SU(N) gauge theories in four dimensions: exploring the approach to N = infinity

TL;DR

This study provides nonperturbative lattice evidence that SU($N$) gauge theories in four dimensions exhibit a smooth large-$N$ limit with confinement and a rising separation of scales in the glueball spectrum. By computing the string tension, low-lying glueball masses, and topological observables for $N=2$–$5$, the authors show mass ratios $m_G/\sqrt{\sigma}$ converge rapidly to their $N\to\infty$ values with leading $O(1/N^2)$ corrections, and that a fixed 't Hooft coupling $\lambda=g^2N$ yields an $N$-independent continuum limit for $N\ge4$. The topological susceptibility approaches a finite large-$N$ limit while the density of small instantons is exponentially suppressed with $N$, consistent with expectations for the $U(1)$ problem in QCD-like theories. Overall, the work supports the view that QCD-like physics at low energies may be governed largely by its large-$N$ limit and outlines concrete steps for more detailed future investigations into the glueball spectrum and $k$-string sectors.

Abstract

We calculate the string tension, K, and some of the lightest glueball masses, M, in 3+1 dimensional SU(N) lattice gauge theories for N=2,3,4,5 . From the continuum extrapolation of the lattice values, we find that the mass ratios, M/sqrt(K), appear to show a rapid approach to the large-N limit, and, indeed, can be described all the way down to SU(2) using just a leading O(1/NxN) correction. We confirm that the smooth large-N limit we find, is obtained by keeping a constant 't Hooft coupling. We also calculate the topological charge of the gauge fields. We observe that, as expected, the density of small-size instantons vanishes rapidly as N increases, while the topological susceptibility appears to have a non-zero N=infinity limit.

Figures (14)

• Figure 1: The (rescaled) average plaquette, $\Box$, the mass gap, $\circ$, and the square root of the string tension, $\Diamond$, over a range of $\beta$ that includes the region of transition between strong and weak coupling, for the SU(4) gauge theory.
• Figure 2: The mass of the lightest periodic flux loop as a function of its length, at $\beta=10.70$ in SU(4). Shown is a linear fit with a string correction as in eqn(\ref{['eqn_poly']}).
• Figure 3: The mass of the lightest periodic flux loop as a function of its length, at $\beta=5.93$ in SU(3). Shown is a linear fit with a string correction as in eqn(\ref{['eqn_poly']}).
• Figure 4: The mass of the lightest scalar glueball, $m_{0^{++}}$, expressed in units of the string tension, $\sigma$, is plotted against the latter in lattice units. The continuum extrapolation, using a leading lattice correction, is shown.
• Figure 5: The mass of the lightest tensor glueball, $m_{2^{++}}$, expressed in units of the string tension, $\sigma$, is plotted against the latter in lattice units. The continuum extrapolation, using a leading lattice correction, is shown.