Skip to main content

\( \mathcal{N} \) = 1 Super-Yang-Mills theory on the lattice with twisted mass fermions

A preprint version of the article is available at arXiv.

Abstract

Super-Yang-Mills theory (SYM) is a central building block for supersymmetric extensions of the Standard Model of particle physics. Whereas the weakly coupled subsector of the latter can be treated within a perturbative setting, the strongly coupled subsector must be dealt with a non-perturbative approach. Such an approach is provided by the lattice formulation. Unfortunately a lattice regularization breaks supersymmetry and consequently the mass degeneracy within a supermultiplet. In this article we investigate the properties of \( \mathcal{N} \) = 1 supersymmetric SU(3) Yang-Mills theory with a lattice Wilson Dirac operator with an additional parity mass, similar as in twisted mass lattice QCD. We show that a special 45° twist effectively removes the mass splitting of the chiral partners. Thus, at finite lattice spacing both chiral and supersymmetry are enhanced resulting in an improved continuum extrapolation. Furthermore, we show that for the non-interacting theory at 45° twist discretization errors of order \( \mathcal{O}(a) \) are suppressed, suggesting that the same happens for the interacting theory as well. As an aside, we demonstrate that the DDαAMG multigrid algorithm accelerates the inversion of the Wilson Dirac operator considerably. On a 163 × 32 lattice, speed-up factors of up to 20 are reached if commonly used algorithms are replaced by the DDαAMG.

References

  1. Particle Data Group collaboration, Review of particle physics, Phys. Rev. D 98 (2018) 030001 [INSPIRE].

  2. E. Witten, Dynamical breaking of supersymmetry, Nucl. Phys. B 188 (1981) 513.

    ADS  MATH  Google Scholar 

  3. S. Dimopoulos and H. Georgi, Softly broken supersymmetry and SU(5), Nucl. Phys. B 193 (1981) 150.

    ADS  Google Scholar 

  4. J. Ellis, J. S. Hagelin, D. V. Nanopoulos, K. Olive and M. Srednicki, Supersymmetric relics from the big bang, Nucl. Phys. B 238 (1984) 453.

    ADS  Google Scholar 

  5. P. Dondi and H. Nicolai, Lattice supersymmetry, Nuovo Cim. A 41 (1977) 1.

    ADS  MathSciNet  Google Scholar 

  6. S. Catterall and S. Karamov, Exact lattice supersymmetry: the two-dimensional N = 2 Wess-Zumino model, Phys. Rev. D 65 (2002) 094501 [hep-lat/0108024] [INSPIRE].

  7. G. Bergner, T. Kaestner, S. Uhlmann and A. Wipf, Low-dimensional supersymmetric lattice models, Annals Phys. 323 (2008) 946 [arXiv:0705.2212] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  8. T. Kastner, G. Bergner, S. Uhlmann, A. Wipf and C. Wozar, Two-dimensional Wess-Zumino models at intermediate couplings, Phys. Rev. D 78 (2008) 095001 [arXiv:0807.1905] [INSPIRE].

    ADS  Google Scholar 

  9. I. Kanamori, F. Sugino and H. Suzuki, Observing dynamical supersymmetry breaking with euclidean lattice simulations, Prog. Theor. Phys. 119 (2008) 797 [arXiv:0711.2132] [INSPIRE].

    ADS  MATH  Google Scholar 

  10. K. Steinhauer and U. Wenger, Loop formulation of supersymmetric Yang-Mills quantum mechanics, JHEP 12 (2014) 044 [arXiv:1410.0235] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  11. R. Flore, D. Korner, A. Wipf and C. Wozar, Supersymmetric nonlinear O(3) σ-model on the lattice, JHEP 11 (2012) 159 [arXiv:1207.6947] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  12. G. Koutsoumbas and I. Montvay, Gluinos on the lattice: quenched calculations, Phys. Lett. B 398 (1997) 130 [hep-lat/9612003] [INSPIRE].

    ADS  Google Scholar 

  13. A. Donini, M. Guagnelli, P. Hernández and A. Vladikas, Towards N = 1 super-Yang-Mills on the lattice, Nucl. Phys. B 523 (1998) 529 [hep-lat/9710065] [INSPIRE].

    ADS  Google Scholar 

  14. DESY-Munster collaboration, Evidence for discrete chiral symmetry breaking in N = 1 supersymmetric Yang-Mills theory, Phys. Lett. B 446 (1999) 209 [hep-lat/9810062] [INSPIRE].

  15. DESY-Munster collaboration, Monte Carlo simulation of SU(2) Yang-Mills theory with light gluinos, Eur. Phys. J. C 11 (1999) 507 [hep-lat/9903014] [INSPIRE].

  16. 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].

    ADS  MATH  Google Scholar 

  17. S. Ali et al., Variational analysis of low-lying states in supersymmetric Yang-Mills theory, JHEP 04 (2019) 150 [arXiv:1901.02416] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  18. DESY-Munster-Roma collaboration, The supersymmetric Ward identities on the lattice, Eur. Phys. J. C 23 (2002) 719 [hep-lat/0111008] [INSPIRE].

  19. 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].

    ADS  Google Scholar 

  20. G. Bergner, C. López and S. Piemonte, Study of center and chiral symmetry realization in thermal \( \mathcal{N} \) = 1 super Yang-Mills theory using the gradient flow, Phys. Rev. D 100 (2019) 074501 [arXiv:1902.08469] [INSPIRE].

    ADS  Google Scholar 

  21. F. Farchioni et al., SUSY Ward identities in 1 loop perturbation theory, Nucl. Phys. B Proc. Suppl. 106 (2002) 941 [hep-lat/0110113] [INSPIRE].

    ADS  Google Scholar 

  22. 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].

    ADS  MathSciNet  MATH  Google Scholar 

  23. S. Musberg, G. Münster and S. Piemonte, Perturbative calculation of the clover term for Wilson fermions in any representation of the gauge group SU(N ), JHEP 05 (2013) 143 [arXiv:1304.5741] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  24. S. Ali et al., Numerical results for the lightest bound states in \( \mathcal{N} \) = 1 supersymmetric SU(3) Yang-Mills theory, Phys. Rev. Lett. 122 (2019) 221601 [arXiv:1902.11127] [INSPIRE].

    ADS  Google Scholar 

  25. S. Ali et al., Analysis of Ward identities in supersymmetric Yang-Mills theory, Eur. Phys. J. C 78 (2018) 404 [arXiv:1802.07067] [INSPIRE].

    ADS  Google Scholar 

  26. H. Neuberger, Vector-like gauge theories with almost massless fermions on the lattice, Phys. Rev. D 57 (1998) 5417 [hep-lat/9710089] [INSPIRE].

    ADS  Google Scholar 

  27. D.B. Kaplan and M. Schmaltz, Supersymmetric Yang-Mills theories from domain wall fermions, Chin. J. Phys. 38 (2000) 543 [hep-lat/0002030] [INSPIRE].

  28. J. Giedt, R. Brower, S. Catterall, G.T. Fleming and P. Vranas, Lattice Super-Yang-Mills using domain wall fermions in the chiral limit, Phys. Rev. D 79 (2009) 025015 [arXiv:0810.5746] [INSPIRE].

    ADS  Google Scholar 

  29. JLQCD collaboration, Lattice study of 4d \( \mathcal{N} \) = 1 super Yang-Mills theory with dynamical overlap gluino, PoS(LATTICE2011)069 [arXiv:1111.2180] [INSPIRE].

  30. S. Ali et al., Continuum limit of SU(3) \( \mathcal{N} \) = 1 supersymmetric Yang-Mills theory and supersymmetric gauge theories on the lattice, PoS LATTICE2019 (2020) 175 [arXiv:2001.09682] [INSPIRE].

  31. G.T. Fleming, J.B. Kogut and P.M. Vranas, SuperYang-Mills on the lattice with domain wall fermions, Phys. Rev. D 64 (2001) 034510 [hep-lat/0008009] [INSPIRE].

    ADS  Google Scholar 

  32. D. August, M. Steinhauser, B.H. Wellegehausen and A. Wipf, Mass spectrum of 2-dimensional \( \mathcal{N} \) = (2, 2) super Yang-Mills theory on the lattice, JHEP 01 (2019) 099 [arXiv:1802.07797] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  33. D. Kadoh and H. Suzuki, SUSY WT identity in a lattice formulation of 2D = (2, 2) SYM, Phys. Lett. B 682 (2010) 466 [arXiv:0908.2274] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  34. S. Catterall, R.G. Jha and A. Joseph, Nonperturbative study of dynamical SUSY breaking in N = (2, 2) Yang-Mills theory, Phys. Rev. D 97 (2018) 054504 [arXiv:1801.00012] [INSPIRE].

    ADS  Google Scholar 

  35. M. Hanada and I. Kanamori, Lattice study of two-dimensional N = (2, 2) super Yang-Mills at large-N , Phys. Rev. D 80 (2009) 065014 [arXiv:0907.4966] [INSPIRE].

    ADS  Google Scholar 

  36. S. Catterall, D.B. Kaplan and M. Ünsal, Exact lattice supersymmetry, Phys. Rept. 484 (2009) 71 [arXiv:0903.4881] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  37. D. Schaich, S. Catterall, P.H. Damgaard and J. Giedt, Latest results from lattice N = 4 supersymmetric Yang–Mills, PoS(LATTICE2016)221 [arXiv:1611.06561] [INSPIRE].

  38. E. Giguère and D. Kadoh, Restoration of supersymmetry in two-dimensional SYM with sixteen supercharges on the lattice, JHEP 05 (2015) 082 [arXiv:1503.04416] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  39. Alpha collaboration, Lattice QCD with a chirally twisted mass term, JHEP 08 (2001) 058 [hep-lat/0101001] [INSPIRE].

  40. R. Frezzotti and G.C. Rossi, Chirally improving Wilson fermions. 1. O(a) improvement, JHEP 08 (2004) 007 [hep-lat/0306014] [INSPIRE].

  41. G. Veneziano and S. Yankielowicz, An effective Lagrangian for the pure N = 1 supersymmetric Yang-Mills theory, Phys. Lett. B 113 (1982) 231.

    ADS  Google Scholar 

  42. G.R. Farrar, G. Gabadadze and M. Schwetz, On the effective action of N = 1 supersymmetric Yang-Mills theory, Phys. Rev. D 58 (1998) 015009 [hep-th/9711166] [INSPIRE].

    ADS  Google Scholar 

  43. G.R. Farrar, G. Gabadadze and M. Schwetz, The spectrum of softly broken N = 1 supersymmetric Yang-Mills theory, Phys. Rev. D 60 (1999) 035002 [hep-th/9806204] [INSPIRE].

    ADS  Google Scholar 

  44. A. Jaffe, Euclidean quantum field theory, Nucl. Phys. B 254 (1985) 31.

    ADS  MathSciNet  Google Scholar 

  45. G. Curci and G. Veneziano, Supersymmetry and the lattice: a reconciliation?, Nucl. Phys. B 292 (1987) 555 [INSPIRE].

    ADS  Google Scholar 

  46. A.D. Kennedy, I. Horvath and S. Sint, A New exact method for dynamical fermion computations with nonlocal actions, Nucl. Phys. B Proc. Suppl. 73 (1999) 834 [hep-lat/9809092] [INSPIRE].

    ADS  MATH  Google Scholar 

  47. J. Nishimura, Four-dimensional N = 1 supersymmetric Yang-Mills theory on the lattice without fine tuning, Phys. Lett. B 406 (1997) 215 [hep-lat/9701013] [INSPIRE].

  48. F. Knechtli, M. Günther and M. Peardon, Lattice quantum chromodynamics: practical essentials, Springer Briefs in Physics, Springer, Germany (2017) [INSPIRE].

  49. K. Demmouche et al., Simulation of 4d N = 1 supersymmetric Yang-Mills theory with Symanzik improved gauge action and stout smearing, Eur. Phys. J. C 69 (2010) 147 [arXiv:1003.2073] [INSPIRE].

    ADS  Google Scholar 

  50. S. Kuberski, Bestimmung von Massen in der supersymmetrischen Yang-Mills-Theorie mit der Variationsmethode, Masterarbeit, Westfälische Wilhelms-Universität Münster, Münster, Germany (2017).

    Google Scholar 

  51. F. Heitger, Darstel lungstheorie der kubischen Gruppe in Anwendung auf Operatoren der N=1 SUSY-Yang-Mills-Theorie auf dem Gitter, Diplomarbeit, Westfälische Wilhelms-Universität Münster, Münster, Germany (1993).

    Google Scholar 

  52. B. Berg and A. Billoire, Glueball spectroscopy in four-dimensional SU(3) lattice gauge theory. 1., Nucl. Phys. B 221 (1983) 109 [INSPIRE].

  53. M. Fierz, Zur Fermischen Theorie des β-Zerfalls, Z. Phys. 104 (1937) 553.

    ADS  MATH  Google Scholar 

  54. P.B. Pal, Representation-independent manipulations with Dirac spinors, physics/0703214 [INSPIRE].

  55. S. Luckmann, Ward-Identitäten in der N = 1 Super-Yang-Mills-Theorie, Diplomarbeit, Westfälische Wilhelms-Universität Münster, Münster, Germany (1997).

    Google Scholar 

  56. R. Kirchner, Ward identities and mass spectrum of N = 1 Super Yang-Mills theory on the lattice, DESY-THESIS-2000-043 (2000).

  57. I. Montvay, Supersymmetric Yang-Mills theory on the lattice, Int. J. Mod. Phys. A 17 (2002) 2377 [hep-lat/0112007] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  58. P. van Nieuwenhuizen and A. Waldron, On Euclidean spinors and Wick rotations, Phys. Lett. B 389 (1996) 29 [hep-th/9608174] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  59. Y. Taniguchi, One loop calculation of SUSY Ward-Takahashi identity on lattice with Wilson fermion, Phys. Rev. D 63 (2000) 014502 [hep-lat/9906026] [INSPIRE].

    ADS  Google Scholar 

  60. T. Kästner, Supersymmetry on a space-time lattice, Dissertation, Friedrich Schiller University Jena, Jena, Germany (2008).

    Google Scholar 

  61. A. Wipf, Statistical approach to quantum field theory: An introduction, Springer, Berlin Germany (2013) [INSPIRE].

  62. 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].

  63. C. Morningstar and M.J. Peardon, Analytic smearing of SU(3) link variables in lattice QCD, Phys. Rev. D 69 (2004) 054501 [hep-lat/0311018] [INSPIRE].

  64. S. Ali et al., The light bound states of \( \mathcal{N} \) = 1 supersymmetric SU(3) Yang-Mills theory on the lattice, JHEP 03 (2018) 113 [arXiv:1801.08062] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  65. N.J. Evans, S.D.H. Hsu and M. Schwetz, Lattice tests of supersymmetric Yang-Mills theory?, hep-th/9707260 [INSPIRE].

  66. RQCD collaboration, Direct determinations of the nucleon and pion σ terms at nearly physical quark masses, Phys. Rev. D 93 (2016) 094504 [arXiv:1603.00827] [INSPIRE].

  67. S. Aoki, O. Bär, S. Takeda and T. Ishikawa, Pseudo scalar meson masses in Wilson chiral perturbation theory for 2 + 1 flavors, Phys. Rev. D 73 (2006) 014511 [hep-lat/0509049] [INSPIRE].

  68. F. Farchioni, I. Montvay, G. Munster, E.E. Scholz, T. Sudmann and J. Wuilloud, Hadron masses in QCD with one quark flavour, Eur. Phys. J. C 52 (2007) 305 [arXiv:0706.1131] [INSPIRE].

    ADS  Google Scholar 

  69. M. Wimmer, Efficient numerical computation of the pfaffian for dense and banded skew-symmetric matrices, ACM Trans. Math. Softw. 38 (2012) 1.

    MathSciNet  MATH  Google Scholar 

  70. C. Alexandrou, S. Bacchio, J. Finkenrath, A. Frommer, K. Kahl and M. Rottmann, Adaptive aggregation-based domain decomposition multigrid for twisted mass fermions, Phys. Rev. D 94 (2016) 114509 [arXiv:1610.02370] [INSPIRE].

    ADS  Google Scholar 

  71. S. Bacchio, DDalphaAMG library including twisted mass fermions, https://github.com/sbacchio/DDalphaAMG.

  72. M. Costa and H. Panagopoulos, Supersymmetric QCD on the lattice: an exploratory study, Phys. Rev. D 96 (2017) 034507 [arXiv:1706.05222] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  73. B. Wellegehausen and A. Wipf, \( \mathcal{N} \) = 1 supersymmetric SU(3) gauge theory — Towards simulations of Super-QCD, PoS(LATTICE2018)210 [arXiv:1811.01784] [INSPIRE].

  74. G. Bergner and S. Piemonte, Supersymmetric and conformal theories on the lattice: from super Yang-Mills towards super QCD, PoS(LATTICE2018)209 [arXiv:1811.01797] [INSPIRE].

  75. D. Weingarten, Mass inequalities for QCD, Phys. Rev. Lett. 51 (1983) 1830 [INSPIRE].

    ADS  Google Scholar 

  76. G. Kilcup and S.R. Sharpe, Phenomenology and lattice QCD, World Scientific, Singapore (1995).

    Google Scholar 

  77. E.V. Shuryak, The QCD vacuum, hadrons and the superdense matter, World Scientific, Singapore (2004) [INSPIRE].

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marc Steinhauser.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 2010.00946

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Steinhauser, M., Sternbeck, A., Wellegehausen, B. et al. \( \mathcal{N} \) = 1 Super-Yang-Mills theory on the lattice with twisted mass fermions. J. High Energ. Phys. 2021, 154 (2021). https://doi.org/10.1007/JHEP01(2021)154

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP01(2021)154

Keywords

  • Lattice QCD
  • Lattice Quantum Field Theory
  • Supersymmetric Gauge Theory