Simultaneous Bootstrap of theρandf0inππScattering

A self-consistent calculation of the parameters of the $\rho$ and $f^0$ resonances is carried out, using an effective-potential approach. The exchange of these resonances in $\pi \pi$ scattering is used to construct a lowest order effective potential at each energy. This is sufficient to produce these same resonances in the scattering process. The requirement that the masses and reduced widths of these output resonances be the same as those of the exchanged resonances can then be used to fix these parameters. With 10% self-consistency, we obtained mass ≈600 MeV, width ≈100 MeV for the $\rho$, and mass ≈1100 MeV, width ≈90 MeV for the $f^0$. The width of the $\rho$ thus comes out considerably narrower than in the usual calculations which ignore the $f^0$.