Nonlinearσmodel in the loop expansion

The nonlinear $\sigma$ model in four dimensions is discussed in the context of the loop expansion. Since the model is perturbatively nonrenormalizable, divergences not of the form of the Lagrangian are of course expected; what is perhaps surprising is that there are divergences which appear not to be invariant under the original nonlinear symmetry. We demonstrate, however, that these apparently noninvariant terms do not contribute to on-mass-shell quantities and may be eliminated order by order by a field redefinition involving space-time derivatives. The linear $\sigma$ model is then examined in detail; it is shown how the nonlinear model, including the apparently noninvariant terms, emerges as the limit of the linear model as the $\sigma$ mass goes to infinity. Finally, we compare our approach with other treatments of the "noninvariant" terms in the nonlinear model.