On the running of the bare coupling in SU(N) lattice gauge theories

TL;DR

This paper investigates the running of the bare coupling in SU(N) lattice gauge theories, analyzing the transition between strong and weak coupling regimes, and compares different schemes to accurately determine the scale parameter Lambda_MSbar.

Contribution

It introduces a method to fit string tension data across coupling regimes and compares coupling schemes, favoring the Parisi mean-field scheme for better accuracy in SU(N) gauge theories.

Findings

The Parisi scheme closely reproduces the Schrödinger Functional benchmark.

A universal relation for Lambda_MSbar/sqrt(sigma) across N>2 is established.

The first order bulk transition simplifies the analysis for N > 4.

Abstract

Interpreting the way that the SU(3) bare lattice coupling runs with the lattice spacing is complicated by the fact that there is a smooth cross-over region in which the strong coupling expansion transforms into a weak-coupling one. For N > 4, however, there is a first order bulk transition that cleanly separates the strong and weak coupling regimes. We find that in this case the calculated string tension can be readily fitted throughout the weak coupling region by a standard 3-loop expression modified by lattice spacing corrections of the expected form. While our fits demand the presence of the latter, they do not constrain the perturbative coupling scheme enough to enable us to extract a usefully accurate value of a(beta) in units of Lambda_MSbar. To resolve this ambiguity we turn to SU(3) where we use the Schrodinger Functional coupling scheme to extract a value of r_0 x Lambda_SF as a benchmark. We then find that the Parisi mean-field improved coupling scheme closely reproduces this result. We also develop a comparison between different schemes that does not rely on the calculation of any physical quantity and which can therefore be applied much further into weak coupling. Again the Parisi scheme is favoured over the others that we compare. Using the mean-field scheme we have fitted the values of the string tension that have been calculated for N=2 to N=8, to obtain Lambda_MSbar/sqrt(sigma) = 0.503(2)(40) + 0.33(3)(3)/N^2 for N > 2, where the first error is statistical and the second is our estimate of the systematic error from all sources.