Skip to main content
SpringerLink
Log in

Flavour-coherent propagators and Feynman rules: covariant cQPA formulation

  • Published:
Journal of High Energy Physics Aims and scope Submit manuscript

Abstract

We present a simplified and generalized derivation of the flavour-coherent propagators and Feynman rules for the fermionic kinetic theory based on coherent quasiparticle approximation (cQPA) [1–7]. The new formulation immediately reveals the composite nature of the cQPA Wightman function as a product of two spectral functions and an effective two-point interaction vertex, which contains all quantum statistical and coherence information. We extend our previous work to the case of nonzero dispersive self-energy, which leads to a broader range of applications. By this scheme, we derive flavoured kinetic equations for local 2-point functions \( S_{\text{k}}^{{ <, > }} \)(t, t), which are reminiscent of the equations of motion for the density matrix. We emphasize that in our approach all the interaction terms are derived from first principles of nonequilibrium quantum field theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Herranen, K. Kainulainen and P.M. Rahkila, Towards a kinetic theory for fermions with quantum coherence, Nucl. Phys. B 810 (2009) 389 [arXiv:0807.1415] [INSPIRE].

    Article  ADS  Google Scholar 

  2. M. Herranen, K. Kainulainen and P.M. Rahkila, Quantum kinetic theory for fermions in temporally varying backgrounds, JHEP 09 (2008) 032 [arXiv:0807.1435] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  3. M. Herranen, K. Kainulainen and P.M. Rahkila, Kinetic theory for scalar fields with nonlocal quantum coherence, JHEP 05 (2009) 119 [arXiv:0812.4029] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  4. M. Herranen, K. Kainulainen and P.M. Rahkila, Coherent quasiparticle approximation (cQPA) and nonlocal coherence, J. Phys. Conf. Ser. 220 (2010) 012007 [arXiv:0912.2490] [INSPIRE].

    Article  ADS  Google Scholar 

  5. M. Herranen, Quantum kinetic theory with nonlocal coherence, Ph.D. thesis, research report 3/2009, University of Jyväskylä, Jyväskylä Finland (2009) [arXiv:0906.3136] [INSPIRE].

  6. M. Herranen, K. Kainulainen and P.M. Rahkila, Coherent quantum Boltzmann equations from cQPA, JHEP 12 (2010) 072 [arXiv:1006.1929] [INSPIRE].

    Article  ADS  Google Scholar 

  7. C. Fidler, M. Herranen, K. Kainulainen and P.M. Rahkila, Flavoured quantum Boltzmann equations from cQPA, arXiv:1108.2309 [INSPIRE].

  8. R. Barbieri and A. Dolgov, Bounds on sterile-neutrinos from nucleosynthesis, Phys. Lett. B 237 (1990) 440 [INSPIRE].

    ADS  Google Scholar 

  9. R. Barbieri and A. Dolgov, Neutrino oscillations in the early universe, Nucl. Phys. B 349 (1991) 743 [INSPIRE].

    Article  ADS  Google Scholar 

  10. K. Kainulainen, Light singlet neutrinos and the primordial nucleosynthesis, Phys. Lett. B 244 (1990) 191 [INSPIRE].

    ADS  Google Scholar 

  11. K. Enqvist, K. Kainulainen and J. Maalampi, Resonant neutrino transitions and nucleosynthesis, Phys. Lett. B 249 (1990) 531 [INSPIRE].

    ADS  Google Scholar 

  12. K. Enqvist, K. Kainulainen and J. Maalampi, Refraction and oscillations of neutrinos in the early universe, Nucl. Phys. B 349 (1991) 754 [INSPIRE].

    Article  ADS  Google Scholar 

  13. K. Enqvist, K. Kainulainen and M.J. Thomson, Stringent cosmological bounds on inert neutrino mixing, Nucl. Phys. B 373 (1992) 498 [INSPIRE].

    Article  ADS  Google Scholar 

  14. D. Boyanovsky and C. Ho, Charged lepton mixing and oscillations from neutrino mixing in the early universe, Astropart. Phys. 27 (2007) 99 [hep-ph/0510214] [INSPIRE].

    Article  ADS  Google Scholar 

  15. C. Ho, D. Boyanovsky and H. de Vega, Neutrino oscillations in the early universe: a real time formulation, Phys. Rev. D 72 (2005) 085016 [hep-ph/0508294] [INSPIRE].

    ADS  Google Scholar 

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

    ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  Google Scholar 

  19. M. Joyce, T. Prokopec and N. Turok, Nonlocal electroweak baryogenesis. Part 2: the classical regime, Phys. Rev. D 53 (1996) 2958 [hep-ph/9410282] [INSPIRE].

    ADS  Google Scholar 

  20. M. Joyce, T. Prokopec and N. Turok, Electroweak baryogenesis from a classical force, Phys. Rev. Lett. 75 (1995) 1695 [Erratum ibid. 75 (1995) 3375] [hep-ph/9408339] [INSPIRE].

    Article  ADS  Google Scholar 

  21. M. Joyce, T. Prokopec and N. Turok, Nonlocal electroweak baryogenesis. Part 1: thin wall regime, Phys. Rev. D 53 (1996) 2930 [hep-ph/9410281] [INSPIRE].

    ADS  Google Scholar 

  22. J.M. Cline, M. Joyce and K. Kainulainen, Supersymmetric electroweak baryogenesis, JHEP 07 (2000) 018 [hep-ph/0006119] [INSPIRE].

    Article  ADS  Google Scholar 

  23. J.M. Cline, M. Joyce and K. Kainulainen, Supersymmetric electroweak baryogenesis in the WKB approximation, Phys. Lett. B 417 (1998) 79 [Erratum ibid. B 448 (1999) 321] [hep-ph/9708393] [INSPIRE].

    ADS  Google Scholar 

  24. J.M. Cline and K. Kainulainen, A new source for electroweak baryogenesis in the MSSM, Phys. Rev. Lett. 85 (2000) 5519 [hep-ph/0002272] [INSPIRE].

    Article  ADS  Google Scholar 

  25. K. Kainulainen, T. Prokopec, M.G. Schmidt and S. Weinstock, First principle derivation of semiclassical force for electroweak baryogenesis, JHEP 06 (2001) 031 [hep-ph/0105295] [INSPIRE].

    Article  ADS  Google Scholar 

  26. K. Kainulainen, T. Prokopec, M.G. Schmidt and S. Weinstock, Semiclassical force for electroweak baryogenesis: three-dimensional derivation, Phys. Rev. D 66 (2002) 043502 [hep-ph/0202177] [INSPIRE].

    ADS  Google Scholar 

  27. K. Kainulainen, T. Prokopec, M.G. Schmidt and S. Weinstock, Quantum Boltzmann equations for electroweak baryogenesis including gauge fields, hep-ph/0201293 [INSPIRE].

  28. T. Prokopec, M.G. Schmidt and S. Weinstock, Transport equations for chiral fermions to order ℏ and electroweak baryogenesis. Part 1, Annals Phys. 314 (2004) 208 [hep-ph/0312110] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  29. T. Prokopec, M.G. Schmidt and S. Weinstock, Transport equations for chiral fermions to order ℏ and electroweak baryogenesis. Part II, Annals Phys. 314 (2004) 267 [hep-ph/0406140] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  30. T. Konstandin, T. Prokopec and M.G. Schmidt, Kinetic description of fermion flavor mixing and CP-violating sources for baryogenesis, Nucl. Phys. B 716 (2005) 373 [hep-ph/0410135] [INSPIRE].

    Article  ADS  Google Scholar 

  31. T. Konstandin, T. Prokopec, M.G. Schmidt and M. Seco, MSSM electroweak baryogenesis and flavor mixing in transport equations, Nucl. Phys. B 738 (2006) 1 [hep-ph/0505103] [INSPIRE].

    Article  ADS  Google Scholar 

  32. V. Cirigliano, C. Lee, M.J. Ramsey-Musolf and S. Tulin, Flavored quantum Boltzmann equations, Phys. Rev. D 81 (2010) 103503 [arXiv:0912.3523] [INSPIRE].

    ADS  Google Scholar 

  33. V. Cirigliano, C. Lee and S. Tulin, Resonant flavor oscillations in electroweak baryogenesis, Phys. Rev. D 84 (2011) 056006 [arXiv:1106.0747] [INSPIRE].

    ADS  Google Scholar 

  34. B. Garbrecht, T. Prokopec and M.G. Schmidt, Coherent baryogenesis, Phys. Rev. Lett. 92 (2004) 061303 [hep-ph/0304088] [INSPIRE].

    Article  ADS  Google Scholar 

  35. B. Garbrecht, T. Prokopec and M.G. Schmidt, SO(10)-GUT coherent baryogenesis, Nucl. Phys. B 736 (2006) 133 [hep-ph/0509190] [INSPIRE].

    Article  ADS  Google Scholar 

  36. M. Fukugita and T. Yanagida, Baryogenesis without grand unification, Phys. Lett. B 174 (1986) 45 [INSPIRE].

    ADS  Google Scholar 

  37. A. Pilaftsis and T.E. Underwood, Resonant leptogenesis, Nucl. Phys. B 692 (2004) 303 [hep-ph/0309342] [INSPIRE].

    Article  ADS  Google Scholar 

  38. A. Pilaftsis and T.E. Underwood, Electroweak-scale resonant leptogenesis, Phys. Rev. D 72 (2005) 113001 [hep-ph/0506107] [INSPIRE].

    ADS  Google Scholar 

  39. A. Abada, S. Davidson, F.-X. Josse-Michaux, M. Losada and A. Riotto, Flavor issues in leptogenesis, JCAP 04 (2006) 004 [hep-ph/0601083] [INSPIRE].

    Article  ADS  Google Scholar 

  40. E. Nardi, Y. Nir, E. Roulet and J. Racker, The importance of flavor in leptogenesis, JHEP 01 (2006) 164 [hep-ph/0601084] [INSPIRE].

    Article  ADS  Google Scholar 

  41. M. Garny, A. Hohenegger, A. Kartavtsev and M. Lindner, Systematic approach to leptogenesis in nonequilibrium QFT: vertex contribution to the CP-violating parameter, Phys. Rev. D 80 (2009) 125027 [arXiv:0909.1559] [INSPIRE].

    ADS  Google Scholar 

  42. M. Garny, A. Hohenegger, A. Kartavtsev and M. Lindner, Systematic approach to leptogenesis in nonequilibrium QFT: self-energy contribution to the CP-violating parameter, Phys. Rev. D 81 (2010) 085027 [arXiv:0911.4122] [INSPIRE].

    ADS  Google Scholar 

  43. M. Beneke, B. Garbrecht, M. Herranen and P. Schwaller, Finite number density corrections to leptogenesis, Nucl. Phys. B 838 (2010) 1 [arXiv:1002.1326] [INSPIRE].

    Article  ADS  Google Scholar 

  44. M. Beneke, B. Garbrecht, C. Fidler, M. Herranen and P. Schwaller, Flavoured leptogenesis in the CTP formalism, Nucl. Phys. B 843 (2011) 177 [arXiv:1007.4783] [INSPIRE].

    Article  ADS  Google Scholar 

  45. A. Anisimov, D. Besak and D. Bödeker, Thermal production of relativistic Majorana neutrinos: strong enhancement by multiple soft scattering, JCAP 03 (2011) 042 [arXiv:1012.3784] [INSPIRE].

    Article  ADS  Google Scholar 

  46. A. Anisimov, W. Buchmüller, M. Drewes and S. Mendizabal, Leptogenesis from quantum interference in a thermal bath, Phys. Rev. Lett. 104 (2010) 121102 [arXiv:1001.3856] [INSPIRE].

    Article  ADS  Google Scholar 

  47. A. Anisimov, W. Buchmüller, M. Drewes and S. Mendizabal, Quantum leptogenesis I, Annals Phys. 326 (2011) 1998 [arXiv:1012.5821] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  48. J.H. Traschen and R.H. Brandenberger, Particle production during out-of-equilibrium phase transitions, Phys. Rev. D 42 (1990) 2491 [INSPIRE].

    ADS  Google Scholar 

  49. Y. Shtanov, J.H. Traschen and R.H. Brandenberger, Universe reheating after inflation, Phys. Rev. D 51 (1995) 5438 [hep-ph/9407247] [INSPIRE].

    ADS  Google Scholar 

  50. J. Berges and J. Serreau, Parametric resonance in quantum field theory, Phys. Rev. Lett. 91 (2003) 111601 [hep-ph/0208070] [INSPIRE].

    Article  ADS  Google Scholar 

  51. J. Berges, J. Pruschke and A. Rothkopf, Instability-induced fermion production in quantum field theory, Phys. Rev. D 80 (2009) 023522 [arXiv:0904.3073] [INSPIRE].

    Article  ADS  Google Scholar 

  52. J.F. Koksma, T. Prokopec and M.G. Schmidt, Decoherence in an interacting quantum field theory: the vacuum case, Phys. Rev. D 81 (2010) 065030 [arXiv:0910.5733] [INSPIRE].

    Article  ADS  Google Scholar 

  53. J.F. Koksma, T. Prokopec and M.G. Schmidt, Entropy and correlators in quantum field theory, Annals Phys. 325 (2010) 1277 [arXiv:1002.0749] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  54. J.F. Koksma, T. Prokopec and M.G. Schmidt, Decoherence in an interacting quantum field theory: thermal case, Phys. Rev. D 83 (2011) 085011 [arXiv:1102.4713] [INSPIRE].

    Article  ADS  Google Scholar 

  55. A. Giraud and J. Serreau, Decoherence and thermalization of a pure quantum state in quantum field theory, Phys. Rev. Lett. 104 (2010) 230405 [arXiv:0910.2570] [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  56. F. Gautier and J. Serreau, Quantum properties of a non-Gaussian state in the large-N approximation, Phys. Rev. D 83 (2011) 125004 [arXiv:1104.0421] [INSPIRE].

    Article  ADS  Google Scholar 

  57. J.S. Schwinger, Brownian motion of a quantum oscillator, J. Math. Phys. 2 (1961) 407 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  58. L. Keldysh, Diagram technique for nonequilibrium processes, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [Sov. Phys. JETP 20 (1965) 1018] [INSPIRE].

    Google Scholar 

  59. P. Danielewicz, Quantum theory of nonequilibrium processes. 1., Annals Phys. 152 (1984) 239 [INSPIRE].

    Article  ADS  Google Scholar 

  60. S. Mrowczynski and U.W. Heinz, Towards a relativistic transport theory of nuclear matter, Annals Phys. 229 (1994) 1 [INSPIRE].

    Article  ADS  Google Scholar 

  61. E. Calzetta and B.L. Hu, Nonequilibrium quantum field theory, Cambridge University Press, Cambridge U.K. (2008).

    Book  MATH  Google Scholar 

  62. C. Greiner and S. Leupold, Stochastic interpretation of Kadanoff-Baym equations and their relation to Langevin processes, Annals Phys. 270 (1998) 328 [hep-ph/9802312] [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  63. A. Anisimov, W. Buchmüller, M. Drewes and S. Mendizabal, Nonequilibrium dynamics of scalar fields in a thermal bath, Annals Phys. 324 (2009) 1234 [arXiv:0812.1934] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  64. L. Kadanoff and G. Baym, Quantum statistical mechanics, Benjamin, New York U.S.A. (1962).

  65. M. Le Bellac, Thermal field theory, Cambridge University Press, Cambridge U.K. (2000).

    Google Scholar 

  66. J. Berges, Introduction to nonequilibrium quantum field theory, AIP Conf. Proc. 739 (2005) 3 [hep-ph/0409233] [INSPIRE].

    Article  ADS  Google Scholar 

  67. J. Luttinger and J.C. Ward, Ground state energy of a many fermion system. 2., Phys. Rev. 118 (1960) 1417 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  68. J.M. Cornwall, R. Jackiw and E. Tomboulis, Effective action for composite operators, Phys. Rev. D 10 (1974) 2428 [INSPIRE].

    ADS  Google Scholar 

  69. K.-C. Chou, Z.-B. Su, B.-L. Hao and L. Yu, Equilibrium and nonequilibrium formalisms made unified, Phys. Rept. 118 (1985) 1 [INSPIRE].

    Article  MathSciNet  ADS  Google Scholar 

  70. E. Calzetta and B. Hu, Nonequilibrium quantum fields: closed time path effective action, Wigner function and Boltzmann equation, Phys. Rev. D 37 (1988) 2878 [INSPIRE].

    MathSciNet  ADS  Google Scholar 

  71. H. Weldon, Dynamical holes in the quark-gluon plasma, Phys. Rev. D 40 (1989) 2410 [INSPIRE].

    ADS  Google Scholar 

  72. H. Weldon, Particles and holes, Physica A 158 (1989) 169 [INSPIRE].

    ADS  Google Scholar 

  73. J. Berges and J. Cox, Thermalization of quantum fields from time reversal invariant evolution equations, Phys. Lett. B 517 (2001) 369 [hep-ph/0006160] [INSPIRE].

    ADS  Google Scholar 

  74. J. Berges, Controlled nonperturbative dynamics of quantum fields out-of-equilibrium, Nucl. Phys. A 699 (2002) 847 [hep-ph/0105311] [INSPIRE].

    ADS  Google Scholar 

  75. G. Aarts, D. Ahrensmeier, R. Baier, J. Berges and J. Serreau, Far from equilibrium dynamics with broken symmetries from the 2P I-1/N expansion, Phys. Rev. D 66 (2002) 045008 [hep-ph/0201308] [INSPIRE].

    ADS  Google Scholar 

  76. J. Berges, S. Borsányi and J. Serreau, Thermalization of fermionic quantum fields, Nucl. Phys. B 660 (2003) 51 [hep-ph/0212404] [INSPIRE].

    Article  ADS  Google Scholar 

  77. R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jap. 12 (1957) 570.

    Article  MathSciNet  ADS  Google Scholar 

  78. R. Kubo, M. Yokota and S. Nakajima, Statistical mechanical theory of irreversible processes. 2. Response to thermal disturbance, J. Phys. Soc. Jap. 12 (1957) 1203.

    Article  MathSciNet  ADS  Google Scholar 

  79. P.C. Martin and J.S. Schwinger, Theory of many particle systems. 1., Phys. Rev. 115 (1959) 1342 [INSPIRE].

    Article  MathSciNet  ADS  MATH  Google Scholar 

  80. J. Ilmavirta, K. Kainulainen and P.M. Rahkila, Neutrino transport in coherent quasiparticle approximation, in preparation.

Download references

Author information

Authors and Affiliations

  1. Institut für Theoretische Teilchenphysik und Kosmologie, RWTH Aachen University, D-52056, Aachen, Germany

    Matti Herranen

  2. Department of Physics, University of Jyväskylä, P.O. Box 35 (YFL), FIN-40014, Jyväskylä, Finland

    Kimmo Kainulainen & Pyry Matti Rahkila

  3. Helsinki Institute of Physics, University of Helsinki, P.O. Box 64, FIN-00014, Helsinki, Finland

    Kimmo Kainulainen & Pyry Matti Rahkila

Authors
  1. Matti Herranen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kimmo Kainulainen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pyry Matti Rahkila
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pyry Matti Rahkila.

Additional information

ArXiv ePrint: 1108.2371

Alexander-von-Humboldt fellow. (Matti Herranen)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Herranen, M., Kainulainen, K. & Rahkila, P.M. Flavour-coherent propagators and Feynman rules: covariant cQPA formulation. J. High Energ. Phys. 2012, 80 (2012). https://doi.org/10.1007/JHEP02(2012)080

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP02(2012)080

Keywords

Search

Navigation