ar X iv : h ep - l at / 0 60 60 19 v 1 2 9 Ju n 20 06 SU ( 3 ) gauge theory at finite temperature in 2 + 1 dimensions

The SU(3) gauge theory in 2+1 dimensions is simple enough from a numerical point of view, so that it is possible with present computers to make continuum extrapolations with controlled systematic errors. Of course, there are some obvious differences between SU(3) gauge theory in 2 + 1 and 3 + 1 dimensions. In 2 + 1 dimensions the coupling constant g has the dimension of a mass, and the theory is superrenormalizable. The tree level potential between heavy quarks is already logarithmically confining: V (r) ∼ log r. There are, however, many similarities. One may introduce a dimensionless “running” coupling constant g3(l) by the definition g 2 3(l) ≡ lg 2 where l is a length scale. Then g 3(l) → 0 for l → 0 and to infinity for l → ∞. This is somewhat analogous to the logarithmically running coupling constant in 3 + 1 dimensional SU(3) gauge theory. In 2 + 1 dimensions the coupling constant g sets the scale, and mi = cig , where ci’s are numerical constants. From Monte Carlo simulations one knows some further properties: There is a linearly rising non-perturbative potential V (r) ≃ σ0r for r large[ 1, 2]. There is a second order phase transition at Tc = 0.55(1)g 2 , with the critical indices of the 2d 3states Potts model[ 2]. Furthermore, the glue ball masses mGB are much bigger than Tc, mGB ≥ 4.4Tc [ 1]. This is all qualitatively similar to 3 + 1 dimensions, where, however, the transition is weakly first order. In the gluon plasma phase T > Tc, one should be able to use perturbation theory. The relevant dimensionless coupling in 2 + 1 dimensions is