Crossing-Symmetry Sum Rules Applied to Brown and Goble's Model forππScattering

We use the sum rules of Roskies to introduce constraints following from crossing symmetry into Brown and Goble's model for low-energy pion-pion scattering. The $S$ and $P$ waves are parametrized in a way consistent with elastic unitarity. We do not modify Brown and Goble's $P$ wave, which exhibits a $\rho$ meson at 750 MeV of width 115 MeV. We introduce a total of five parameters into the $S$ waves; these are fixed in a well-defined way by the five sum rules relating the $S$ and $P$ waves. The resulting isospin-0 $S$-wave phase shift passes through 90° at 800 MeV. The isospin-2 $S$-wave phase shift is small and negative. The $S$-wave scattering lengths are determined to be $a_0^{}{}_{}{}^0=0.105\mathrm{}m_{\pi }^{}{}_{}{}^{-1\mathrm{}}$ and $a_0^{}{}_{}{}^2=-0.051\mathrm{}m_{\pi }^{}{}_{}{}^{-1\mathrm{}}$. The ${\pi }^0{\pi }^0$ amplitude satisfies Martin's inequalities, which follow from crossing symmetry and the positivity of the crossed-channel absorptive part.