Complex Langevin simulations and the QCD phase diagram: recent developments

Abstract

In this review we present the current state-of-the-art on complex Langevin simulations and their implications for the QCD phase diagram. After a short summary of the complex Langevin method, we present and discuss recent developments. Here we focus on the explicit computation of boundary terms, which provide an observable that can be used to check one of the criteria of correctness explicitly. We also present the method of Dynamic Stabilization and elaborate on recent results for fully dynamical QCD.

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Fig. 1

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.]

Notes

  1. 1.

    It has been recently shown in a study of 2D U(1) gauge theory on a torus with a \(\theta \)-term that correct convergence can be obtained even when the unitarity norm is large [59]. In that study, it was also found that observables only thermalise after the unitarity norm saturates.

References

  1. 1.

    G. Parisi, Ys. Wu, Sci. Sin. 24, 483 (1981)

  2. 2.

    F. Karsch, H.W. Wyld, Phys. Rev. Lett. 55, 2242 (1985)

    ADS  Google Scholar 

  3. 3.

    P.H. Damgaard, H. Hüffel, Phys. Rep. 152, 227 (1987)

    ADS  MathSciNet  Google Scholar 

  4. 4.

    J. Ambjorn, S.K. Yang, Phys. Lett. B 165, 140 (1985)

    ADS  Google Scholar 

  5. 5.

    J.R. Klauder, W.P. Petersen, J. Stat. Phys. 39, 53 (1985)

    ADS  Google Scholar 

  6. 6.

    H.Q. Lin, J.E. Hirsch, Phys. Rev. B 34, 1964 (1986)

    ADS  Google Scholar 

  7. 7.

    J. Ambjorn, M. Flensburg, C. Peterson, Nucl. Phys. B 275, 375 (1986)

    ADS  Google Scholar 

  8. 8.

    J. Berges, I.O. Stamatescu, Phys. Rev. Lett. 95, 202003 (2005). arXiv:hep-lat/0508030

    ADS  Google Scholar 

  9. 9.

    J. Berges, S. Borsányi, D. Sexty, I.O. Stamatescu, Phys. Rev. D 75, 045007 (2007). arXiv:hep-lat/0609058

    ADS  Google Scholar 

  10. 10.

    J. Berges, D. Sexty, Nucl. Phys. B 799, 306 (2008). arXiv:0708.0779

    ADS  Google Scholar 

  11. 11.

    C. Pehlevan, G. Guralnik, Nucl. Phys. B 811, 519 (2009). arXiv:0710.3756

    ADS  Google Scholar 

  12. 12.

    G. Aarts, I.O. Stamatescu, JHEP 09, 018 (2008). arXiv:0807.1597

    ADS  Google Scholar 

  13. 13.

    G. Aarts, Phys. Rev. Lett. 102, 131601 (2009). arXiv:0810.2089

    ADS  Google Scholar 

  14. 14.

    G. Guralnik, C. Pehlevan, Nucl. Phys. B 822, 349 (2009). arXiv:0902.1503

    ADS  Google Scholar 

  15. 15.

    G. Aarts, E. Seiler, I.O. Stamatescu, Phys. Rev. D 81, 054508 (2010). arXiv:0912.3360

    ADS  Google Scholar 

  16. 16.

    G. Aarts, F.A. James, JHEP 08, 020 (2010). arXiv:1005.3468

    ADS  Google Scholar 

  17. 17.

    G. Aarts, K. Splittorff, JHEP 08, 017 (2010). arXiv:1006.0332

    ADS  Google Scholar 

  18. 18.

    G. Aarts, F.A. James, E. Seiler, I.O. Stamatescu, Phys. Lett. B 687, 154 (2010). arXiv:0912.0617

    ADS  Google Scholar 

  19. 19.

    G. Aarts, F.A. James, E. Seiler, I.O. Stamatescu, Eur. Phys. J. C 71, 1756 (2011). arXiv:1101.3270

    ADS  Google Scholar 

  20. 20.

    K. Nagata, J. Nishimura, S. Shimasaki, Phys. Rev. D 94, 114515 (2016). arXiv:1606.07627

    ADS  Google Scholar 

  21. 21.

    E. Seiler, D. Sexty, I.O. Stamatescu, Phys. Lett. B 723, 213 (2013). arXiv:1211.3709

    ADS  Google Scholar 

  22. 22.

    G. Aarts, L. Bongiovanni, E. Seiler, D. Sexty, I.O. Stamatescu, Eur. Phys. J. A 49, 89 (2013). arXiv:1303.6425

    ADS  Google Scholar 

  23. 23.

    G. Aarts, F.A. James, JHEP 01, 118 (2012). arXiv:1112.4655

    ADS  Google Scholar 

  24. 24.

    J.M. Pawlowski, C. Zielinski, Phys. Rev. D 87, 094503 (2013). arXiv:1302.1622

    ADS  Google Scholar 

  25. 25.

    J.M. Pawlowski, C. Zielinski, Phys. Rev. D 87, 094509 (2013). arXiv:1302.2249

    ADS  Google Scholar 

  26. 26.

    A. Mollgaard, K. Splittorff, Phys. Rev. D 88, 116007 (2013). arXiv:1309.4335

    ADS  Google Scholar 

  27. 27.

    A. Mollgaard, K. Splittorff, Phys. Rev. D 91, 036007 (2015). arXiv:1412.2729

    ADS  Google Scholar 

  28. 28.

    J. Bloch, J. Glesaaen, J.J.M. Verbaarschot, S. Zafeiropoulos, JHEP 03, 015 (2018). arXiv:1712.07514

    ADS  Google Scholar 

  29. 29.

    D. Sexty, Phys. Lett. B 729, 108 (2014). arXiv:1307.7748

    ADS  Google Scholar 

  30. 30.

    G. Aarts, E. Seiler, D. Sexty, I.O. Stamatescu, Phys. Rev. D 90, 114505 (2014). arXiv:1408.3770

    ADS  Google Scholar 

  31. 31.

    G. Aarts, F. Attanasio, B. Jäger, D. Sexty, JHEP 09, 087 (2016). arXiv:1606.05561

    ADS  Google Scholar 

  32. 32.

    J. Nishimura, A. Tsuchiya, JHEP 06, 077 (2019). arXiv:1904.05919

    ADS  Google Scholar 

  33. 33.

    T. Hayata, A. Yamamoto, Phys. Rev. A 92, 043628 (2015). arXiv:1411.5195

    ADS  Google Scholar 

  34. 34.

    C. Berger, J. Drut, PoS LATTICE2018, 244 (2018)

    Google Scholar 

  35. 35.

    F. Attanasio, J.E. Drut, Phys. Rev. A 101, 033617 (2020). arXiv:1908.02715

    ADS  MathSciNet  Google Scholar 

  36. 36.

    A.C. Loheac, J.E. Drut, Phys. Rev. D 95, 094502 (2017). arXiv:1702.04666

    ADS  Google Scholar 

  37. 37.

    A.C. Loheac, J. Braun, J.E. Drut, Phys. Rev. D 98, 054507 (2018). arXiv:1804.10257

    ADS  Google Scholar 

  38. 38.

    L. Rammelmüller, A.C. Loheac, J.E. Drut, J. Braun, Phys. Rev. Lett. 121, 173001 (2018). arXiv:1807.04664

    ADS  Google Scholar 

  39. 39.

    L. Rammelmüller, W.J. Porter, J.E. Drut, J. Braun, Phys. Rev. D 96, 094506 (2017). arXiv:1708.03149

    ADS  Google Scholar 

  40. 40.

    L. Rammelmüller, J.E. Drut, J. Braun (2020). arXiv:2003.06853

  41. 41.

    C. Shill, J. Drut, Phys. Rev. A 98, 053615 (2018). arXiv:1808.07836

    ADS  Google Scholar 

  42. 42.

    P. Basu, K. Jaswin, A. Joseph, Phys. Rev. D 98, 034501 (2018). arXiv:1802.10381

    ADS  Google Scholar 

  43. 43.

    A. Joseph, A. Kumar, Phys. Rev. D 100, 074507 (2019). arXiv:1908.04153

    ADS  MathSciNet  Google Scholar 

  44. 44.

    G. Aarts, P. Giudice, E. Seiler, Ann. Phys. 337, 238 (2013). arXiv:1306.3075

    ADS  Google Scholar 

  45. 45.

    G. Aarts, F.A. James, J.M. Pawlowski, E. Seiler, D. Sexty, I.O. Stamatescu, JHEP 03, 073 (2013). arXiv:1212.5231

    ADS  Google Scholar 

  46. 46.

    Z. Cai, Y. Di, X. Dong, Commun. Comput. Phys. 27, 1344 (2020). arXiv:1905.11683

    Google Scholar 

  47. 47.

    K. Nagata, J. Nishimura, S. Shimasaki, PTEP 2016, 013B01 (2016). arXiv:1508.02377

    Google Scholar 

  48. 48.

    J. Nishimura, S. Shimasaki, Phys. Rev. D 92, 011501 (2015). arXiv:1504.08359

    ADS  Google Scholar 

  49. 49.

    G. Aarts, E. Seiler, D. Sexty, I.O. Stamatescu, JHEP 05, 044 (2017). arXiv:1701.02322

    ADS  Google Scholar 

  50. 50.

    K. Splittorff, Phys. Rev. D 91, 034507 (2015). arXiv:1412.0502

    ADS  MathSciNet  Google Scholar 

  51. 51.

    J. Greensite, Phys. Rev. D 90, 114507 (2014). arXiv:1406.4558

    ADS  Google Scholar 

  52. 52.

    E. Seiler (2020). arXiv:2006.04714

  53. 53.

    K. Nagata, J. Nishimura, S. Shimasaki, JHEP 05, 004 (2018). arXiv:1802.01876

    ADS  Google Scholar 

  54. 54.

    S. Tsutsui, Y. Ito, H. Matsufuru, J. Nishimura, S. Shimasaki, A. Tsuchiya (2019). arXiv:1912.00361

  55. 55.

    M. Scherzer, E. Seiler, D. Sexty, I.O. Stamatescu, Phys. Rev. D 99, 014512 (2019). arXiv:1808.05187

    ADS  MathSciNet  Google Scholar 

  56. 56.

    Z. Cai, X. Dong, Y. Kuang (2020). arXiv:2007.10198

  57. 57.

    M. Scherzer, E. Seiler, D. Sexty, I.O. Stamatescu, Phys. Rev. D 101, 014501 (2020). arXiv:1910.09427

    ADS  MathSciNet  Google Scholar 

  58. 58.

    D. Banerjee, S. Chandrasekharan, Phys. Rev. D 81, 125007 (2010). arXiv:1001.3648

    ADS  Google Scholar 

  59. 59.

    M. Hirasawa, A. Matsumoto, J. Nishimura, A. Yosprakob (2020). arXiv:2004.13982

  60. 60.

    M. Scherzer, PhD thesis (Heidelberg, 2019)

  61. 61.

    F. Attanasio, B. Jäger, Eur. Phys. J. C 79, 16 (2019). arXiv:1808.04400

    ADS  Google Scholar 

  62. 62.

    K. Nagata, J. Nishimura, S. Shimasaki, Phys. Rev. D 98, 114513 (2018). arXiv:1805.03964

    ADS  MathSciNet  Google Scholar 

  63. 63.

    D. Sexty, Phys. Rev. D 100, 074503 (2019). arXiv:1907.08712

    ADS  MathSciNet  Google Scholar 

  64. 64.

    Y. Ito, H. Matsufuru, J. Nishimura, S. Shimasaki, A. Tsuchiya, S. Tsutsui, PoS LATTICE2018, 146 (2018). arXiv:1811.12688

    Google Scholar 

  65. 65.

    J.B. Kogut, D.K. Sinclair, Phys. Rev. D 100, 054512 (2019). arXiv:1903.02622

    ADS  MathSciNet  Google Scholar 

  66. 66.

    Y. Ito, H. Matsufuru, Y. Namekawa, J. Nishimura, S. Shimasaki, A. Tsuchiya, S. Tsutsui (2020), arXiv:2007.08778

  67. 67.

    M. Scherzer, D. Sexty, I.O. Stamatescu (2020). arXiv:2004.05372

  68. 68.

    C. Morningstar, M.J. Peardon, Phys. Rev. D 69, 054501 (2004). arXiv:hep-lat/0311018

    ADS  Google Scholar 

  69. 69.

    J. Kogut, D. Sinclair, Phys. Rev. D 77, 114503 (2008). arXiv:0712.2625

    ADS  Google Scholar 

  70. 70.

    J. Bloch, O. Schenk, EPJ Web Conf. 175, 07003 (2018). arXiv:1707.08874

    Google Scholar 

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Acknowledgements

We are grateful for discussion and collaboration with Gert Aarts and Ion-Olimpiu Stamatescu. We thank Dénes Sexty for providing one of the figures to this manuscript. The work of F.A. was supported by US DOE Grant No. DE-FG02-97ER41014 MOD27.

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Correspondence to Benjamin Jäger.

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Communicated by Laura Tolos.

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Attanasio, F., Jäger, B. & Ziegler, F.P.G. Complex Langevin simulations and the QCD phase diagram: recent developments. Eur. Phys. J. A 56, 251 (2020). https://doi.org/10.1140/epja/s10050-020-00256-z

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