Group theoretic approach to fermion production


We propose a universal group theoretic description of the fermion production through any type of interaction to scalar or pseudo-scalar. Our group theoretic approach relies on the group SU(2) × U(1), corresponding to the freedom in choosing representations of the gamma matrices in Clifford algebra, under which a part of the Dirac spinor function transforms like a fundamental representation. In terms of a new SO(3) (∼ SU(2)) vector constructed out of spinor functions, we show that fermion production mechanism can be analogous to the classical dynamics of a vector precessing with the angular velocity. In our group theoretic approach, the equation of motion takes a universal form for any system, and choosing a different type of interaction or a different basis amounts to selecting the corresponding angular velocity. The expression of the particle number density is greatly simplified, compared to the traditional approach, and it provides us with a simple geometric interpretation of the fermion production dynamics. For the purpose of the demonstration, we focus on the fermion production through the derivative coupling to the pseudo-scalar.

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  1. [1]

    L. Kofman, A.D. Linde and A.A. Starobinsky, Towards the theory of reheating after inflation, Phys. Rev. D 56 (1997) 3258 [hep-ph/9704452] [INSPIRE].

  2. [2]

    M.M. Anber and L. Sorbo, Naturally inflating on steep potentials through electromagnetic dissipation, Phys. Rev. D 81 (2010) 043534 [arXiv:0908.4089] [INSPIRE].

    ADS  Google Scholar 

  3. [3]

    P. Adshead and E.I. Sfakianakis, Fermion production during and after axion inflation, JCAP 11 (2015) 021 [arXiv:1508.00891] [INSPIRE].

    ADS  Article  Google Scholar 

  4. [4]

    P. Adshead, L. Pearce, M. Peloso, M.A. Roberts and L. Sorbo, Phenomenology of fermion production during axion inflation, JCAP 06 (2018) 020 [arXiv:1803.04501] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  5. [5]

    S.Y. Khlebnikov and I.I. Tkachev, Relic gravitational waves produced after preheating, Phys. Rev. D 56 (1997) 653 [hep-ph/9701423] [INSPIRE].

  6. [6]

    R. Easther and E.A. Lim, Stochastic gravitational wave production after inflation, JCAP 04 (2006) 010 [astro-ph/0601617] [INSPIRE].

  7. [7]

    R. Easther, J.T. Giblin Jr. and E.A. Lim, Gravitational Wave Production At The End Of Inflation, Phys. Rev. Lett. 99 (2007) 221301 [astro-ph/0612294] [INSPIRE].

  8. [8]

    R. Easther, J.T. Giblin and E.A. Lim, Gravitational Waves From the End of Inflation: Computational Strategies, Phys. Rev. D 77 (2008) 103519 [arXiv:0712.2991] [INSPIRE].

    ADS  Google Scholar 

  9. [9]

    J. García-Bellido and D.G. Figueroa, A stochastic background of gravitational waves from hybrid preheating, Phys. Rev. Lett. 98 (2007) 061302 [astro-ph/0701014] [INSPIRE].

  10. [10]

    J.F. Dufaux, A. Bergman, G.N. Felder, L. Kofman and J.-P. Uzan, Theory and Numerics of Gravitational Waves from Preheating after Inflation, Phys. Rev. D 76 (2007) 123517 [arXiv:0707.0875] [INSPIRE].

    ADS  Google Scholar 

  11. [11]

    J.-F. Dufaux, D.G. Figueroa and J. García-Bellido, Gravitational Waves from Abelian Gauge Fields and Cosmic Strings at Preheating, Phys. Rev. D 82 (2010) 083518 [arXiv:1006.0217] [INSPIRE].

  12. [12]

    L. Bethke, D.G. Figueroa and A. Rajantie, Anisotropies in the Gravitational Wave Background from Preheating, Phys. Rev. Lett. 111 (2013) 011301 [arXiv:1304.2657] [INSPIRE].

    ADS  Article  Google Scholar 

  13. [13]

    D.G. Figueroa and T. Meriniemi, Stochastic Background of Gravitational Waves from FermionsTheory and Applications, JHEP 10 (2013) 101 [arXiv:1306.6911] [INSPIRE].

    ADS  Article  Google Scholar 

  14. [14]

    L. Bethke, D.G. Figueroa and A. Rajantie, On the Anisotropy of the Gravitational Wave Background from Massless Preheating, JCAP 06 (2014) 047 [arXiv:1309.1148] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  15. [15]

    D.G. Figueroa, J. García-Bellido and F. Torrentí, Gravitational wave production from the decay of the standard model Higgs field after inflation, Phys. Rev. D 93 (2016) 103521 [arXiv:1602.03085] [INSPIRE].

  16. [16]

    D.G. Figueroa and F. Torrentí, Gravitational wave production from preheating: parameter dependence, JCAP 10 (2017) 057 [arXiv:1707.04533] [INSPIRE].

  17. [17]

    P. Adshead, J.T. Giblin and Z.J. Weiner, Gravitational waves from gauge preheating, Phys. Rev. D 98 (2018) 043525 [arXiv:1805.04550] [INSPIRE].

    ADS  Google Scholar 

  18. [18]

    A. Hook and G. Marques-Tavares, Relaxation from particle production, JHEP 12 (2016) 101 [arXiv:1607.01786] [INSPIRE].

    ADS  Article  Google Scholar 

  19. [19]

    N. Fonseca, E. Morgante and G. Servant, Higgs relaxation after inflation, JHEP 10 (2018) 020 [arXiv:1805.04543] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  20. [20]

    P.W. Graham, D.E. Kaplan and S. Rajendran, Cosmological Relaxation of the Electroweak Scale, Phys. Rev. Lett. 115 (2015) 221801 [arXiv:1504.07551] [INSPIRE].

    ADS  Article  Google Scholar 

  21. [21]

    P.B. Greene and L. Kofman, Preheating of fermions, Phys. Lett. B 448 (1999) 6 [hep-ph/9807339] [INSPIRE].

  22. [22]

    P.B. Greene and L. Kofman, On the theory of fermionic preheating, Phys. Rev. D 62 (2000) 123516 [hep-ph/0003018] [INSPIRE].

  23. [23]

    M. Peloso and L. Sorbo, Preheating of massive fermions after inflation: Analytical results, JHEP 05 (2000) 016 [hep-ph/0003045] [INSPIRE].

  24. [24]

    J. García-Bellido, S. Mollerach and E. Roulet, Fermion production during preheating after hybrid inflation, JHEP 02 (2000) 034 [hep-ph/0002076] [INSPIRE].

  25. [25]

    S. Tsujikawa, B.A. Bassett and F. Viniegra, Multifield fermionic preheating, JHEP 08 (2000) 019 [hep-ph/0006354] [INSPIRE].

  26. [26]

    B. Garbrecht, T. Prokopec and M.G. Schmidt, Particle number in kinetic theory, Eur. Phys. J. C 38 (2004) 135 [hep-th/0211219] [INSPIRE].

    ADS  Article  Google Scholar 

  27. [27]

    V. Domcke and K. Mukaida, Gauge Field and Fermion Production during Axion Inflation, JCAP 11 (2018) 020 [arXiv:1806.08769] [INSPIRE].

    ADS  Article  Google Scholar 

  28. [28]

    M. Bastero-Gil and A. Mazumdar, Gravitino production in hybrid inflationary models, Phys. Rev. D 62 (2000) 083510 [hep-ph/0002004] [INSPIRE].

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Correspondence to Minho Son.

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ArXiv ePrint: 1808.00939

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Min, U., Son, M. & Suh, H.G. Group theoretic approach to fermion production. J. High Energ. Phys. 2019, 72 (2019).

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  • Cosmology of Theories beyond the SM
  • Beyond Standard Model