The Low Energy Expansion for Pion-Pion Scattering and Crossing Symmetry in Dispersion Relations

We show that a suitable setting for comparison of the low-energy representation for pion-pion scattering amplitudes, with dispersive representation for these amplitudes, is provided by certain manifestly crossing symmetric dispersion relations. We begin with a discussion of fixed-t dispersion relations and discuss the origin of crossing constraints that arise in this context when we consider resonance saturation with certain $l\geq 2$ states. We demonstrate that the approach advocated here does not require us to enforce such constraints. Our results are contrasted with those from fixed-t dispersion relations. We finally discuss the numerical import of our results.