Lattice artefacts and the running of the coupling constant

Abstract

We study the running of the Lüscher-Weisz-Wolff (LWW) coupling constant in the two dimensional O(3) nonlinear $\sigma$ model. To investigate the continuum limit we refine the lattice spacing from the $\frac{1}{16}$ value used by LWW up to $\frac{1}{160}$. We find that the lattice artefacts are much larger than estimated by LWW and that most likely the coupling constant runs slower than predicted by perturbation theory. A precise determination of the running in the continuum limit would require a controlled ansatz of extrapolation, which, we argue, is not presently available.

Comments: 4 pages, 4 figures. To address the criticism that we are studying a different quantitiy than Luscher, Weisz and Wolff originally did, we introduced a new equation (2), a new paragraph discussing this issue and a new figure comparing the results obtained with our prescription to that obtained with the original one of Luscher, Weisz and Wolff