Consistency of the chiral pion-pion scattering amplitudes with axiomatic constraints

The pion-pion scattering amplitudes provided by second-order chiral perturbation theory are confronted with known rigorous constraints derived from the axioms of quantum field theory. We mainly test constraints restricting the ${\mathrm{\pi }}^0$ ${\mathrm{\pi }}^0$ S- and D-wave amplitudes in the unphysical interval 0≤s≤4${\mathit{m}}_{\mathrm{\pi }}^2$. These constraints impose significant lower bounds for a linear combination of coupling constants specifying the second-order chiral Lagrangian. The accepted value of this combination is consistent with these bounds. The ${\mathrm{\pi }}^{0\mathrm{-}}$ ${\mathrm{\pi }}^0$ S and D waves are strongly correlated by a set of constraints.