Unsubtracted Pion-Pion Dispersion Relation

The de Alfaro-Fubini-Furlan-Rossetti assumption on the rapid falloff of amplitudes corresponding to pure $I=2\mathrm{}$, $t$-channel exchange is applied to pion-pion forward dispersion relations, and a sum rule is obtained for the pion-pion scattering lengths. The sum rule is fairly well satisfied using the $\rho$, $f$, and low-energy $s$-wave contributions; but the inclusion of the recently discovered $g_1$ resonance requires the existence of at least one new $I=0\mathrm{}$ resonance to satisfy the sum rule.