Crossing-Symmetry Restrictions on Dispersion Relations, and Sum Rules forππScattering Lengths

We study the constraints crossing symmetry imposes on fixed-variable dispersion relations for $\pi \pi$ scattering. We show that the sum rules relating $2a_0^0-5a_0^2-18a_1^1,a_2^0$, and $a_2^2$ to the total cross sections, which were derived by Wanders using the Mandelstam representation, follow from twice-subtracted dispersion relations. These sum rules are good physical-region constraints to supplement the unphysical-region constraints of Martin and Roskies in the study of models for low-energy $\pi \pi$ scattering. Using a restriction on the absorptive parts following from crossing symmetry, we transform Wander's sum rule for the $I=0\mathrm{}$, $l=2\mathrm{}$ scattering length into a form which is manifestly positive. Keeping only the $S$- and $P$- wave contributions, we obtain a lower bound for $a_2^0$. If the $\rho$-trajectory intercept is less than 1, we show that ${\lim }_{s\to \infty }\mathrm{Re}{T}^I\frac{(s,0,4\mathrm{}-s)}{s}$ is determined by the total cross sections. If, in addition, the leading isospin-2 trajectory has intercept less than zero, then even without imposing elastic unitarity, the $I=0\mathrm{} S$ wave is determined by the absorptive parts without the freedom of adding an arbitrary constant.