ρ-f⁰ Double Bootstrap by the Balázs Method

We have studied the simultaneous bootstrap of the ρ and f⁰ mesons, using the Balázs method. When we include the exchange of an f⁰ with physical parameters in the input force, we find that we obtain an approximately self-consistent ρ with a mass of 710 MeV and full width of 100 MeV; the degree of self-consistency is somewhat improved if the width is reduced to 66 MeV. One can obtain a simultaneous boot-strap of both the ρ and f⁰ with a ρ mass and full width of 710 and 46 MeV, and f⁰ mass and full width of 1344 and 112 MeV, respectively. We believe that the double-bootstrap results are somewhat affected by our approximate treatment of inelastic effects, which involves a discontinuity in the inelasticity function R_l^I near the position of the f⁰. These results are based upon taking n, the number of poles approximating the left-hand cut, equal to 7. The use of a large number of poles is important, and it is not at all clear from the n dependence of our calculations that we have yet obtained a limiting form of the results as n becomes large, even with n=7.