Heavy quark chemical equilibration rate as a transport coefficient

  • D. Bödeker
  • M. Laine


Motivated by indications that heavy (charm and bottom) quarks interact strongly at temperatures generated in heavy ion collision experiments, we suggest a non- perturbative definition of a heavy quark chemical equilibration rate as a transport coefficient. Within leading-order perturbation theory (corresponding to 3-loop level), the definition is argued to reduce to an expression obtained from the Boltzmann equation. Around T ~ 400 MeV, an order-of-magnitude estimate for charm yields a rate \(\Gamma_{\text{chem}}^{ - 1} \gtrsim {6}0{{\text{fm}} \left/ {\text{c}} \right.}\) which remains too slow to play a practical role in current experiments. However, the rate increases rapidly with T and, due to non-linear effects, also if the initial state contains an overabundance of heavy quarks.


Thermal Field Theory Quark-Gluon Plasma Heavy Quark Physics 


  1. [1]
    S. Hofmann, D.J. Schwarz and H. Stöcker, Damping scales of neutralino cold dark matter, Phys. Rev. D 64 (2001) 083507 [astro-ph/0104173] [INSPIRE].ADSGoogle Scholar
  2. [2]
    L.S. Brown and R. Sawyer, Finite temperature corrections to weak rates prior to nucleosynthesis, Phys. Rev. D 63 (2001) 083503 [astro-ph/0006370] [INSPIRE].ADSGoogle Scholar
  3. [3]
    B. Svetitsky, Diffusion of charmed quarks in the quark-gluon plasma, Phys. Rev. D 37 (1988) 2484 [INSPIRE].ADSGoogle Scholar
  4. [4]
    E. Braaten and M.H. Thoma, Energy loss of a heavy quark in the quark-gluon plasma, Phys. Rev. D 44 (1991) 2625 [INSPIRE].ADSGoogle Scholar
  5. [5]
    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
  6. [6]
    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
  7. [7]
    L.D. Landau and E.M. Lifshitz, Statistical physics, 3rd Edition, Butterworth-Heinemann, Oxford, U.K. (1980), see §118.Google Scholar
  8. [8]
    T.S. Biró and J. Zimányi, Quarkochemistry in relativistic heavy ion collisions, Phys. Lett. B 113 (1982) 6 [INSPIRE].ADSGoogle Scholar
  9. [9]
    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
  10. [10]
    PHENIX collaboration, A. Adare et al., Heavy quark production in p + p and energy loss and flow of heavy quarks in Au+Au collisions at \(\sqrt {{{s_{NN}}}} = {2}00\;GeV\), Phys. Rev. C 84 (2011) 044905 [arXiv:1005.1627] [INSPIRE].ADSGoogle Scholar
  11. [11]
    ALICE collaboration, A. Dainese, Heavy-flavour production in Pb-Pb collisions at the LHC, measured with the ALICE detector, J. Phys. G 38 (2011) 124032 [arXiv:1106.4042] [INSPIRE].ADSGoogle Scholar
  12. [12]
    H.B. Meyer, The errant life of a heavy quark in the quark-gluon plasma, New J. Phys. 13 (2011) 035008 [arXiv:1012.0234] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    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].
  14. [14]
    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
  15. [15]
    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
  16. [16]
    A. Andronic, P. Braun-Munzinger, K. Redlich and J. Stachel, Statistical hadronization of heavy quarks in ultra-relativistic nucleus-nucleus collisions, Nucl. Phys. A 789 (2007) 334 [nucl-th/0611023] [INSPIRE].ADSGoogle Scholar
  17. [17]
    G. Torrieri and J. Noronha, Flavoring the quark-gluon plasma with charm, Phys. Lett. B 690 (2010) 477 [arXiv:1004.0237] [INSPIRE].ADSGoogle Scholar
  18. [18]
    J. Casalderrey-Solana and D. Teaney, Heavy quark diffusion in strongly coupled \(\mathcal{N} = 4\) Yang-Mills, Phys. Rev. D 74 (2006) 085012 [hep-ph/0605199] [INSPIRE].ADSGoogle Scholar
  19. [19]
    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
  20. [20]
    J. Bernstein, L.S. Brown and G. Feinberg, The cosmological heavy neutrino problem revisited, Phys. Rev. D 32 (1985) 3261 [INSPIRE].ADSGoogle Scholar
  21. [21]
    M. Glück, J. Owens and E. Reya, Gluon contribution to hadronic J/ψ production, Phys. Rev. D 17 (1978) 2324 [INSPIRE].ADSGoogle Scholar
  22. [22]
    Particle Data Group collaboration, K. Nakamura et al., Review of particle physics, J. Phys. G 37 (2010) 075021 [INSPIRE].ADSGoogle Scholar
  23. [23]
    P. Gondolo and G. Gelmini, Cosmic abundances of stable particles: improved analysis, Nucl. Phys. B 360 (1991) 145 [INSPIRE].ADSCrossRefGoogle Scholar
  24. [24]
    A. Hryczuk and R. Iengo, The one-loop and Sommerfeld electroweak corrections to the Wino dark matter annihilation, JHEP 01 (2012) 163 [Erratum ibid. 06 (2012) 137] [arXiv:1111.2916] [INSPIRE].ADSCrossRefGoogle Scholar
  25. [25]
    A. Vuorinen, Quark number susceptibilities of hot QCD up to g 6 ln g, Phys. Rev. D 67 (2003) 074032 [hep-ph/0212283] [INSPIRE].ADSGoogle Scholar
  26. [26]
    C. DeTar et al., QCD thermodynamics with nonzero chemical potential at N t = 6 and effects from heavy quarks, Phys. Rev. D 81 (2010) 114504 [arXiv:1003.5682] [INSPIRE].ADSGoogle Scholar
  27. [27]
    H.-T. Ding et al., Charmonium correlation and spectral functions at finite temperature, PoS(LATTICE 2010)180 [arXiv:1011.0695] [INSPIRE].
  28. [28]
    S. Borsanyi et al., The QCD equation of state and the effects of the charm, PoS(LATTICE 2011)201 [arXiv:1204.0995] [INSPIRE].
  29. [29]
    M. Laine and Y. Schröder, Quark mass thresholds in QCD thermodynamics, Phys. Rev. D 73 (2006) 085009 [hep-ph/0603048] [INSPIRE].ADSGoogle Scholar
  30. [30]
    S.L. Adler, J.C. Collins and A. Duncan, Energy-momentum-tensor trace anomaly in spin 1/2 quantum electrodynamics, Phys. Rev. D 15 (1977) 1712 [INSPIRE].ADSGoogle Scholar
  31. [31]
    J.C. Collins, A. Duncan and S.D. Joglekar, Trace and dilatation anomalies in gauge theories, Phys. Rev. D 16 (1977) 438 [INSPIRE].ADSGoogle Scholar
  32. [32]
    Y. Burnier, M. Laine, J. Langelage and L. Mether, Colour-electric spectral function at next-to-leading order, JHEP 08 (2010) 094 [arXiv:1006.0867] [INSPIRE].ADSCrossRefGoogle Scholar
  33. [33]
    M. Laine, A. Vuorinen and Y. Zhu, Next-to-leading order thermal spectral functions in the perturbative domain, JHEP 09 (2011) 084 [arXiv:1108.1259] [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    Y. Burnier and M. Laine, Towards flavour diffusion coefficient and electrical conductivity without ultraviolet contamination, Eur. Phys. J. C 72 (2012) 1902 [arXiv:1201.1994] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    M. Laine, Heavy flavour kinetic equilibration in the confined phase, JHEP 04 (2011) 124 [arXiv:1103.0372] [INSPIRE].ADSCrossRefGoogle Scholar
  36. [36]
    M. He, R.J. Fries and R. Rapp, Thermal relaxation of charm in hadronic matter, Phys. Lett. B 701 (2011) 445 [arXiv:1103.6279] [INSPIRE].ADSGoogle Scholar
  37. [37]
    S. Ghosh, S.K. Das, S. Sarkar and J.-e. Alam, Dragging D mesons by hot hadrons, Phys. Rev. D 84 (2011) 011503 [arXiv:1104.0163] [INSPIRE].ADSGoogle Scholar
  38. [38]
    L.M. Abreu, D. Cabrera, F.J. Llanes-Estrada and J.M. Torres-Rincon, Charm diffusion in a pion gas implementing unitarity, chiral and heavy quark symmetries, Annals Phys. 326 (2011) 2737 [arXiv:1104.3815] [INSPIRE].ADSzbMATHCrossRefGoogle Scholar
  39. [39]
    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

Copyright information

© SISSA, Trieste, Italy 2012

Authors and Affiliations

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

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