Low-energy expansion for pion-pion scattering and crossing symmetry in dispersion relations

We show that a suitable setting for a comparison of the low-energy representation for pion-pion scattering amplitudes, with a dispersive representation for these amplitudes, is provided by certain 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 $l>~2$ 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.