Chiral perturbation theory to one loop

Abstract

The Green's functions of QCD are expanded in powers of the external momenta and of the quark masses. The Ward identities of chiral symmetry determine the expansion up to and including terms of order p 4 (at fixed ratio m quark p 2 ) in terms of a few constants, which may be identified with the coupling constants of a unique effective low energy Lagrangian. The low energy representation of several Green's functions and form factors and of the ππ scattering amplitude are then calculated. The values of the low energy coupling constants are extracted from available experimental data. The corrections of order M π 2 to the ππ scattering lengths and effective ranges turn out to be substantial and the improved low energy theorems agree very well with the measured phase shifts. The observed differences between the data and the uncorrected soft pion theorems may even be used to measure the scalar radius of the pion, which plays a central role in the low energy expansion.