Sommerfeld effect in heavy quark chemical equilibration

  • D. Bödeker
  • M. LaineEmail author


The chemical equilibration of heavy quarks in a quark-gluon plasma proceeds via annihilation or pair creation. For temperatures T much below the heavy quark mass M, when kinetically equilibrated heavy quarks move very slowly, the annihilation in the colour singlet channel is enhanced because the quark and antiquark attract each other which increases their probability to meet, whereas the octet contribution is suppressed. This is the so-called Sommerfeld effect. It has not been taken into account in previous calculations of the chemical equilibration rate, which are therefore incomplete for T\( \alpha_s^2M \). We compute the leading-order equilibration rate in this regime; there is a large enhancement in the singlet channel, but the rate is dominated by the octet channel, and therefore the total effect is small. In the course of the computation we demonstrate how operators that represent the annihilation of heavy quarks in non-relativistic QCD can be incorporated into the imaginary-time formalism.


Thermal Field Theory Resummation Quark-Gluon Plasma Heavy Quark Physics 


  1. [1]
    G.D. Moore and D. Teaney, How much do heavy quarks thermalize in a heavy ion collision?, Phys. Rev. C 71 (2005) 064904 [hep-ph/0412346] [INSPIRE].ADSGoogle Scholar
  2. [2]
    S. Caron-Huot and G.D. Moore, Heavy quark diffusion in QCD and \( \mathcal{N}=4 \) SYM at next-to-leading order, JHEP 02 (2008) 081 [arXiv:0801.2173] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S. Caron-Huot, M. Laine and G.D. Moore, A way to estimate the heavy quark thermalization rate from the lattice, JHEP 04 (2009) 053 [arXiv:0901.1195] [INSPIRE].ADSCrossRefGoogle Scholar
  4. [4]
    A. Francis, O. Kaczmarek, M. Laine and J. Langelage, Towards a non-perturbative measurement of the heavy quark momentum diffusion coefficient, PoS(LATTICE 2011)202 [arXiv:1109.3941] [INSPIRE].
  5. [5]
    D. Banerjee, S. Datta, R. Gavai and P. Majumdar, Heavy quark momentum diffusion coefficient from lattice QCD, Phys. Rev. D 85 (2012) 014510 [arXiv:1109.5738] [INSPIRE].ADSGoogle Scholar
  6. [6]
    T.S. Biró and J. Zimányi, Quarkochemistry in relativistic heavy ion collisions, Phys. Lett. B 113 (1982) 6 [INSPIRE].ADSGoogle Scholar
  7. [7]
    J. Rafelski and B. Müller, Strangeness production in the quark-gluon plasma, Phys. Rev. Lett. 48 (1982) 1066 [Erratum ibid. 56 (1986) 2334] [INSPIRE].ADSCrossRefGoogle Scholar
  8. [8]
    T. Matsui, B. Svetitsky and L.D. McLerran, Strangeness production in ultrarelativistic heavy ion collisions. 1. Chemical kinetics in the quark-gluon plasma, Phys. Rev. D 34 (1986) 783 [Erratum ibid. D 37 (1988) 844] [INSPIRE].ADSGoogle Scholar
  9. [9]
    A. Sommerfeld, Über die Beugung und Bremsung der Elektronen (in German), Annalen Phys. 403 (1931) 257.ADSCrossRefGoogle Scholar
  10. [10]
    L.D. Landau and E.M. Lifshitz, Quantum mechanics, non-relativistic theory, third edition, § 136, Butterworth-Heinemann, Oxford U.K. (1977).Google Scholar
  11. [11]
    V.S. Fadin, V.A. Khoze and T. Sjöstrand, On the threshold behavior of heavy top production, Z. Phys. C 48 (1990) 613 [INSPIRE].ADSGoogle Scholar
  12. [12]
    M. Beneke, P. Falgari, S. Klein, J. Piclum, C. Schwinn, M. Ubiali and F. Yan, The total top-pair production cross section at NNLL, arXiv:1208.5578 [INSPIRE].
  13. [13]
    U. Langenfeld, S.-O. Moch and T. Pfoh, QCD threshold corrections for gluino pair production at hadron colliders, JHEP 11 (2012) 070 [arXiv:1208.4281] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    W. Beenakker, S. Brensing, M. Krämer, A. Kulesza, E. Laenen and I. Niessen, NNLL resummation for squark-antisquark production, arXiv:1112.5057 [INSPIRE].
  15. [15]
    J. Hisano, S. Matsumoto, M.M. Nojiri and O. Saito, Non-perturbative effect on dark matter annihilation and gamma ray signature from galactic center, Phys. Rev. D 71 (2005) 063528 [hep-ph/0412403] [INSPIRE].ADSGoogle Scholar
  16. [16]
    J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Non-perturbative effect on thermal relic abundance of dark matter, Phys. Lett. B 646 (2007) 34 [hep-ph/0610249] [INSPIRE].ADSGoogle Scholar
  17. [17]
    J.L. Feng, M. Kaplinghat and H.-B. Yu, Sommerfeld enhancements for thermal relic dark matter, Phys. Rev. D 82 (2010) 083525 [arXiv:1005.4678] [INSPIRE].ADSGoogle Scholar
  18. [18]
    A. Hryczuk, R. Iengo and P. Ullio, Relic densities including Sommerfeld enhancements in the MSSM, JHEP 03 (2011) 069 [arXiv:1010.2172] [INSPIRE].ADSCrossRefGoogle Scholar
  19. [19]
    A. Strumia, Sommerfeld corrections to type-II and III leptogenesis, Nucl. Phys. B 809 (2009) 308 [arXiv:0806.1630] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    D. Bödeker and M. Laine, Heavy quark chemical equilibration rate as a transport coefficient, JHEP 07 (2012) 130 [arXiv:1205.4987] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    W.E. Caswell and G.P. Lepage, Effective Lagrangians for bound state problems in QED, QCD and other field theories, Phys. Lett. B 167 (1986) 437 [INSPIRE].ADSGoogle Scholar
  22. [22]
    G.T. Bodwin, E. Braaten and G.P. Lepage, Rigorous QCD analysis of inclusive annihilation and production of heavy quarkonium, Phys. Rev. D 51 (1995) 1125 [Erratum ibid. D 55 (1997)5853] [hep-ph/9407339] [INSPIRE].ADSGoogle Scholar
  23. [23]
    J. Frenkel and J.C. Taylor, Hard thermal QCD, forward scattering and effective actions, Nucl. Phys. B 374 (1992) 156 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  24. [24]
    E. Braaten and R.D. Pisarski, Simple effective Lagrangian for hard thermal loops, Phys. Rev. D 45 (1992) 1827 [INSPIRE].ADSGoogle Scholar
  25. [25]
    M. Laine, A resummed perturbative estimate for the quarkonium spectral function in hot QCD, JHEP 05 (2007) 028 [arXiv:0704.1720] [INSPIRE].ADSCrossRefGoogle Scholar
  26. [26]
    R.D. Pisarski, Scattering amplitudes in hot gauge theories, Phys. Rev. Lett. 63 (1989) 1129 [INSPIRE].ADSCrossRefGoogle Scholar
  27. [27]
    I. Kuznetsova, D. Habs and J. Rafelski, Pion and muon production in e , e + , γ plasma, Phys. Rev. D 78 (2008) 014027 [arXiv:0803.1588] [INSPIRE].ADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  1. 1.Faculty of PhysicsUniversity of BielefeldBielefeldGermany
  2. 2.Institute for Theoretical Physics, Albert Einstein CenterUniversity of BernBernSwitzerland

Personalised recommendations