Can a logarithmically running coupling mimic a string tension?

It is shown that a Coulomb potential using a running coupling slightly modified from the perturbative form can produce an interquark potential that appears nearly linear over a large distance range. Recent high-statistics SU(2) lattice gauge theory data fit well to this potential without the need for a linear string-tension term. This calls into question the accuracy of string tension measurements which are based on the assumption of a constant coefficient for the Coulomb term. It also opens up the possibility of obtaining an effectively confining potential from gluon exchange alone. It is surprising that the interquark potential for pure-gauge SU(2) and SU(3) lattice gauge theories fits as well as it does to a simple linear+Coulomb law. Even the extremely high statistics SU(2) results of the UKQCD collaboration at β = 4/g = 2.85, which includes distances to R/a = 24 and with a relatively small physical lattice spacing, a, (a ≃ 6.56 GeV) which probes well both short and long distances, requires no additional terms to fit the data [1]. What is surprising about this is that one of the most definite predictions of perturbation theory, backed up by high-energy scattering experiments, is that the effective coupling is a running coupling, so one would expect the coefficient of the Coulomb term, which can be taken to be a renormalized coupling, to depend upon distance. At weak couplings corresponding to short distances this should match the logarithmic dependence given by renormalization group improved low-order perturbation theory. If lattice gauge theory is to be successfully matched onto perturbation theory, then at least the short distance part of the potential should be allowed to run. This has been tried and gives indications of a reasonable match to perturbation theory [2]. For longer distances (say R/a ≃ 6 on the above lattice) the coupling is generally assumed to stop running, to allow an accurate determination of the string tension. However, the fits which show a running coupling at shorter distances show no indication that the running is slowing down. The stopping of the running coupling has been justified by the strong-coupling string model of Lüscher which predicts the coefficient of 1/R in the potential to be the constant value of π/12 [3]. The problem with this is that the couplings for which this string picture become valid are probably much stronger than those of the simulations being discussed here [4]. There is very little