Abstract
We investigate the lattice regularization of \( \mathcal{N} \) = 4 supersymmetric Yang-Mills theory, by stochastically computing the eigenvalue mode number of the fermion operator. This provides important insight into the non-perturbative renormalization group flow of the lattice theory, through the definition of a scale-dependent effective mass anomalous dimension. While this anomalous dimension is expected to vanish in the conformal continuum theory, the finite lattice volume and lattice spacing generically lead to non-zero values, which we use to study the approach to the continuum limit. Our numerical results, comparing multiple lattice volumes, ’t Hooft couplings, and numbers of colors, confirm convergence towards the expected continuum result, while quantifying the increasing significance of lattice artifacts at larger couplings.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
J. M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
H. Osborn, Topological Charges for N = 4 Supersymmetric Gauge Theories and Monopoles of Spin 1, Phys. Lett. B 83 (1979) 321 [INSPIRE].
H. Elvang and Y.-t. Huang, Scattering Amplitudes in Gauge Theory and Gravity, Cambridge University Press (2015) [INSPIRE].
S. Catterall, D. B. Kaplan and M. Ünsal, Exact lattice supersymmetry, Phys. Rept. 484 (2009) 71 [arXiv:0903.4881] [INSPIRE].
G. Bergner and S. Catterall, Supersymmetry on the lattice, Int. J. Mod. Phys. A 31 (2016) 1643005 [arXiv:1603.04478] [INSPIRE].
D. Schaich, Progress and prospects of lattice supersymmetry, PoS LATTICE2018 (2019) 005 [arXiv:1810.09282] [INSPIRE].
S. Catterall, J. Giedt and A. Joseph, Twisted supersymmetries in lattice \( \mathcal{N} \) = 4 super Yang-Mills theory, JHEP 10 (2013) 166 [arXiv:1306.3891] [INSPIRE].
S. Catterall and J. Giedt, Real space renormalization group for twisted lattice \( \mathcal{N} \) = 4 super Yang-Mills, JHEP 11 (2014) 050 [arXiv:1408.7067] [INSPIRE].
S. Catterall, E. Dzienkowski, J. Giedt, A. Joseph and R. Wells, Perturbative renormalization of lattice N = 4 super Yang-Mills theory, JHEP 04 (2011) 074 [arXiv:1102.1725] [INSPIRE].
S. Catterall and D. Schaich, Lifting flat directions in lattice supersymmetry, JHEP 07 (2015) 057 [arXiv:1505.03135] [INSPIRE].
D. J. Weir, S. Catterall and D. Mehta, Eigenvalue spectrum of lattice \( \mathcal{N} \) = 4 super Yang-Mills, PoS LATTICE2013 (2014) 093 [arXiv:1311.3676] [INSPIRE].
F. Sugino, A Lattice formulation of superYang-Mills theories with exact supersymmetry, JHEP 01 (2004) 015 [hep-lat/0311021] [INSPIRE].
F. Sugino, SuperYang-Mills theories on the two-dimensional lattice with exact supersymmetry, JHEP 03 (2004) 067 [hep-lat/0401017] [INSPIRE].
S. Catterall, A Geometrical approach to N = 2 super Yang-Mills theory on the two dimensional lattice, JHEP 11 (2004) 006 [hep-lat/0410052] [INSPIRE].
A. G. Cohen, D. B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a Euclidean space-time lattice. 1. A Target theory with four supercharges, JHEP 08 (2003) 024 [hep-lat/0302017] [INSPIRE].
A. G. Cohen, D. B. Kaplan, E. Katz and M. Ünsal, Supersymmetry on a Euclidean space-time lattice. 2. Target theories with eight supercharges, JHEP 12 (2003) 031 [hep-lat/0307012] [INSPIRE].
D. B. Kaplan and M. Ünsal, A Euclidean lattice construction of supersymmetric Yang-Mills theories with sixteen supercharges, JHEP 09 (2005) 042 [hep-lat/0503039] [INSPIRE].
M. Ünsal, Twisted supersymmetric gauge theories and orbifold lattices, JHEP 10 (2006) 089 [hep-th/0603046] [INSPIRE].
S. Catterall, From Twisted Supersymmetry to Orbifold Lattices, JHEP 01 (2008) 048 [arXiv:0712.2532] [INSPIRE].
P. H. Damgaard and S. Matsuura, Geometry of Orbifolded Supersymmetric Lattice Gauge Theories, Phys. Lett. B 661 (2008) 52 [arXiv:0801.2936] [INSPIRE].
M. Hanada and I. Kanamori, Absence of sign problem in two-dimensional N = (2, 2) super Yang-Mills on lattice, JHEP 01 (2011) 058 [arXiv:1010.2948] [INSPIRE].
S. Catterall, P. H. Damgaard, T. Degrand, R. Galvez and D. Mehta, Phase Structure of Lattice N = 4 Super Yang-Mills, JHEP 11 (2012) 072 [arXiv:1209.5285] [INSPIRE].
S. Catterall, D. Schaich, P. H. Damgaard, T. DeGrand and J. Giedt, N = 4 Supersymmetry on a Space-Time Lattice, Phys. Rev. D 90 (2014) 065013 [arXiv:1405.0644] [INSPIRE].
D. Schaich and T. DeGrand, Parallel software for lattice N = 4 supersymmetric Yang-Mills theory, Comput. Phys. Commun. 190 (2015) 200 [arXiv:1410.6971] [INSPIRE].
S. Catterall, J. Giedt and G. C. Toga, Lattice \( \mathcal{N} \) = 4 super Yang-Mills at strong coupling, JHEP 12 (2020) 140 [arXiv:2009.07334] [INSPIRE].
M. A. Clark and A. D. Kennedy, Accelerating dynamical fermion computations using the rational hybrid Monte Carlo (RHMC) algorithm with multiple pseudofermion fields, Phys. Rev. Lett. 98 (2007) 051601 [hep-lat/0608015] [INSPIRE].
A. Patella, A precise determination of the psibar-psi anomalous dimension in conformal gauge theories, Phys. Rev. D 86 (2012) 025006 [arXiv:1204.4432] [INSPIRE].
A. Cheng, A. Hasenfratz, G. Petropoulos and D. Schaich, Scale-dependent mass anomalous dimension from Dirac eigenmodes, JHEP 07 (2013) 061 [arXiv:1301.1355] [INSPIRE].
A. Cheng, A. Hasenfratz, G. Petropoulos and D. Schaich, Determining the mass anomalous dimension through the eigenmodes of Dirac operator, PoS LATTICE2013 (2014) 088 [arXiv:1311.1287] [INSPIRE].
Z. Fodor, K. Holland, J. Kuti, D. Nógrádi and C. H. Wong, The chiral condensate from the Dirac spectrum in BSM gauge theories, PoS LATTICE2013 (2014) 089 [arXiv:1402.6029] [INSPIRE].
L. Giusti, C. Hölbling, M. Lüscher and H. Wittig, Numerical techniques for lattice QCD in the E-regime, Comput. Phys. Commun. 153 (2003) 31 [hep-lat/0212012] [INSPIRE].
Z. Fodor, K. Holland, J. Kuti, S. Mondal, D. Nogradi and C. H. Wong, New approach to the Dirac spectral density in lattice gauge theory applications, PoS LATTICE2015 (2016) 310 [arXiv:1605.08091] [INSPIRE].
G. Bergner, P. Giudice, G. Münster, I. Montvay and S. Piemonte, Spectrum and mass anomalous dimension of SU(2) adjoint QCD with two Dirac flavors, Phys. Rev. D 96 (2017) 034504 [arXiv:1610.01576] [INSPIRE].
A. Stathopoulos and J. R. McCombs, PRIMME: preconditioned iterative multimethod eigensolver-methods and software description, ACM Trans. Math. Softw. 37 (2010) 21.
D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, emcee: The MCMC Hammer, Publ. Astron. Soc. Pac. 125 (2013) 306 [arXiv:1202.3665] [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: 2102.06775
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
Bergner, G., Schaich, D. Eigenvalue spectrum and scaling dimension of lattice \( \mathcal{N} \) = 4 supersymmetric Yang-Mills. J. High Energ. Phys. 2021, 260 (2021). https://doi.org/10.1007/JHEP04(2021)260
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP04(2021)260