Abstract
This work is a step towards merging the ideas that arise from semi-classical methods in continuum QFT with analytic/numerical lattice field theory. In this context, we consider Yang-Mills theories coupled to fermions transforming in the adjoint representation of the gauge group. These theories have the remarkable property that confinement and discrete chiral symmetry breaking can persist at weak coupling on ℝ3 × S1 up to small (non-thermal) compactification radii. This work presents a lattice investigation of a gauge theory coupled to a single adjoint Majorana fermion, the \( \mathcal{N}=1 \) Supersymmetric Yang-Mills theory (SYM), and opens the prospect to understand analytically a number of non-perturbative phenomena, such as confinement, mass gap, chiral and center symmetry realizations, both on the lattice and in the continuum. We study the compactification of \( \mathcal{N}=1 \) SYM on the lattice with periodic and thermal boundary conditions applied to the fermion field. We provide numerical evidences for the conjectured absence of phase transitions with periodic boundary conditions for sufficiently light lattice fermions (stability of center-symmetry), for the suppression of the chiral transition, and we provide also a diagnostic for Abelian vs. non-Abelian confinement, based on per-site Polyakov loop eigenvalue distribution functions. We identify the role of the lattice artefacts that become relevant in the very small radius regime, and we resolve some puzzles in the naive comparison between continuum and lattice.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.M. Polyakov, Quark Confinement and Topology of Gauge Groups, Nucl. Phys. B 120 (1977) 429 [INSPIRE].
I. Affleck, J.A. Harvey and E. Witten, Instantons and (Super)Symmetry Breaking in (2+1)-Dimensions, Nucl. Phys. B 206 (1982) 413 [INSPIRE].
M. Ünsal, Abelian duality, confinement and chiral symmetry breaking in QCD(adj), Phys. Rev. Lett. 100 (2008) 032005 [arXiv:0708.1772] [INSPIRE].
M. Ünsal, Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80 (2009) 065001 [arXiv:0709.3269] [INSPIRE].
M. Ünsal and L.G. Yaffe, Center-stabilized Yang-Mills theory: Confinement and large N volume independence, Phys. Rev. D 78 (2008) 065035 [arXiv:0803.0344] [INSPIRE].
M. Shifman and M. Ünsal, QCD-like Theories on R 3 × S 1: A Smooth Journey from Small to Large r(S 1 ) with Double-Trace Deformations, Phys. Rev. D 78 (2008) 065004 [arXiv:0802.1232] [INSPIRE].
N. Seiberg and E. Witten, Gauge dynamics and compactification to three-dimensions, in The mathematical beauty of physics: A memorial volume for Claude Itzykson, proceedings, conference, Saclay, France, 5-7 June 1996, pp. 333-366, (1996), hep-th/9607163 [INSPIRE].
K.-M. Lee and P. Yi, Monopoles and instantons on partially compactified D-branes, Phys. Rev. D 56 (1997) 3711 [hep-th/9702107] [INSPIRE].
T.C. Kraan and P. van Baal, Periodic instantons with nontrivial holonomy, Nucl. Phys. B 533 (1998) 627 [hep-th/9805168] [INSPIRE].
N.M. Davies, T.J. Hollowood, V.V. Khoze and M.P. Mattis, Gluino condensate and magnetic monopoles in supersymmetric gluodynamics, Nucl. Phys. B 559 (1999) 123 [hep-th/9905015] [INSPIRE].
P.C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion and renormalon effects, JHEP 08 (2012) 063 [arXiv:1206.1890] [INSPIRE].
E. Poppitz and T. Sulejmanpasic, (S)QCD on \( {\mathbb{R}}^3 \times {\mathbb{S}}^1 \): Screening of Polyakov loop by fundamental quarks and the demise of semi-classics, JHEP 09 (2013) 128 [arXiv:1307.1317] [INSPIRE].
G.V. Dunne and M. Ünsal, Generating nonperturbative physics from perturbation theory, Phys. Rev. D 89 (2014) 041701 [arXiv:1306.4405] [INSPIRE].
G.V. Dunne and M. Ünsal, New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence, Ann. Rev. Nucl. Part. Sci. 66 (2016) 245 [arXiv:1601.03414] [INSPIRE].
G.V. Dunne and M. Ünsal, Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model, JHEP 11 (2012) 170 [arXiv:1210.2423] [INSPIRE].
A. Cherman, D. Dorigoni and M. Ünsal, Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles, JHEP 10 (2015) 056 [arXiv:1403.1277] [INSPIRE].
D.J. Gross, R.D. Pisarski and L.G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys. 53 (1981) 43 [INSPIRE].
E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].
E. Poppitz, T. Schäfer and M. Ünsal, Continuity, Deconfinement and (Super) Yang-Mills Theory, JHEP 10 (2012) 115 [arXiv:1205.0290] [INSPIRE].
Y. Hosotani, Dynamical Mass Generation by Compact Extra Dimensions, Phys. Lett. B 126 (1983) 309 [INSPIRE].
P. Kovtun, M. Ünsal and L.G. Yaffe, Volume independence in large N c QCD-like gauge theories, JHEP 06 (2007) 019 [hep-th/0702021] [INSPIRE].
S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge University Press, Cambridge (1985).
T. Schäfer and E.V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323 [hep-ph/9610451] [INSPIRE].
M.M. Anber and E. Poppitz, Two-flavor adjoint QCD, Phys. Rev. D 98 (2018) 034026 [arXiv:1805.12290] [INSPIRE].
M.M. Anber, E. Poppitz and B. Teeple, Deconfinement and continuity between thermal and (super) Yang-Mills theory for all gauge groups, JHEP 09 (2014) 040 [arXiv:1406.1199] [INSPIRE].
H. Kouno, T. Misumi, K. Kashiwa, T. Makiyama, T. Sasaki and M. Yahiro, Differences and similarities between fundamental and adjoint matters in SU(N) gauge theories, Phys. Rev. D 88 (2013) 016002 [arXiv:1304.3274] [INSPIRE].
K. Kashiwa and T. Misumi, Phase structure and Hosotani mechanism in gauge theories with compact dimensions revisited, JHEP 05 (2013) 042 [arXiv:1302.2196] [INSPIRE].
M.M. Anber, S. Collier, E. Poppitz, S. Strimas-Mackey and B. Teeple, Deconfinement in \( \mathcal{N}=1 \) super Yang-Mills theory on \( {\mathbb{R}}^3\times {\mathbb{S}}^1 \) via dual-Coulomb gas and “affine” XY-model, JHEP 11 (2013) 142 [arXiv:1310.3522] [INSPIRE].
G. Bergner and S. Piemonte, Compactified \( \mathcal{N}=1 \) supersymmetric Yang-Mills theory on the lattice: continuity and the disappearance of the deconfinement transition, JHEP 12 (2014) 133 [arXiv:1410.3668] [INSPIRE].
I. Montvay and G. Münster, Quantum fields on a lattice, Cambridge University Press (1994).
G. Bergner, P. Giudice, G. Münster, I. Montvay and S. Piemonte, The light bound states of supersymmetric SU(2) Yang-Mills theory, JHEP 03 (2016) 080 [arXiv:1512.07014] [INSPIRE].
G. Bergner, I. Montvay, G. Münster, U.D. Özugurel and D. Sandbrink, Towards the spectrum of low-lying particles in supersymmetric Yang-Mills theory, JHEP 11 (2013) 061 [arXiv:1304.2168] [INSPIRE].
G. Münster and H. Stüwe, The mass of the adjoint pion in \( \mathcal{N}=1 \) supersymmetric Yang-Mills theory, JHEP 05 (2014) 034 [arXiv:1402.6616] [INSPIRE].
G. Bergner, P. Giudice, G. Münster, S. Piemonte and D. Sandbrink, Phase structure of the \( \mathcal{N}=1 \) supersymmetric Yang-Mills theory at finite temperature, JHEP 11 (2014) 049 [arXiv:1405.3180] [INSPIRE].
A.M. Ferrenberg, J. Xu and D.P. Landau, Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model, Phys. Rev. E 97 (2018) 043301 [arXiv:1806.03558] [INSPIRE].
B. Bringoltz, Large-N volume reduction of lattice QCD with adjoint Wilson fermions at weak-coupling, JHEP 06 (2009) 091 [arXiv:0905.2406] [INSPIRE].
E. Poppitz and M. Ünsal, Comments on large-N volume independence, JHEP 01 (2010) 098 [arXiv:0911.0358] [INSPIRE].
M. Ünsal and L.G. Yaffe, Large-N volume independence in conformal and confining gauge theories, JHEP 08 (2010) 030 [arXiv:1006.2101] [INSPIRE].
G. Cossu, H. Hatanaka, Y. Hosotani and J.-I. Noaki, Polyakov loops and the Hosotani mechanism on the lattice, Phys. Rev. D 89 (2014) 094509 [arXiv:1309.4198] [INSPIRE].
H. Georgi and S.L. Glashow, Unified weak and electromagnetic interactions without neutral currents, Phys. Rev. Lett. 28 (1972) 1494 [INSPIRE].
M.M. Anber, The abelian confinement mechanism revisited: new aspects of the Georgi-Glashow model, Annals Phys. 341 (2014) 21 [arXiv:1308.0027] [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
G. ’t Hooft, Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories, Nucl. Phys. B 190 (1981) 455 [INSPIRE].
M.R. Douglas and S.H. Shenker, Dynamics of SU(N) supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 271 [hep-th/9503163] [INSPIRE].
E. Poppitz and M.E. Shalchian T., String tensions in deformed Yang-Mills theory, JHEP 01 (2018) 029 [arXiv:1708.08821] [INSPIRE].
L. O’Raifeartaigh, A. Wipf and H. Yoneyama, The Constraint Effective Potential, Nucl. Phys. B 271 (1986) 653 [INSPIRE].
C.P. Korthals Altes, Constrained effective potential in hot QCD, Nucl. Phys. B 420 (1994) 637 [hep-th/9310195] [INSPIRE].
A. Dumitru, Y. Guo and C.P. Korthals Altes, Two-loop perturbative corrections to the thermal effective potential in gluodynamics, Phys. Rev. D 89 (2014) 016009 [arXiv:1305.6846] [INSPIRE].
A. Athenodorou, E. Bennett, G. Bergner and B. Lucini, Infrared regime of SU(2) with one adjoint Dirac flavor, Phys. Rev. D 91 (2015) 114508 [arXiv:1412.5994] [INSPIRE].
G. Bergner and S. Piemonte, Running coupling from gluon and ghost propagators in the Landau gauge: Yang-Mills theories with adjoint fermions, Phys. Rev. D 97 (2018) 074510 [arXiv:1709.07367] [INSPIRE].
D. Smith, A. Dumitru, R. Pisarski and L. von Smekal, Effective potential for SU(2) Polyakov loops and Wilson loop eigenvalues, Phys. Rev. D 88 (2013) 054020 [arXiv:1307.6339] [INSPIRE].
C. Gattringer, Linking confinement to spectral properties of the Dirac operator, Phys. Rev. Lett. 97 (2006) 032003 [hep-lat/0605018] [INSPIRE].
G. Cossu and M. D’Elia, Finite size phase transitions in QCD with adjoint fermions, JHEP 07 (2009) 048 [arXiv:0904.1353] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1806.10894
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bergner, G., Piemonte, S. & Ünsal, M. Adiabatic continuity and confinement in supersymmetric Yang-Mills theory on the lattice. J. High Energ. Phys. 2018, 92 (2018). https://doi.org/10.1007/JHEP11(2018)092
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2018)092