SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology

TL;DR

This study presents a comprehensive lattice investigation of SU(N) gauge theories in 3+1 dimensions across N=2..12, computing the low-lying glueball spectrum in all R^{PC} sectors, fundamental and k=2 string tensions, and select topological observables. By extrapolating to the continuum and to N→∞, the authors quantify how glueball masses and string tensions scale with N, identify continuum spins in many channels, and extract the running coupling and Λ_{\overline{MS}} while examining topology and topological susceptibility. The work demonstrates that SU(3) is close to SU(∞) for the quantities studied, provides analytic interpolations for a(β) and Λ-values, and discusses topological freezing and methods to mitigate its impact. Overall, the results offer a robust large-N perspective on nonperturbative Yang–Mills dynamics and deliver practical inputs for phenomenology and model-building that relies on constant-physics extrapolations. The paper also highlights methodological advances in handling finite-volume effects, operator bases, and topology within large-N lattice simulations.

Abstract

We calculate the low-lying glueball spectrum, some string tensions and some properties of topology and the running coupling for SU(N) lattice gauge theories in 3+1 dimensions. We do so for N = 2,3,...12, using lattice simulations with the Wilson plaquette action, and for glueball states in all the representations of the cubic rotation group, for both values of parity and charge conjugation. We extrapolate these results to the continuum limit of each theory and then to N=infinity. For a number of these states we are able to identify their continuum spins with very little ambiguity. We calculate the fundamental string tension and k=2 string tension and investigate the N dependence of the ratio. Using the string tension as the scale, we calculate the running of a lattice coupling and confirm that g(a)**2 varies as 1/N for constant physics as N->oo. We fit our calculated values of the string tension with the 3-loop beta-function, and extract a value for Lambda-MSbar, in units of the string tension, for all our values of N, including SU(3). We calculate the topological charge Q for N=2,..,6 where it fluctuates sufficiently for a plausible estimate of the continuum topological susceptibility. We also calculate the renormalisation of the lattice topological charge, ZQ(beta), for all our SU(N) gauge theories, using a standard definition of the charge, and we provide interpolating formulae, which may be useful in estimating the renormalisation of the lattice theta parameter. We provide quantitative results for how the topological charge `freezes' with decreasing lattice spacing and with increasing N, and show how we cicumvent this issue in our calculations.

Figures (30)

• Figure 1: Effective energies of the ground state of a fundamental $k=1$ flux tube winding around a spatial torus, extracted from the best correlator $C(t)$ between $t=an_t$ and $t=a(n_t+1)$. For $SU(8)$ and at $\beta=44.10, 44.85, 45.50, 46.10, 46.70, 47.75$ in descending order. Lines are our estimates of the $t\to\infty$ asymptotic energies. The nearly invisible bands around those lines denote the errors on those estimates.
• Figure 2: Effective energy of the ground state of a fundamental $k=1$ flux tube winding around a spatial torus, as in Fig.\ref{['fig_EeffK1_SU8']}, for $\beta=47.75$ in $SU(8)$, with a rescaling sufficient to expose the errors. The solid line is the best estimate from a fit to the correlation function, and the dashed lines show the $\pm 1$ standard deviation fits.
• Figure 3: $k=2$ string tensions, $\sigma_k$, in $SU(N)$ gauge theories for $N=4,5,6,8,10,12$ in ascending order, in units of the $k=1$ fundamental string tension, $\sigma_f$. Lines are extrapolations to the continuum limits.
• Figure 4: Continuum limit of $k=2$ string tension, $\sigma_k$, in units of the $k=1$ fundamental string tension, $\sigma_f$, for our $SU(N)$ gauge theories. Solid line is the best fit in powers of $1/N$ and dashed line is the best fit in powers of $1/N^2$, with the constraint that the ratio is 2 at $N=\infty$.
• Figure 5: Running (mean-field improved) 't Hooft coupling on the lattice scale $a$, expressed in units of the string tension, for $SU(2)$, $\bullet$, $SU(3)$, $\circ$, $SU(4)$, $\blacksquare$, $SU(5)$, $\square$, $SU(6)$, $\blacklozenge$, $SU(8)$, $\lozenge$, $SU(10)$, $\blacktriangle$, $SU(12)$, $\vartriangle$. Solid and dashed lines are (improved) perturbative fits to $SU(8)$ and $SU(3)$ respectively.