The puzzle of apparent linear lattice artifacts in the 2d non-linear sigma-model and Symanzik's solution

TL;DR

This paper investigates the unexpected linear lattice artifacts in the 2d O(n) non-linear sigma-model, explaining them through logarithmic corrections within Symanzik's effective action framework, supported by Monte Carlo simulations.

Contribution

It provides a detailed calculation showing how large logarithmic corrections cause apparent linear artifacts, resolving a longstanding puzzle in lattice field theory.

Findings

Logarithmic corrections (ln(a))^3 explain the observed artifacts.

Monte Carlo data supports the presence of these corrections.

Theoretical framework aligns with numerical results.

Abstract

Lattice artifacts in the 2d O(n) non-linear sigma-model are expected to be of the form O(a^2), and hence it was (when first observed) disturbing that some quantities in the O(3) model with various actions show parametrically stronger cutoff dependence, apparently O(a), up to very large correlation lengths. In a previous letter we described the solution to this puzzle. Based on the conventional framework of Symanzik's effective action, we showed that there are logarithmic corrections to the O(a^2) artifacts which are especially large, (ln(a))^3, for n=3 and that such artifacts are consistent with the data. In this paper we supply the technical details of this computation. Results of Monte Carlo simulations using various lattice actions for O(3) and O(4) are also presented.