Dispersion relation bounds forππscattering

Axiomatic principles such as analyticity, unitarity, and crossing symmetry constrain the second derivative of the $\pi \pi$ 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 ${\overline{l}}_1$ and ${\overline{l}}_2$. We compare our predictions with those derived previously in the literature using similar methods. We compute the one-loop $\pi \pi$ scattering amplitude in the linear sigma model (LSM) using the $\overline{\mathrm{MS}}$ scheme, a result hitherto absent in the literature. The LSM values for ${\overline{l}}_1$ and ${\overline{l}}_2$ violate the bounds for small values of $m_{\sigma }/m_{\pi }$. We show how this can occur, while still being consistent with the axiomatic principles.