Cutoff Effects in O(N) Nonlinear Sigma Models

TL;DR

This paper investigates cutoff effects in nonlinear O(N) sigma models, revealing that observed anomalies are due to large cutoff effects with significant logarithmic components, which diminish at larger N, supported by exact large N analysis.

Contribution

The study provides a detailed analysis of cutoff effects in O(N) sigma models, demonstrating their consistency with Symanzik's asymptotic form and offering exact large N results for nonperturbative control.

Findings

Cutoff effects follow Symanzik's asymptotic form.

Large N results show diminishing cutoff effects.

Observed anomalies are due to large cutoff effects, not true anomalies.

Abstract

In the nonlinear O(N) sigma model at N=3 unexpected cutoff effects have been found before with standard discretizations and lattice spacings. Here the situation is analyzed further employing additional data for the step scaling function of the finite volume massgap at N=3,4,8 and a large N-study of the leading as well as next-to-leading terms in 1/N. The latter exact results are demonstrated to follow Symanzik's form of the asymptotic cutoff dependence. At the same time, when fuzzed with artificial statistical errors and then fitted like the Monte Carlo results, a picture similar to N=3 emerges. We hence cannot conclude a truly anomalous cutoff dependence but only relatively large cutoff effects, where the logarithmic component is important. Their size shrinks at larger N, but the structure remains similar. The large N results are particularly interesting as we here have exact nonperturbative control over an asymptotically free model both in the continuum limit and on the lattice.