Srednicki — Quantum Field Theory · Problem 83.4

Tree-level pion scattering is governed by the chiral Lagrangian's four-point interaction and s-, t-, and u-channel tree diagrams. The key insight is that isospin symmetry lets you write the amplitude in terms of three independent kinematic structures (corresponding to the three Mandelstam channels), each weighted by a pole or contact term. You'll combine the contributions from all three channels, then express the result using $s$, $t$, $u$ and $m_\pi$ to reveal the underlying crossing symmetry and soft-pion structure of the amplitude.

Tree-level pion scattering is governed by the chiral Lagrangian's four-point interaction and s-, t-, and u-channel tree diagrams. The key insight is that isospin symmetry lets you write the amplitude in terms of three independent kinematic structures (corresponding to the three Mandelstam channels), each weighted by a pole or contact term. You'll combine the contributions from all three channels, then express the result using $s$, $t$, $u$ and $m_\pi$ to reveal the underlying crossing symmetry and soft-pion structure of the amplitude.