One-loop effective action in the σ-model for a periodic chiral field ansatz

One-fermion and one-boson loop contributions to the effective action in the Gell-Mann - Levy σ-model are calculated analytically for the case of a periodic chiral field ansatz. Accuracy and limits of validity of various approximate methods used in the effective action formalism are tested quantitatively on this example, in particular the convergence of the gradient expansion. Vacuum instability of the theory at the one-loop level is found, in agreement with earlier results for solitons. 39 refs., 7 figs. (author)