Advertisement

Journal of High Energy Physics

, 2018:151 | Cite as

Complex Langevin analysis of the spontaneous symmetry breaking in dimensionally reduced super Yang-Mills models

  • Konstantinos N. Anagnostopoulos
  • Takehiro Azuma
  • Yuta Ito
  • Jun NishimuraEmail author
  • Stratos Kovalkov Papadoudis
Open Access
Regular Article - Theoretical Physics
First Online:

Abstract

In recent years the complex Langevin method (CLM) has proven a powerful method in studying statistical systems which suffer from the sign problem. Here we show that it can also be applied to an important problem concerning why we live in four-dimensional spacetime. Our target system is the type IIB matrix model, which is conjectured to be a nonperturbative definition of type IIB superstring theory in ten dimensions. The fermion determinant of the model becomes complex upon Euclideanization, which causes a severe sign problem in its Monte Carlo studies. It is speculated that the phase of the fermion determinant actually induces the spontaneous breaking of the SO(10) rotational symmetry, which has direct consequences on the aforementioned question. In this paper, we apply the CLM to the 6D version of the type IIB matrix model and show clear evidence that the SO(6) symmetry is broken down to SO(3). Our results are consistent with those obtained previously by the Gaussian expansion method.

Keywords

Matrix Models 1/N Expansion 
Download to read the full article text

Notes

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.

References

  1. [1]
    AuroraScience collaboration, M. Cristoforetti et al., New approach to the sign problem in quantum field theories: high density QCD on a Lefschetz thimble, Phys. Rev. D 86 (2012) 074506 [arXiv:1205.3996] [INSPIRE].
  2. [2]
    M. Cristoforetti, F. Di Renzo, A. Mukherjee and L. Scorzato, Monte Carlo simulations on the Lefschetz thimble: taming the sign problem, Phys. Rev. D 88 (2013) 051501 [arXiv:1303.7204] [INSPIRE].
  3. [3]
    H. Fujii, D. Honda, M. Kato, Y. Kikukawa, S. Komatsu and T. Sano, Hybrid Monte Carlo on Lefschetz thimbles — A study of the residual sign problem, JHEP 10 (2013) 147 [arXiv:1309.4371] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A. Alexandru et al., Sign problem and Monte Carlo calculations beyond Lefschetz thimbles, JHEP 05 (2016) 053 [arXiv:1512.08764] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    G. Parisi, On complex probabilities, Phys. Lett. 131B (1983) 393 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  6. [6]
    J.R. Klauder, Coherent state Langevin equations for canonical quantum systems with applications to the quantized Hall effect, Phys. Rev. A 29 (1984) 2036 [INSPIRE].
  7. [7]
    G. Parisi and Y.S. Wu, Perturbation theory without gauge fixing, Sci. Sin. 24 (1981) 483 [INSPIRE].MathSciNetGoogle Scholar
  8. [8]
    G. Aarts, E. Seiler and I.-O. Stamatescu, The complex Langevin method: when can it be trusted?, Phys. Rev. D 81 (2010) 054508 [arXiv:0912.3360] [INSPIRE].
  9. [9]
    G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Complex Langevin: etiology and diagnostics of its main problem, Eur. Phys. J. C 71 (2011) 1756 [arXiv:1101.3270] [INSPIRE].
  10. [10]
    J. Nishimura and S. Shimasaki, New insights into the problem with a singular drift term in the complex Langevin method, Phys. Rev. D 92 (2015) 011501 [arXiv:1504.08359] [INSPIRE].
  11. [11]
    K. Nagata, J. Nishimura and S. Shimasaki, Argument for justification of the complex Langevin method and the condition for correct convergence, Phys. Rev. D 94 (2016) 114515 [arXiv:1606.07627] [INSPIRE].ADSGoogle Scholar
  12. [12]
    L.L. Salcedo, Does the complex Langevin method give unbiased results?, Phys. Rev. D 94 (2016) 114505 [arXiv:1611.06390] [INSPIRE].
  13. [13]
    G. Aarts, E. Seiler, D. Sexty and I.-O. Stamatescu, Complex Langevin dynamics and zeroes of the fermion determinant, JHEP 05 (2017) 044 [Erratum ibid. 01 (2018) 128] [arXiv:1701.02322] [INSPIRE].
  14. [14]
    E. Seiler, D. Sexty and I.-O. Stamatescu, Gauge cooling in complex Langevin for QCD with heavy quarks, Phys. Lett. B 723 (2013) 213 [arXiv:1211.3709] [INSPIRE].
  15. [15]
    K. Nagata, J. Nishimura and S. Shimasaki, Justification of the complex Langevin method with the gauge cooling procedure, PTEP 2016 (2016) 013B01 [arXiv:1508.02377] [INSPIRE].
  16. [16]
    S. Tsutsui and T.M. Doi, Improvement in complex Langevin dynamics from a view point of Lefschetz thimbles, Phys. Rev. D 94 (2016) 074009 [arXiv:1508.04231] [INSPIRE].
  17. [17]
    K. Nagata, J. Nishimura and S. Shimasaki, Gauge cooling for the singular-drift problem in the complex Langevin method — A test in random matrix theory for finite density QCD, JHEP 07 (2016) 073 [arXiv:1604.07717] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    Y. Ito and J. Nishimura, The complex Langevin analysis of spontaneous symmetry breaking induced by complex fermion determinant, JHEP 12 (2016) 009 [arXiv:1609.04501] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    J. Bloch, Reweighting complex Langevin trajectories, Phys. Rev. D 95 (2017) 054509 [arXiv:1701.00986] [INSPIRE].
  20. [20]
    T.M. Doi and S. Tsutsui, Modifying partition functions: a way to solve the sign problem, Phys. Rev. D 96 (2017) 094511 [arXiv:1709.05806] [INSPIRE].
  21. [21]
    N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A large-N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/9612115] [INSPIRE].
  22. [22]
    J. Nishimura and G. Vernizzi, Spontaneous breakdown of Lorentz invariance in IIB matrix model, JHEP 04 (2000) 015 [hep-th/0003223] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    J. Nishimura and G. Vernizzi, Brane world generated dynamically from string type IIB matrices, Phys. Rev. Lett. 85 (2000) 4664 [hep-th/0007022] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    K.N. Anagnostopoulos and J. Nishimura, New approach to the complex action problem and its application to a nonperturbative study of superstring theory, Phys. Rev. D 66 (2002) 106008 [hep-th/0108041] [INSPIRE].
  25. [25]
    H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99 (1998) 713 [hep-th/9802085] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  26. [26]
    J. Nishimura and F. Sugino, Dynamical generation of four-dimensional space-time in the IIB matrix model, JHEP 05 (2002) 001 [hep-th/0111102] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  27. [27]
    J. Nishimura, T. Okubo and F. Sugino, Systematic study of the SO(10) symmetry breaking vacua in the matrix model for type IIB superstrings, JHEP 10 (2011) 135 [arXiv:1108.1293] [INSPIRE].
  28. [28]
    J. Nishimura, Exactly solvable matrix models for the dynamical generation of space-time in superstring theory, Phys. Rev. D 65 (2002) 105012 [hep-th/0108070] [INSPIRE].
  29. [29]
    A. Mollgaard and K. Splittorff, Complex Langevin dynamics for chiral random matrix theory, Phys. Rev. D 88 (2013) 116007 [arXiv:1309.4335] [INSPIRE].
  30. [30]
    J. Nishimura, T. Okubo and F. Sugino, Gaussian expansion analysis of a matrix model with the spontaneous breakdown of rotational symmetry, Prog. Theor. Phys. 114 (2005) 487 [hep-th/0412194] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  31. [31]
    K.N. Anagnostopoulos, T. Azuma and J. Nishimura, A general approach to the sign problem: the factorization method with multiple observables, Phys. Rev. D 83 (2011) 054504 [arXiv:1009.4504] [INSPIRE].
  32. [32]
    K.N. Anagnostopoulos, T. Azuma and J. Nishimura, A practical solution to the sign problem in a matrix model for dynamical compactification, JHEP 10 (2011) 126 [arXiv:1108.1534] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    T. Aoyama, J. Nishimura and T. Okubo, Spontaneous breaking of the rotational symmetry in dimensionally reduced super Yang-Mills models, Prog. Theor. Phys. 125 (2011) 537 [arXiv:1007.0883] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  34. [34]
    K.N. Anagnostopoulos, T. Azuma and J. Nishimura, Monte Carlo studies of the spontaneous rotational symmetry breaking in dimensionally reduced super Yang-Mills models, JHEP 11 (2013) 009 [arXiv:1306.6135] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  35. [35]
    K.N. Anagnostopoulos, T. Azuma and J. Nishimura, Monte Carlo studies of dynamical compactification of extra dimensions in a model of nonperturbative string theory, PoS(LATTICE 2015)307 [arXiv:1509.05079] [INSPIRE].
  36. [36]
    S.-W. Kim, J. Nishimura and A. Tsuchiya, Expanding (3 + 1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9 + 1)-dimensions, Phys. Rev. Lett. 108 (2012) 011601 [arXiv:1108.1540] [INSPIRE].
  37. [37]
    Y. Ito, S.-W. Kim, Y. Koizuka, J. Nishimura and A. Tsuchiya, A renormalization group method for studying the early universe in the Lorentzian IIB matrix model, PTEP 2014 (2014) 083B01 [arXiv:1312.5415] [INSPIRE].
  38. [38]
    Y. Ito, J. Nishimura and A. Tsuchiya, Power-law expansion of the Universe from the bosonic Lorentzian type IIB matrix model, JHEP 11 (2015) 070 [arXiv:1506.04795] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  39. [39]
    Y. Ito, J. Nishimura and A. Tsuchiya, Universality and the dynamical space-time dimensionality in the Lorentzian type IIB matrix model, JHEP 03 (2017) 143 [arXiv:1701.07783] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    W. Krauth, H. Nicolai and M. Staudacher, Monte Carlo approach to M-theory, Phys. Lett. B 431 (1998) 31 [hep-th/9803117] [INSPIRE].
  41. [41]
    P. Austing and J.F. Wheater, Convergent Yang-Mills matrix theories, JHEP 04 (2001) 019 [hep-th/0103159] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  42. [42]
    J. Ambjørn et al., Large-N dynamics of dimensionally reduced 4D SU(N) super Yang-Mills theory, JHEP 07 (2000) 013 [hep-th/0003208] [INSPIRE].
  43. [43]
    T. Hotta, J. Nishimura and A. Tsuchiya, Dynamical aspects of large-N reduced models, Nucl. Phys. B 545 (1999) 543 [hep-th/9811220] [INSPIRE].
  44. [44]
    J. Ambjørn et al., Monte Carlo studies of the IIB matrix model at large-N , JHEP 07 (2000) 011 [hep-th/0005147] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    G. Aarts, F.A. James, E. Seiler and I.-O. Stamatescu, Adaptive stepsize and instabilities in complex Langevin dynamics, Phys. Lett. B 687 (2010) 154 [arXiv:0912.0617] [INSPIRE].
  46. [46]
    K.N. Anagnostopoulos, T. Azuma, Y. Ito, J. Nishimura and S.K. Papadoudis, work in progress.Google Scholar
  47. [47]
    K. Nagata, J. Nishimura and S. Shimasaki, Complex Langevin simulation of QCD at finite density and low temperature using the deformation technique, arXiv:1710.07416 [INSPIRE].
  48. [48]
    G.G. Batrouni et al., Langevin simulations of lattice field theories, Phys. Rev. D 32 (1985) 2736 [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Physics DepartmentNational Technical UniversityAthensGreece
  2. 2.Institute for Fundamental SciencesSetsunan UniversityNeyagawaJapan
  3. 3.KEK Theory Center, High Energy Accelerator Research OrganizationTsukubaJapan
  4. 4.Department of Particle and Nuclear Physics, School of High Energy Accelerator ScienceGraduate University for Advanced Studies (SOKENDAI)TsukubaJapan

Personalised recommendations