Consistency of the Chiral Pion-Pion Scattering Amplitudes with Axiomatic Constraints

TL;DR

This work tests second-order chiral perturbation theory for $\pi\pi$ scattering against axiomatic QFT constraints (analyticity, positivity, crossing) within the low-energy triangle $\Delta$ in the Mandelstam plane. By employing fixed-$t$ dispersion relations, positivity of absorptive parts, and crossing, the authors derive lower bounds on the combination $\bar l$ of second-order couplings and reveal strong correlations between the $\pi^0$-$\pi^0$ $S$- and $D$-waves. The analysis shows that $\bar l$ must exceed a modest value (around 6) and that the physical value $\bar l\approx 21$ not only satisfies these bounds but is also consistent with sum-rule estimates and $S$–$D$-wave gating, highlighting robust internal consistency of the CPT framework with fundamental axioms. Overall, the study demonstrates that the second-order CPT amplitudes are compatible with axiomatic constraints and reveals tight S–D wave correlations that constrain higher-order corrections and guide phenomenology.

Abstract

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