ar X iv : h ep - l at / 9 41 20 77 v 1 1 4 D ec 1 99 4 1 The Perturbative Method Fails in Non - Abelian Models ∗

It is shown that perturbation theory in 2D nonlinear σ-models as well gauge theories in dimension D ≥ 2 produces answers that depend on boundary conditions even after the infinite volume limit has been taken. This unphysical phenomenon occurs only in the non-Abelian versions of those models, starting at O(1/β 2). It is not present in the true (nonperturbatively defined) models and represents a failure of the perturbative method. It is related to a hitherto unnoticed type of low-lying excitation, dubbed super-instanton, that dominates the low-temperature (= weak coupling) regime of these models.