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

Abstract

We calculate the low-lying glueball spectrum, several 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 2 ≤ N ≤ 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 = ∞. 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 g2(a) ∝ 1/N for constant physics as N → ∞. We fit our calculated values of a√σ with the 3-loop β-function, and extract a value for $$ {\Lambda}_{\overline{MS}} $$ Λ MS ¯ , in units of the string tension, for all our values of N, including SU(3). We use these fits to provide analytic formulae for estimating the string tension at a given lattice coupling. We calculate the topological charge Q for N ≤ 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(β), 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 θ parameter. We provide quantitative results for how the topological charge ‘freezes’ with decreasing lattice spacing and with increasing N. Although we are able to show that within our typical errors our glueball and string tension results are insensitive to the freezing of Q at larger N and β, we choose to perform our calculations with a typical distribution of Q imposed upon the fields so as to further reduce any potential systematic errors.

Sp(4) gauge theories and beyond the standard model physics

“…1 (in magenta for the groundstates, and in yellow for the excitations). While we notice that the authors of [15] quote significantly smaller errors, the agreement with our measurements is within the error bars (as measured on our own data) in most of the cases, with the largest discrepancies being at most around two standard deviations. This comparison provides evidence of the universality of the large-N glueball spectrum across the SU(N) and the Sp(2N) series.…”

“…This enables us to perform a broader comparison of the two extrapolations. The results of [15] are also reported in Fig. 1 (in magenta for the groundstates, and in yellow for the excitations).…”

“…1 the more constraining measurements of [13] (in cyan for the groundstates and in green for the excitation in the A + 1 channel), which are in full agreement with ours. After our work was published, a comprehensive study of the large-N SU(N) glueball spectrum was published [15]. This enables us to perform a broader comparison of the two extrapolations.…”

The torelon spectrum and the world-sheet axion

“…The above consist of N = 3 at β = 6.0625 (a 0.09fm) and β = 6.338 (a 0.06fm), of N = 5 at β = 17.630 (a 0.09fm) and β = 18.375 (a 0.06fm) as well as for N = 6 at β = 25.550 (a 0.09fm). Critical slowing down [17,18], as one moves towards the continuum and the large-N limit, prohibits the investigation of gauge groups with N ≥ 6 and a < 0.09fm. Nevertheless, the above configuration of measurements is enough to determine whether significant lattice artifacts as well as 1/N 2 corrections are affecting our statistically more accurate N = 3 calculations.…”

A lattice study of $ππ$ scattering at large $N_\text{c}$

“…A first-principle method that can quantify such corrections is the lattice formulation, where the number of colors can be varied [7,8]. Indeed, various groups have successfully studied the N c scaling of some observables, such as the hadron or glueball spectrum, as well as matrix elements [9][10][11][12][13][14][15][16][17][18][19]-see Ref. [20] for a recent review.…”