Skip to main content

Real-time fermions for baryogenesis simulations


We study how to numerically simulate quantum fermions out of thermal equilibrium, in the context of electroweak baryogenesis. We find that by combining the lattice implementation of Aarts and Smit [1] with the “low cost” fermions of Borsanyi and Hind-marsh [2], we are able to describe the dynamics of a classical bosonic system coupled to quantum fermions, that correctly reproduces anomalous baryon number violation. To demonstrate the method, we apply it to the 1 + 1 dimensional axial U(1) model, and perform simulations of a fast symmetry breaking transition. Compared to solving all the quantum mode equations as in [1], we find that this statistical approach may lead to a significant gain in computational time, when applied to 3 + 1dimensionalphysics.

This is a preview of subscription content, access via your institution.


  1. [1]

    G. Aarts and J. Smit, Real-time dynamics with fermions on a lattice, Nucl. Phys. B 555 (1999) 355 [hep-ph/9812413] [SPIRES].

    ADS  Article  Google Scholar 

  2. [2]

    S. Borsányi and M. Hindmarsh, Low-cost fermions in classical field simulations, Phys. Rev. D 79 (2009) 065010 [arXiv:0809.4711] [SPIRES].

    ADS  Google Scholar 

  3. [3]

    V.A. Kuzmin, V.A. Rubakov and M.E. Shaposhnikov, On the anomalous electroweak baryon number nonconservation in the early universe, Phys. Lett. B 155 (1985) 36 [SPIRES].

    ADS  Google Scholar 

  4. [4]

    A.G. Cohen, D.B. Kaplan and A.E. Nelson, Progress in electroweak baryogenesis, Ann. Rev. Nucl. Part. Sci. 43 (1993) 27 [hep-ph/9302210] [SPIRES].

    ADS  Article  Google Scholar 

  5. [5]

    G.D. Moore and K. Rummukainen, Classical sphaleron rate on fine lattices, Phys. Rev. D 61 (2000) 105008 [hep-ph/9906259] [SPIRES].

    ADS  Google Scholar 

  6. [6]

    M. D’Onofrio, K. Rummukainen and A. Tranberg, T he sphaleron rate at the electroweak crossover, PoS(LATTICE 2010)048 [arXiv:1011.2074] [SPIRES].

  7. [7]

    K. Kajantie, M. Laine, K. Rummukainen and M.E. Shaposhnikov, Generic rules for high temperature dimensional reduction and their application to the Standard Model, Nucl. Phys. B 458 (1996) 90 [hep-ph/9508379] [SPIRES].

    ADS  Article  Google Scholar 

  8. [8]

    J.García-Bellido, D.Y. Grigoriev, A. Kusenko and M.E. Shaposhnikov, Non-equilibrium electroweak baryogenesis from preheating after inflation, Phys. Rev. D 60 (1999) 123504 [hep-ph/9902449] [SPIRES].

    ADS  Google Scholar 

  9. [9]

    L.M. Krauss and M. Trodden, Baryogenesis below the electroweak scale, Phys. Rev. Lett. 83 (1999) 1502 [hep-ph/9902420] [SPIRES].

    ADS  Article  Google Scholar 

  10. [10]

    A. Rajantie, P.M. Saffin and E.J. Copeland, Electroweak preheating on a lattice, Phys. Rev. D 63 (2001) 123512 [hep-ph/0012097] [SPIRES].

    ADS  Google Scholar 

  11. [11]

    E.J. Copel and, D. Lyth, A. Rajantie and M. Trodden, Hybrid inflation and baryogenesis at the TeV scale, Phys. Rev. D 64 (2001) 043506 [hep-ph/0103231] [SPIRES].

    ADS  Google Scholar 

  12. [12]

    A. Tranberg and J. Smit, Baryon asymmetry from electroweak tachyonic preheating, JHEP 11 (2003) 016 [hep-ph/0310342] [SPIRES].

    MathSciNet  ADS  Article  Google Scholar 

  13. [13]

    A. Tranberg, A. Hernandez, T. Konstandin and M.G. Schmidt, Cold electroweak baryogenesis with Standard Model CP-violation, Phys. Lett. B 690 (2010) 207 [arXiv:0909.4199] [SPIRES].

    ADS  Google Scholar 

  14. [14]

    M.E. Shaposhnikov, Baryon asymmetry of the universe in standard electroweak theory, Nucl. Phys. B 287 (1987) 757 [SPIRES].

    ADS  Article  Google Scholar 

  15. [15]

    V.A. Rubakov and M.E. Shaposhnikov, Electroweak baryon number non-conservation in the early universe and in high-energy collisions, Usp. Fiz. Nauk 166 (1996) 493 [Phys. Usp. 39 (1996) 461] [ hep-ph/9603208] [SPIRES].

    Article  Google Scholar 

  16. [16]

    J.Smit, Effective CP-violation in the Standard Model, JHEP 09 (2004) 067 [hep-ph/0407161] [SPIRES].

    ADS  Article  Google Scholar 

  17. [17]

    A. Hernandez, T. Konstandin and M.G. Schmidt, Sizable CP-violation in the bosonized Standard Model, Nucl. Phys. B 812 (2009) 290 [arXiv:0810.4092] [SPIRES].

    ADS  Article  Google Scholar 

  18. [18]

    C. Garcia-Recio and L.L. Salcedo, CP violation in the effective action of the Standard Model, JHEP 07 (2009) 015 [arXiv:0903.5494] [SPIRES].

    ADS  Article  Google Scholar 

  19. [19]

    L.L. Salcedo, Leading order one-loop CP and P violating effective action in the Standard Model, Phys. Lett. B 700 (2011) 331 [arXiv:1102.2400] [SPIRES].

    ADS  Google Scholar 

  20. [20]

    A. Tranberg, Standard Model CP-violation and cold electroweak baryogenesis, arXiv:1009.2358 [SPIRES].

  21. [21]

    G. Aarts and J. Smit, Particle production and effective thermalization in inhomogeneous mean field theory, Phys. Rev. D 61 (2000) 025002 [hep-ph/9906538] [SPIRES].

    ADS  Google Scholar 

  22. [22]

    J. Berges, D. Gelfand and J. Pruschke, Quantum theory of fermion preheating, arXiv:1012.4632 [SPIRES].

  23. [23]

    J. Smit and A. Tranberg, Chern-Simons number asymmetry from CP-violation at electroweak tachyonic preheating, JHEP 12 (2002) 020 [hep-ph/0211243] [SPIRES].

    ADS  Article  Google Scholar 

  24. [24]

    J. García-Bellido, M. Garcia Perez and A. Gonzalez-Arroyo, Symmetry breaking and false vacuum decay after hybrid inflation, Phys. Rev. D 67 (2003) 103501 [hep-ph/0208228] [SPIRES].

    ADS  Google Scholar 

  25. [25]

    K. Kajantie, M. Karjalainen, M. Laine, J. Peisa and A. Rajantie, Thermodynamics of gauge-invariant U(1) vortices from lattice Monte Carlo simulations, Phys. Lett. B 428 (1998) 334 [hep-ph/9803367] [SPIRES].

    ADS  Google Scholar 

  26. [26]

    A. Tranberg, J. Smit and M. Hindmarsh, Simulations of cold electroweak baryogenesis: finite time quenches, JHEP 01 (2007) 034 [hep-ph/0610096] [SPIRES].

    ADS  Article  Google Scholar 

  27. [27]

    A. Tranberg and J. Smit, Simulations of cold electroweak baryogenesis: dependence on Higgs mass and strength of CP-violation, JHEP 08 (2006) 012 [hep-ph/0604263] [SPIRES].

    ADS  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Anders Tranberg.

Additional information

ArXiv ePrint: 1105.5546

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Saffin, P.M., Tranberg, A. Real-time fermions for baryogenesis simulations. J. High Energ. Phys. 2011, 66 (2011).

Download citation


  • Lattice Gauge Field Theories
  • Nonperturbative Effects
  • Thermal Field Theory
  • Lattice Quantum Field Theory