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 \( {\Lambda}_{\overline{MS}} \), 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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B. Lucini and M. Teper, SU(N) gauge theories in four-dimensions: Exploring the approach to N = ∞, JHEP 06 (2001) 050 [hep-lat/0103027] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, Glueballs and k-strings in SU(N) gauge theories: Calculations with improved operators, JHEP 06 (2004) 012 [hep-lat/0404008] [INSPIRE].
B. Lucini, A. Rago and E. Rinaldi, Glueball masses in the large N limit, JHEP 08 (2010) 119 [arXiv:1007.3879] [INSPIRE].
E. Bennett et al., Glueballs and strings in Sp(2N) Yang-Mills theories, Phys. Rev. D 103 (2021) 054509 [arXiv:2010.15781] [INSPIRE].
C. Michael and M. Teper, The Glueball Spectrum in SU(3), Nucl. Phys. B 314 (1989) 347 [INSPIRE].
UKQCD collaboration, A Comprehensive lattice study of SU(3) glueballs, Phys. Lett. B 309 (1993) 378 [hep-lat/9304012] [INSPIRE].
C. J. Morningstar and M. J. Peardon, The Glueball spectrum from an anisotropic lattice study, Phys. Rev. D 60 (1999) 034509 [hep-lat/9901004] [INSPIRE].
H. B. Meyer and M. J. Teper, Glueball Regge trajectories and the Pomeron: A Lattice study, Phys. Lett. B 605 (2005) 344 [hep-ph/0409183] [INSPIRE].
H. B. Meyer, Glueball Regge trajectories, other thesis, 2004 [hep-lat/0508002] [INSPIRE].
Y. Chen et al., Glueball spectrum and matrix elements on anisotropic lattices, Phys. Rev. D 73 (2006) 014516 [hep-lat/0510074] [INSPIRE].
A. Athenodorou and M. Teper, The glueball spectrum of SU(3) gauge theory in 3 + 1 dimensions, JHEP 11 (2020) 172 [arXiv:2007.06422] [INSPIRE].
M. G. Pérez, A. González-Arroyo and M. Okawa, Meson spectrum in the large N limit, JHEP 04 (2021) 230 [arXiv:2011.13061] [INSPIRE].
M. García Pérez, A. González-Arroyo, M. Koren and M. Okawa, The spectrum of 2 + 1 dimensional Yang-Mills theory on a twisted spatial torus, JHEP 07 (2018) 169 [arXiv:1807.03481] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, The High temperature phase transition in SU(N) gauge theories, JHEP 01 (2004) 061 [hep-lat/0307017] [INSPIRE].
B. Lucini, M. Teper and U. Wenger, Properties of the deconfining phase transition in SU(N) gauge theories, JHEP 02 (2005) 033 [hep-lat/0502003] [INSPIRE].
N. Husung, P. Marquard and R. Sommer, Asymptotic behavior of cutoff effects in Yang-Mills theory and in Wilson’s lattice QCD, Eur. Phys. J. C 80 (2020) 200 [arXiv:1912.08498] [INSPIRE].
M. Teper, An Improved Method for Lattice Glueball Calculations, Phys. Lett. B 183 (1987) 345 [INSPIRE].
M. Teper, The Scalar and Tensor Glueball Masses in Lattice Gauge Theory, Phys. Lett. B 185 (1987) 121 [INSPIRE].
M. Lüscher, Volume Dependence of the Energy Spectrum in Massive Quantum Field Theories. 1. Stable Particle States, Commun. Math. Phys. 104 (1986) 177 [INSPIRE].
G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432 [Erratum ibid. 18 (1978) 2199] [INSPIRE].
S. Coleman, The uses of instantons, in Aspects of Symmetry. Selected Erice Lectures, chapter 7, pp. 265–350, Cambridge University Press (1985) [DOI].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
S. Coleman, 1/N, in Aspects of Symmetry. Selected Erice Lectures, chapter 8, pp. 351–402, Cambridge University Press (1985) [DOI].
E. Witten, Baryons in the 1/n Expansion, Nucl. Phys. B 160 (1979) 57 [INSPIRE].
E. Witten, Instantons, the Quark Model, and the 1/n Expansion, Nucl. Phys. B 149 (1979) 285 [INSPIRE].
M. J. Teper, Instantons and the 1/N Expansion, Z. Phys. C 5 (1980) 233 [INSPIRE].
L. Del Debbio, H. Panagopoulos, P. Rossi and E. Vicari, k string tensions in SU(N) gauge theories, Phys. Rev. D 65 (2002) 021501 [hep-th/0106185] [INSPIRE].
E. Witten, Theta dependence in the large N limit of four-dimensional gauge theories, Phys. Rev. Lett. 81 (1998) 2862 [hep-th/9807109] [INSPIRE].
L. Del Debbio, H. Panagopoulos and E. Vicari, θ dependence of SU(N) gauge theories, JHEP 08 (2002) 044 [hep-th/0204125] [INSPIRE].
S. Aoki, H. Fukaya, S. Hashimoto and T. Onogi, Finite volume QCD at fixed topological charge, Phys. Rev. D 76 (2007) 054508 [arXiv:0707.0396] [INSPIRE].
O. Aharony and Z. Komargodski, The Effective Theory of Long Strings, JHEP 05 (2013) 118 [arXiv:1302.6257] [INSPIRE].
S. Dubovsky, R. Flauger and V. Gorbenko, Flux Tube Spectra from Approximate Integrability at Low Energies, J. Exp. Theor. Phys. 120 (2015) 399 [arXiv:1404.0037] [INSPIRE].
M. Lüscher and P. Weisz, String excitation energies in SU(N) gauge theories beyond the free-string approximation, JHEP 07 (2004) 014 [hep-th/0406205] [INSPIRE].
J. M. Drummond, Universal subleading spectrum of effective string theory, hep-th/0411017 [INSPIRE].
A. Athenodorou, B. Bringoltz and M. Teper, Closed flux tubes and their string description in D = 3 + 1 SU(N) gauge theories, JHEP 02 (2011) 030 [arXiv:1007.4720] [INSPIRE].
B. Lucini and M. Teper, Confining strings in SU(N) gauge theories, Phys. Rev. D 64 (2001) 105019 [hep-lat/0107007] [INSPIRE].
A. Hasenfratz and P. Hasenfratz, The Connection Between the Lambda Parameters of Lattice and Continuum QCD, Phys. Lett. B 93 (1980) 165 [INSPIRE].
R. F. Dashen and D. J. Gross, The Relationship Between Lattice and Continuum Definitions of the Gauge Theory Coupling, Phys. Rev. D 23 (1981) 2340 [INSPIRE].
J. Ambjørn, P. Olesen and C. Peterson, Stochastic Confinement and Dimensional Reduction. 1. Four-Dimensional SU(2) Lattice Gauge Theory, Nucl. Phys. B 240 (1984) 189 [INSPIRE].
G. P. Lepage, Redesigning lattice QCD, Lect. Notes Phys. 479 (1997) 1 [hep-lat/9607076] [INSPIRE].
G. Parisi, Recent Progresses in Gauge Theories, AIP Conf. Proc. 68 (1980) 1531 [INSPIRE].
C. Allton, M. Teper and A. Trivini, On the running of the bare coupling in SU(N) lattice gauge theories, JHEP 07 (2008) 021 [arXiv:0803.1092] [INSPIRE].
M. Lüscher, R. Sommer, P. Weisz and U. Wolff, A Precise determination of the running coupling in the SU(3) Yang-Mills theory, Nucl. Phys. B 413 (1994) 481 [hep-lat/9309005] [INSPIRE].
S. Capitani, M. Lüscher, R. Sommer and H. Wittig, Non-perturbative quark mass renormalization in quenched lattice QCD, Nucl. Phys. B 544 (1999) 669 [Erratum ibid. 582 (2000) 762] [hep-lat/9810063] [INSPIRE].
C. R. Allton, Lattice Monte Carlo data versus perturbation theory, hep-lat/9610016 [INSPIRE].
R. Sommer, A New way to set the energy scale in lattice gauge theories and its applications to the static force and αs in SU(2) Yang-Mills theory, Nucl. Phys. B 411 (1994) 839 [hep-lat/9310022] [INSPIRE].
R. Sommer, Scale setting in lattice QCD, PoS LATTICE2013 (2014) 015 [arXiv:1401.3270] [INSPIRE].
K.-I. Ishikawa, I. Kanamori, Y. Murakami, A. Nakamura, M. Okawa and R. Ueno, Non-perturbative determination of the Λ-parameter in the pure SU(3) gauge theory from the twisted gradient flow coupling, JHEP 12 (2017) 067 [arXiv:1702.06289] [INSPIRE].
N. Husung, M. Koren, P. Krah and R. Sommer, SU(3) Yang-Mills theory at small distances and fine lattices, EPJ Web Conf. 175 (2018) 14024 [arXiv:1711.01860] [INSPIRE].
K. Ishikawa, G. Schierholz and M. Teper, Calculation of the Glueball Mass Spectrum of SU(2) and SU(3) Nonabelian Lattice Gauge Theories I: Introduction and SU(2), Z. Phys. C 19 (1983) 327 [INSPIRE].
K. Ishikawa, A. Sato, G. Schierholz and M. Teper, Calculation of the Glueball Mass Spectrum of SU(2) and SU(3) Nonabelian Lattice Gauge Theories II: SU(3), Z. Phys. C 21 (1983) 167 [INSPIRE].
U. M. Heller, SU(3) lattice gauge theory in the fundamental adjoint plane and scaling along the Wilson axis, Phys. Lett. B 362 (1995) 123 [hep-lat/9508009] [INSPIRE].
P. Weisz, Renormalization and lattice artifacts, in Les Houches Summer School: Session 93: Modern perspectives in lattice QCD: Quantum field theory and high performance computing, pp. 93–160 (2010) [arXiv:1004.3462] [INSPIRE].
P. Conkey, S. Dubovsky and M. Teper, Glueball spins in D = 3 Yang-Mills, JHEP 10 (2019) 175 [arXiv:1909.07430] [INSPIRE].
P. Di Vecchia, K. Fabricius, G. C. Rossi and G. Veneziano, Preliminary Evidence for UA(1) Breaking in QCD from Lattice Calculations, Nucl. Phys. B 192 (1981) 392 [INSPIRE].
M. Campostrini, A. Di Giacomo and H. Panagopoulos, The Topological Susceptibility on the Lattice, Phys. Lett. B 212 (1988) 206 [INSPIRE].
M. Teper, Instantons in the Quantized SU(2) Vacuum: A Lattice Monte Carlo Investigation, Phys. Lett. B 162 (1985) 357 [INSPIRE].
UKQCD collaboration, Topological structure of the SU(3) vacuum, Phys. Rev. D 58 (1998) 014505 [hep-lat/9801008] [INSPIRE].
M. Lüscher, Properties and uses of the Wilson flow in lattice QCD, JHEP 08 (2010) 071 [Erratum ibid. 03 (2014) 092] [arXiv:1006.4518] [INSPIRE].
M. Lüscher and P. Weisz, Perturbative analysis of the gradient flow in non-abelian gauge theories, JHEP 02 (2011) 051 [arXiv:1101.0963] [INSPIRE].
M. Lüscher, Future applications of the Yang-Mills gradient flow in lattice QCD, PoS LATTICE2013 (2014) 016 [arXiv:1308.5598] [INSPIRE].
C. Bonati and M. D’Elia, Comparison of the gradient flow with cooling in SU(3) pure gauge theory, Phys. Rev. D 89 (2014) 105005 [arXiv:1401.2441] [INSPIRE].
C. Alexandrou, A. Athenodorou and K. Jansen, Topological charge using cooling and the gradient flow, Phys. Rev. D 92 (2015) 125014 [arXiv:1509.04259] [INSPIRE].
C. Alexandrou et al., Comparison of topological charge definitions in Lattice QCD, Eur. Phys. J. C 80 (2020) 424 [arXiv:1708.00696] [INSPIRE].
G. Cossu, D. Lancastera, B. Lucini, R. Pellegrini and A. Rago, Ergodic sampling of the topological charge using the density of states, Eur. Phys. J. C 81 (2021) 375 [arXiv:2102.03630] [INSPIRE].
G. Veneziano, U(1) Without Instantons, Nucl. Phys. B 159 (1979) 213 [INSPIRE].
E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269 [INSPIRE].
C. Bonanno, C. Bonati and M. D’Elia, Large-N SU(N) Yang-Mills theories with milder topological freezing, JHEP 03 (2021) 111 [arXiv:2012.14000] [INSPIRE].
M. Hasenbusch, Fighting topological freezing in the two-dimensional CPN-1 model, Phys. Rev. D 96 (2017) 054504 [arXiv:1706.04443] [INSPIRE].
M. Lüscher and F. Palombi, Universality of the topological susceptibility in the SU(3) gauge theory, JHEP 09 (2010) 110 [arXiv:1008.0732] [INSPIRE].
C. Bonati, M. D’Elia, P. Rossi and E. Vicari, θ dependence of 4D SU(N) gauge theories in the large-N limit, Phys. Rev. D 94 (2016) 085017 [arXiv:1607.06360] [INSPIRE].
T. van Ritbergen, J. A. M. Vermaseren and S. A. Larin, The Four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].
C. Christou, A. Feo, H. Panagopoulos and E. Vicari, The three loop β-function of SU(N) lattice gauge theories with Wilson fermions, Nucl. Phys. B 525 (1998) 387 [Erratum ibid. 608 (2001) 479] [hep-lat/9801007] [INSPIRE].
B. Alles, A. Feo and H. Panagopoulos, Asymptotic scaling corrections in QCD with Wilson fermions from the three loop average plaquette, Phys. Lett. B 426 (1998) 361 [Erratum ibid. 553 (2003) 337] [hep-lat/9801003] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2106.00364
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Athenodorou, A., Teper, M. SU(N) gauge theories in 3+1 dimensions: glueball spectrum, string tensions and topology. J. High Energ. Phys. 2021, 82 (2021). https://doi.org/10.1007/JHEP12(2021)082
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2021)082