SU(N)gauge theories in 2+1 dimensions

We calculate the mass spectra and string tensions of SU(2), SU(3), SU(4) and SU(5) gauge theories in 2+1 dimensions. We do so by simulating the corresponding lattice theories and then extrapolating dimensionless mass ratios to the continuum limit. We find that such mass ratios are, to a first approximation, independent of the number of colors, $N_c,$ and that the remaining dependence can be accurately reproduced by a simple ${O(1/N}_c^2)$ correction. This provides us with a prediction of these mass ratios for all $\mathrm{SU}{(N}_c)$ theories in 2+1 dimensions and demonstrates that these theories are already “close” to $N_c=\infty$ for $N_c>~2.$ We find that the theory retains a non-zero confining string tension as $N_c\to \infty$ and that the dimensionful coupling $g^2$ is proportional to ${1/N}_c$ at large $N_c,$ when expressed in units of the dynamical length scale of the theory. During the course of these calculations we study in detail the effects of including over-relaxation in the Monte Carlo calculation, of using a mean-field improved coupling to extrapolate to the continuum limit, and the use of space-time asymmetric lattice actions to resolve heavy glueball correlators.