Low-Energy Theorems for Renormalizable Field Theories

Abstract

A general method is described whereby the low-energy behavior of collision processes can be investigated systematically. The basic procedure is the construction of the renormalized Green's functions for the participating particles in the presence of external sources. For processes involving the collision of a boson with a fermion,—photon scattering, meson scattering, and photoproduction of mesons—one requires the propagation function of a nucleon in the presence of external electromagnetic and meson fields, as well as the propagator for a meson in an electromagnetic field. The rather direct relation of these functions to the various transition matrices is derived. The possible forms of the Green's functions are limited by the principles of Lorentz, gauge, and charge conjugation invariance. Their detailed structure is determined in part by the requirement that they describe particles of known mass, charge, magnetic moment, and mesic coupling constant. These conditions are sufficient to imply a number of theorems. For example, it is shown that the amplitude for photon-proton scattering is given correctly up to terms of first order in the photon energy by a Born approximation calculation, if one assigns to the proton its experimental charge and magnetic moment. If one includes the meson-nucleon coupling constant in this set of parameters, then perturbation theory is also valid for the leading terms in a momentum-energy expansion of the $P$-wave in meson-nucleon scattering, the $S$- and $P$-waves associated with the nucleon current in photomeson production, and the entire meson current for the same process.