SO(N) gauge theories in 2 + 1 dimensions: glueball spectra and confinement

Abstract

Abstract:We calculate the spectrum of light glueballs and the string tension in a number of SO(N ) lattice gauge theories in 2+1 dimensions, with N in the range 3 ≤ N ≤ 16. After extrapolating to the continuum limit and then to N = ∞ we compare to the spectrum and string tension of the SU(N → ∞) gauge theory and find that the most reliably and precisely calculated physical quantities are consistent in that limit. We also compare the glueball spectra of those pairs of SO(N ) and SU(N ) theories that possess the same Lie algebra, i.e. SO(3) and SU(2), SO(4) and SU(2)×SU(2), SO(6) and SU(4), and find that for the very lightest glueballs the spectra are consistent within each such pair, as are the string tensions and the couplings. Where there are apparent discrepancies they are typically for heavier glueballs, where the systematic errors are much harder to control. We calculate the SO(N ) string tensions with a particular focus on the confining properties of SO(2N + 1) theories which, unlike SO(2N ) theories, possess a trivial centre. We find that for both the light glueballs and for the string tension SO(2N ) and SO(2N + 1) gauge theories appear to form a single smooth sequence.

The deconfining phase transition of SO(N) gauge theories in 2+1 dimensions

“…Of course that assumes that these theories are linearly confining at low T . While we shall provide some evidence for confinement at low T in this paper (see in particular the discussion in section 3.3), the explicit evidence for the confinement being linear is given in our companion paper on the glueball spectra and string tensions [12], where we show that the energy of closed flux tubes is (roughly) proportional to their length for JHEP03(2016)072 both odd and even N . Of course such numerical evidence possesses intrinsic limitations: we cannot distinguish between confinement that is exact and confinement to a very good approximation.…”

“…The implications of the Lie algebra equivalence between certain SO(N ) and SU(N ′ ) groups are discussed in more detail in [12,13]. Here we merely summarise some points that are relevant to our present calculations.…”

“…Of course, if the decay width is small enough, then just as for the mass of a narrow resonance, we can estimate a string tension. More importantly, glueball masses and T c should be the same within SU (2) and SO (3), and the coupling g 2 should satisfy [12,14] In this paper we do not calculate T c for SO(3) because the strong-to-weak coupling transition in our lattice theory occurs at such a small value of the lattice spacing that we would need to use very large lattices (in lattice units) and this would be computationally quite expensive. Of course, if one assumes that the physics of SU (2) and SO (3) is the same, as described above, then one can infer the value of T c in SO(3) from the known value in SU (2), and compare it to the values obtained in SO(N > 3).…”

“…In this paper we will show that SO(N ) gauge theories in 2 + 1 dimensions possess a deconfining phase transition at a finite temperature T = T c , just like the deconfining transition in SU(N ) gauge theories. We will calculate its value and determine its nature for N = 4, 5,6,7,8,9,12,16. This will enable us to extrapolate to N = ∞ where we can compare to the SU(∞) extrapolated value [1].…”

“…There are companion papers, both published [13] and in progress [12], that contain our results for the mass spectrum and string tension of SO(N ) gauge theories. The latter paper describes the Monte Carlo algorithm in more detail, as well as providing more discussion of the 'bulk' transition.…”