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 $\pi^0$-$\pi^0$ $S$- and $D$-wave amplitudes in the unphysical interval $0\leq s\leq 4m_\pi^2$. 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 $\pi^0$-$\pi^0$ $S$- and $D$-waves are strongly correlated by a set of constraints.