A Sum Rule Based on Unitarity

Unitarity, analyticity, Regge asymptotic behavior, and a resonance approximation are combined to derive a new sum rule. The sum rule is very convergent; the contribution of high-mass resonances is suppressed by a decreasing weight function. The spin-flip and non-spin-flip residues of the $\rho$ meson in the $\pi \pi \to N\overline{N}$ amplitude are evaluated at the mass of the $\rho$, and in conjunction with the first-moment finiteenergy sum rule, a calculation of the $\rho$-meson mass is performed. The results are in good agreement with experiment. A calculation of the $\rho$ and $f_0$ resonance parameters in the $\pi \pi \to \pi \pi$ amplitude is also discussed.