Positivity constraints on chiral perturbation theory pion-pion scattering amplitudes

We test the positivity property of the chiral perturbation theory (ChPT) pion-pion scattering amplitudes within the Mandelstam triangle. In the one-loop approximation $\mathcal{O}{(p}^{4})$ the positivity constrains only the coefficients ${b}_{3}$ and ${b}_{4};$ namely, one obtains that ${b}_{4}$ and the linear combination ${b}_{3}{+3b}_{4}$ are positive quantities. The two-loop approximation gives inequalities involving all six arbitrary parameters entering ChPT amplitude, but the corrections to the one-loop approximation results are small. ChPT amplitudes pass unexpectedly well all the positivity tests, giving strong support to the idea that ChPT is a good theory of low-energy pion-pion scattering.