The non-linear sigma model: 3-loop renormalization and lattice scaling

We present a perturbative calculation of the first non-universal terms of the renormalization group functions for the non-linear σ-model regularized on a lattice, for a choice of different lattice actions. Such terms, hitherto unknown, fix the leading corrections to universal asymptotic scaling. However, by comparing with Monte Carlo simulations, one sees that the actual deviations from universal scaling are due to non-perturbative effects, and not to perturbative analytic corrections. The latter are nonetheless relevant to determine the ratios of scaling operators to Λ-parameters.