $\gamma \mathrm{}\left(t\right)\mathrm{}$ for $t<\mathrm{}0\mathrm{}$, with strip widths in the range 150 to 300 $m_{\pi }^{}{}_{}{}^2$. A particular example is given in detail. The absence of the force from other trajectories, and the rapid variation of $\alpha \mathrm{}\left(t\right)\mathrm{}$ and $\gamma \mathrm{}\left(t\right)\mathrm{}$ for $t>~0$, mean that our results can not represent the real $\rho$ trajectory, but at least they confirm the viability of the methods used.