ρππRegge Residue Function from a New Class of Sum Rules, and a New Representation forππAmplitudes

A new class of sum rules is developed within the context of the $\pi \pi$ system. One of these sum rules is used to extract the $\rho \pi \pi$ Regge residue function from the recent $\pi \pi$ data of Carroll et al., for a wide range of momentum transfer. The residue has a zero near $t=-0.5\mathrm{}$ Ge${\mathrm{V}}^2$, in agreement with duality and with a new prediction which we make from the dual absorption model together with $\pi p$ and $\overline{p}p$ data. The $\pi \pi$ charge-exchange amplitude is shown to be dominated by $\rho$ exchange above 1.0 GeV. We also derive a new representation which expresses $\pi \pi$ amplitudes in terms of a single subtraction parameter and integrals over physical-region absorptive parts. The representation is valid over a substantial portion of the physical region, and constitutes a powerful tool for studying the $\pi \pi$ interaction. Finally, we show that a unitarized Veneziano model proposed earlier by the present author is an excellent approximation to nature below about 800 MeV, despite its neglect of Pomeranchon exchange and $K\overline{K}$ production.