Double Bootstrap of theϱandf0

We obtain a "double bootstrap" of the $\rho$ and $f^0$ mesons, in which the input force consists of both $\rho$ and $f^0$ exchange, and both particles are obtained as ouptut resonances, with corresponding input and output values of masses and widths being equal to within about 1%. The values obtained for the resonance parameters are $m_{\rho }=750\mathrm{}$ MeV, ${\Gamma }_{\rho }=162\mathrm{}$ MeV, $m_f=1240\mathrm{}$ MeV, and ${\Gamma }_f=100\mathrm{}$ MeV, in reasonable agreement with experiment. The $\rho$ width is somewhat larger than most measured values, but much smaller than generally obtained in bootstrap calculations in which only $\rho$ exchange is included as an input force. Our calculations are carried out using the equivalent-potential method, and are free of arbitrary parameters. The model yields a second $I=0\mathrm{}$, $j=2\mathrm{}$ resonance, presumably to be associated with the $f^{0\prime }$, and also a broad second $P$-wave resonance. The latter may correspond to the ${\rho }^{\prime }$ Regge trajectory hypothesized in several Regge-pole analyses of high-energy data, especially the polarization in $\pi N$ charge exchange. The parameters of the predicted resonance do not agree with those of any known resonance, but it might be difficult to observe because of its width. The output Regge trajectories predicted by the model are roughly linear. The $\rho$ trajectory has a slope about half the generally accepted experimental value of 1 Be${\mathrm{V}}^{-2\mathrm{}}$. We comment in passing that general considerations, based only on the crossing matrix, make it somewhat difficult to reconcile the latter value with the absence of an $I=0\mathrm{}$ $D$-wave resonance at an energy less than 1250 MeV.