Dispersion relation bounds for ππ scattering

Axiomatic principles such as analyticity, unitarity, and crossing symmetry constrain the second derivative of the ππ scattering amplitudes in some channels to be positive in a region of the Mandelstam plane. Since this region lies in the domain of validity of chiral perturbation theory, we can use these positivity conditions to bound linear combinations of l1 and l2. We compare our predictions with those derived previously in the literature using similar methods. We compute the one-loop ππ scattering amplitude in the linear sigma model (LSM) using the MS scheme, a result hitherto absent in the literature. The LSM values for l1 and l2 violate the bounds for small values of mσ/mπ. We show how this can occur, while still being consistent with the axiomatic principles.