The paper by Gasser and Leutwyler explores the expansion of Green's functions in QCD up to and including terms of order \( p^4 \) (at fixed ratio \( m_quark/p^2 \)) using Ward identities of chiral symmetry. They identify a few constants that can be identified with coupling constants of an effective low-energy Lagrangian. The authors calculate the low-energy representation of various Green's functions, form factors, and the ππ scattering amplitude, extracting the values of the low-energy coupling constants from experimental data. They find that corrections of order \( M^2 \) to the ππ scattering lengths and effective ranges are significant, and the improved low-energy theorems agree well with measured phase shifts. The observed differences between data and uncorrected soft pion theorems can be used to measure the scalar radius of the pion, which plays a central role in the low-energy expansion. The paper also discusses the symmetries of Green's functions, anomalies, and the general form of the effective Lagrangian to order \( p^4 \). It addresses the renormalization of the nonlinear σ-model and the evaluation of the Feynman path integral in the one-loop approximation.The paper by Gasser and Leutwyler explores the expansion of Green's functions in QCD up to and including terms of order \( p^4 \) (at fixed ratio \( m_quark/p^2 \)) using Ward identities of chiral symmetry. They identify a few constants that can be identified with coupling constants of an effective low-energy Lagrangian. The authors calculate the low-energy representation of various Green's functions, form factors, and the ππ scattering amplitude, extracting the values of the low-energy coupling constants from experimental data. They find that corrections of order \( M^2 \) to the ππ scattering lengths and effective ranges are significant, and the improved low-energy theorems agree well with measured phase shifts. The observed differences between data and uncorrected soft pion theorems can be used to measure the scalar radius of the pion, which plays a central role in the low-energy expansion. The paper also discusses the symmetries of Green's functions, anomalies, and the general form of the effective Lagrangian to order \( p^4 \). It addresses the renormalization of the nonlinear σ-model and the evaluation of the Feynman path integral in the one-loop approximation.