Some Applications of a Simple Approach to Unitarization in Hard-Meson Calculations

Several applications of a simple approach to unitarization in the case of the three-point functions of hard-meson theory are made. For the first of these, effective-range formulas are obtained for the pion form factor in the timelike region for both hard-pion current-algebra ($\delta =-\frac{1}{2}$) and Veneziano models. Next, the extension of this procedure to the case of several coupled channels is used to obtain an effective-range formula for the pion form factor which includes the small effect of the closed $K\overline{K}$ channel. Finally, an expression for the pion form factor is derived which allows for the inclusion of some nonresonant $p$-wave ($I=1\mathrm{}$) $\pi \pi$ scattering. Since this component is parametrized via the introduction of an additional (nonresonant) $p$-wave scattering length, the resulting expression for the pion form factor generates a one-parameter family of fits to the experimental data. The data appear to be consistent with a small positive nonresonant scattering length and favor a value of ${\Gamma }_{\rho }$ slightly less than 124 MeV.