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

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 = 5, 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 alpha(beta) in units of Lambda((MS) over bar). To resolve this ambiguity we turn to SU(3) where we use the Schrodinger Functional coupling scheme to extract a value of r(0)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 a(2)sigma that have been calculated for 2 <= N <= 8, to obtain Lambda MS/root sigma = 0.503(2)(40) + 0.33(3)(3)/N-2 for N = 3, where the first error is statistical and the second is our estimate of the systematic error from all sources.