SU(N) gauge theories in 2+1 dimensions

TL;DR

This work performs a comprehensive lattice study of pure SU($N_c$) gauge theories in 2+1 dimensions for $N_c=2,3,4,5$, extracting continuum-extrapolated string tensions and glueball spectra. The authors employ anisotropic lattices, mean-field improved couplings, and a robust variational framework with smeared, blocked operators to obtain ground and excited state masses across multiple $J^{PC}$ channels, while carefully controlling finite-volume effects. They find that mass ratios are largely independent of $N_c$, with leading corrections scaling as $O(1/N_c^2)$, and that the string tension scales as $ surdsigma o g^2 N_c imes ext{const}$, indicating a smooth large-$N_c$ limit with confinement persisting at $N_c o afty$. The results provide a precise, nonperturbative benchmark for large-$N_c$ dynamics in 2+1D and offer a stringent testbed for models of glueball structure and flux-tube phenomenology, with implications for understanding the $N_c o afty$ limit in higher dimensions as well.

Abstract

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 colours and that the remaining dependence can be accurately reproduced by a simple O(1/N.N) correction. This provides us with a prediction of these mass ratios for all SU(N) theories in 2+1 dimensions and demonstrates that these theories are already `close' to N=infinity for N=2. We find that the theory retains a non-zero confining string tension as N goes to infinity and that the dimensionful coupling g.g is proportional to 1/N at large N, 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, 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.

Figures (13)

• Figure 1: Mass of periodic flux loop, $am_P$, against its length, $L$, at $\beta=6$. The straight line is to guide the eye.
• Figure 2: Mass of periodic flux loop of length, $L$, at $\beta=9$; divided by $L$ to expose the correction to the linear rise. Curve is fit using eqn(\ref{['C1']}).
• Figure 3: The values of $\beta a \surd\sigma$ plotted against $1/\beta$ for SU(2). Also shown is the leading-order strong coupling prediction at low $\beta$, and a leading-order continuum extrapolation at high $\beta$.
• Figure 4: The values of $\beta a \surd\sigma$ plotted against $1/\beta$ for SU(3). Also shown is the strong coupling prediction to $O(\beta)$ at low $\beta$, and a leading-order continuum extrapolation at high $\beta$.
• Figure 5: As in Fig.\ref{['fig_Kbetasu2']} but using the mean-field improved coupling, $\beta_I$, in place of $\beta$.