Simple Approach to Unitarization in Hard-Meson Calculations

The hard-pion effective-range formula for the pion form factor of Brehm, Golowich, and Prasad is derived from unitarity and (vector) meson dominance assuming a real tree form factor as input. The result is generalized and applied to the $\pi -A_1-\rho$ system. The suitably continued finite $\rho$-width form factors are used to calculate the soft ${\pi }^{+}-{\pi }^0$ mass difference when (a) both tree form factors are unsubtracted in broken chiral symmetry ($\delta m\simeq 5.7$ MeV) and when (b) both tree form factors are subtracted ($\delta =-\frac{1}{2}$) ($\delta m\simeq 5.2$ MeV).