The operator product expansion, non-perturbative couplings and the Landau pole: lessons from the O(N) σ-model

We obtain the operator product expansion of the self-energy in the O(N) non-linear σ-model to all orders in the coupling and the large momentum, and to next-to-leading order in 1/N. In the light of this result we discuss recent suggestions that there may be additional power corrections from short distances, associated with defining the coupling constant non-perturbatively. The non-linear σ-model provides no evidence for such `non-standardʹ power corrections. We also find that the OPE converges for sufficiently large external momentum, presumably because there are no multi-particle thresholds at arbitrarily high energies in the 1/N expansion.