Free energy of a heavy quark-antiquark pair in a thermal medium from AdS/CFT

  • Carlo Ewerz
  • Olaf Kaczmarek
  • Andreas Samberg
Open Access
Regular Article - Theoretical Physics


We study the free energy of a heavy quark-antiquark pair in a thermal medium using the AdS/CFT correspondence. We point out that a commonly used prescription for calculating this quantity leads to a temperature dependence in conflict with general properties of the free energy. The problem originates from a particular way of subtracting divergences. We argue that the commonly used prescription gives rise to the binding energy rather than the free energy. We consider a different subtraction procedure and show that the resulting free energy is well-behaved and in qualitative agreement with results from lattice QCD. The free energy and the binding energy of the quark pair are computed for \( \mathcal{N}=4 \) supersymmetric Yang-Mills theory and several non-conformal theories. We also calculate the entropy and the internal energy of the pair in these theories. Using the consistent subtraction, we further study the free energy, entropy, and internal energy of a single heavy quark in the thermal medium for various theories. Also here the results are found to be in qualitative agreement with lattice QCD results.


AdS-CFT Correspondence Gauge-gravity correspondence Holography and quark-gluon plasmas Heavy Quark Physics 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  3. [3]
    E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    J. Casalderrey-Solana, H. Liu, D. Mateos, K. Rajagopal and U.A. Wiedemann, Gauge/String Duality, Hot QCD and Heavy Ion Collisions, Cambridge University Press, Cambridge U.K. (2014).Google Scholar
  5. [5]
    O. DeWolfe, S.S. Gubser, C. Rosen and D. Teaney, Heavy ions and string theory, Prog. Part. Nucl. Phys. 75 (2014) 86 [arXiv:1304.7794] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    BRAHMS collaboration, I. Arsene et al., Quark-gluon plasma and color glass condensate at RHIC? The Perspective from the BRAHMS experiment, Nucl. Phys. A 757 (2005) 1 [nucl-ex/0410020] [INSPIRE].
  7. [7]
    PHOBOS collaboration, B.B. Back et al., The PHOBOS perspective on discoveries at RHIC, Nucl. Phys. A 757 (2005) 28 [nucl-ex/0410022] [INSPIRE].
  8. [8]
    STAR collaboration, J. Adams et al., Experimental and theoretical challenges in the search for the quark gluon plasma: The STAR Collaboration’s critical assessment of the evidence from RHIC collisions, Nucl. Phys. A 757 (2005) 102 [nucl-ex/0501009] [INSPIRE].
  9. [9]
    PHENIX collaboration, K. Adcox et al., Formation of dense partonic matter in relativistic nucleus-nucleus collisions at RHIC: Experimental evaluation by the PHENIX collaboration, Nucl. Phys. A 757 (2005) 184 [nucl-ex/0410003] [INSPIRE].
  10. [10]
    ALICE collaboration, Elliptic flow of charged particles in Pb-Pb collisions at \( \sqrt{s_{N\ N}}=2.76 \) TeV, Phys. Rev. Lett. 105 (2010) 252302 [arXiv:1011.3914] [INSPIRE].
  11. [11]
    ATLAS collaboration, Measurement of the pseudorapidity and transverse momentum dependence of the elliptic flow of charged particles in lead-lead collisions at \( \sqrt{s_{N\ N}}=2.76 \) TeV with the ATLAS detector, Phys. Lett. B 707 (2012) 330 [arXiv:1108.6018] [INSPIRE].
  12. [12]
    CMS collaboration, Measurement of the elliptic anisotropy of charged particles produced in PbPb collisions at \( {\sqrt{s}}_{N\ N}=2.76 \) TeV, Phys. Rev. C 87 (2013) 014902 [arXiv:1204.1409] [INSPIRE].
  13. [13]
    T. Matsui and H. Satz, J/ψ Suppression by Quark-Gluon Plasma Formation, Phys. Lett. B 178 (1986) 416 [INSPIRE].
  14. [14]
    N. Brambilla, A. Pineda, J. Soto and A. Vairo, Effective field theories for heavy quarkonium, Rev. Mod. Phys. 77 (2005) 1423 [hep-ph/0410047] [INSPIRE].
  15. [15]
    N. Brambilla, J. Ghiglieri, A. Vairo and P. Petreczky, Static quark-antiquark pairs at finite temperature, Phys. Rev. D 78 (2008) 014017 [arXiv:0804.0993] [INSPIRE].
  16. [16]
    L.D. McLerran and B. Svetitsky, Quark Liberation at High Temperature: A Monte Carlo Study of SU(2) Gauge Theory, Phys. Rev. D 24 (1981) 450 [INSPIRE].
  17. [17]
    O. Kaczmarek, F. Karsch, P. Petreczky and F. Zantow, Heavy quark anti-quark free energy and the renormalized Polyakov loop, Phys. Lett. B 543 (2002) 41 [hep-lat/0207002] [INSPIRE].
  18. [18]
    P. Petreczky and K. Petrov, Free energy of a static quark anti-quark pair and the renormalized Polyakov loop in three flavor QCD, Phys. Rev. D 70 (2004) 054503 [hep-lat/0405009] [INSPIRE].
  19. [19]
    O. Kaczmarek, F. Karsch, F. Zantow and P. Petreczky, Static quark anti-quark free energy and the running coupling at finite temperature, Phys. Rev. D 70 (2004) 074505 [Erratum ibid. D 72 (2005) 059903] [hep-lat/0406036] [INSPIRE].
  20. [20]
    O. Kaczmarek and F. Zantow, Static quark anti-quark interactions in zero and finite temperature QCD. I. Heavy quark free energies, running coupling and quarkonium binding, Phys. Rev. D 71 (2005) 114510 [hep-lat/0503017] [INSPIRE].
  21. [21]
    S.-J. Rey, S. Theisen and J.-T. Yee, Wilson-Polyakov loop at finite temperature in large N gauge theory and anti-de Sitter supergravity, Nucl. Phys. B 527 (1998) 171 [hep-th/9803135] [INSPIRE].
  22. [22]
    A. Brandhuber, N. Itzhaki, J. Sonnenschein and S. Yankielowicz, Wilson loops in the large N limit at finite temperature, Phys. Lett. B 434 (1998) 36 [hep-th/9803137] [INSPIRE].
  23. [23]
    H. Boschi-Filho, N.R.F. Braga and C.N. Ferreira, Heavy quark potential at finite temperature from gauge/string duality, Phys. Rev. D 74 (2006) 086001 [hep-th/0607038] [INSPIRE].
  24. [24]
    O. Andreev and V.I. Zakharov, On Heavy-Quark Free Energies, Entropies, Polyakov Loop and AdS/QCD, JHEP 04 (2007) 100 [hep-ph/0611304] [INSPIRE].
  25. [25]
    J. Noronha and A. Dumitru, The Heavy Quark Potential as a Function of Shear Viscosity at Strong Coupling, Phys. Rev. D 80 (2009) 014007 [arXiv:0903.2804] [INSPIRE].
  26. [26]
    T. Hayata, K. Nawa and T. Hatsuda, Time-dependent heavy-quark potential at finite temperature from gauge-gravity duality, Phys. Rev. D 87 (2013) 101901 [arXiv:1211.4942] [INSPIRE].
  27. [27]
    S.I. Finazzo and J. Noronha, Estimates for the Thermal Width of Heavy Quarkonia in Strongly Coupled Plasmas from Holography, JHEP 11 (2013) 042 [arXiv:1306.2613] [INSPIRE].ADSCrossRefGoogle Scholar
  28. [28]
    S.I. Finazzo and J. Noronha, Debye screening mass near deconfinement from holography, Phys. Rev. D 90 (2014) 115028 [arXiv:1411.4330] [INSPIRE].
  29. [29]
    B.K. Patra and H. Khanchandani, Heavy Quark Potential at Finite Temperature in a Dual Gravity Closer to Large N QCD, Phys. Rev. D 91 (2015) 066008 [arXiv:1412.5003] [INSPIRE].
  30. [30]
    Y. Yang and P.-H. Yuan, Confinement-deconfinement phase transition for heavy quarks in a soft wall holographic QCD model, JHEP 12 (2015) 161 [arXiv:1506.05930] [INSPIRE].ADSMathSciNetMATHGoogle Scholar
  31. [31]
    D. Bak, A. Karch and L.G. Yaffe, Debye screening in strongly coupled \( \mathcal{N}=4 \) supersymmetric Yang-Mills plasma, JHEP 08 (2007) 049 [arXiv:0705.0994] [INSPIRE].
  32. [32]
    A. Karch, E. Katz, D.T. Son and M.A. Stephanov, Linear confinement and AdS/QCD, Phys. Rev. D 74 (2006) 015005 [hep-ph/0602229] [INSPIRE].
  33. [33]
    O. Andreev and V.I. Zakharov, The Spatial String Tension, Thermal Phase Transition and AdS/QCD, Phys. Lett. B 645 (2007) 437 [hep-ph/0607026] [INSPIRE].
  34. [34]
    K. Kajantie, T. Tahkokallio and J.-T. Yee, Thermodynamics of AdS/QCD, JHEP 01 (2007) 019 [hep-ph/0609254] [INSPIRE].
  35. [35]
    C. Csáki and M. Reece, Toward a systematic holographic QCD: A Braneless approach, JHEP 05 (2007) 062 [hep-ph/0608266] [INSPIRE].
  36. [36]
    U. Gürsoy and E. Kiritsis, Exploring improved holographic theories for QCD: Part I, JHEP 02 (2008) 032 [arXiv:0707.1324] [INSPIRE].CrossRefGoogle Scholar
  37. [37]
    S.S. Gubser and A. Nellore, Mimicking the QCD equation of state with a dual black hole, Phys. Rev. D 78 (2008) 086007 [arXiv:0804.0434] [INSPIRE].
  38. [38]
    S.S. Gubser, A. Nellore, S.S. Pufu and F.D. Rocha, Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics, Phys. Rev. Lett. 101 (2008) 131601 [arXiv:0804.1950] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    J. Noronha and A. Dumitru, Thermal Width of the ϒ at Large ’t Hooft Coupling, Phys. Rev. Lett. 103 (2009) 152304 [arXiv:0907.3062] [INSPIRE].
  40. [40]
    G. Grignani, T. Harmark, A. Marini, N.A. Obers and M. Orselli, Thermal string probes in AdS and finite temperature Wilson loops, JHEP 06 (2012) 144 [arXiv:1201.4862] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  41. [41]
    J. Armas and M. Blau, Black probes of Schrödinger spacetimes, JHEP 08 (2014) 140 [arXiv:1405.1301] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  42. [42]
    J.J. Friess, S.S. Gubser, G. Michalogiorgakis and S.S. Pufu, Stability of strings binding heavy-quark mesons, JHEP 04 (2007) 079 [hep-th/0609137] [INSPIRE].ADSCrossRefGoogle Scholar
  43. [43]
    T. Appelquist, M. Dine and I.J. Muzinich, The Static Potential in Quantum Chromodynamics, Phys. Lett. B 69 (1977) 231 [INSPIRE].
  44. [44]
    W. Fischler, Quark-anti-Quark Potential in QCD, Nucl. Phys. B 129 (1977) 157 [INSPIRE].
  45. [45]
    L.D. McLerran and B. Svetitsky, A Monte Carlo Study of SU(2) Yang-Mills Theory at Finite Temperature, Phys. Lett. B 98 (1981) 195 [INSPIRE].
  46. [46]
    Y. Burnier, O. Kaczmarek and A. Rothkopf, Quarkonium at finite temperature: Towards realistic phenomenology from first principles, JHEP 12 (2015) 101 [arXiv:1509.07366] [INSPIRE].ADSGoogle Scholar
  47. [47]
    M. Laine, O. Philipsen, P. Romatschke and M. Tassler, Real-time static potential in hot QCD, JHEP 03 (2007) 054 [hep-ph/0611300] [INSPIRE].
  48. [48]
    A. Beraudo, J.P. Blaizot and C. Ratti, Real and imaginary-time \( Q\overline{Q} \) correlators in a thermal medium, Nucl. Phys. A 806 (2008) 312 [arXiv:0712.4394] [INSPIRE].
  49. [49]
    Y. Burnier, O. Kaczmarek and A. Rothkopf, Static quark-antiquark potential in the quark-gluon plasma from lattice QCD, Phys. Rev. Lett. 114 (2015) 082001 [arXiv:1410.2546] [INSPIRE].
  50. [50]
    Y. Burnier and A. Rothkopf, A gauge invariant Debye mass and the complex heavy-quark potential, Phys. Lett. B 753 (2016) 232 [arXiv:1506.08684] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  51. [51]
    S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large-N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].
  52. [52]
    J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    J.L. Albacete, Y.V. Kovchegov and A. Taliotis, Heavy Quark Potential at Finite Temperature in AdS/CFT Revisited, Phys. Rev. D 78 (2008) 115007 [arXiv:0807.4747] [INSPIRE].
  54. [54]
    M. Bianchi, D.Z. Freedman and K. Skenderis, Holographic renormalization, Nucl. Phys. B 631 (2002) 159 [hep-th/0112119] [INSPIRE].
  55. [55]
    A.N. Atmaja, J. de Boer and M. Shigemori, Holographic Brownian Motion and Time Scales in Strongly Coupled Plasmas, Nucl. Phys. B 880 (2014) 23 [arXiv:1002.2429] [INSPIRE].
  56. [56]
    M. Taylor and W. Woodhead, Renormalized entanglement entropy, JHEP 08 (2016) 165 [arXiv:1604.06808] [INSPIRE].ADSMathSciNetCrossRefMATHGoogle Scholar
  57. [57]
    Y. Kinar, E. Schreiber and J. Sonnenschein, \( Q\overline{Q} \) potential from strings in curved space-time: Classical results, Nucl. Phys. B 566 (2000) 103 [hep-th/9811192] [INSPIRE].
  58. [58]
    H. Liu, K. Rajagopal and U.A. Wiedemann, Wilson loops in heavy ion collisions and their calculation in AdS/CFT, JHEP 03 (2007) 066 [hep-ph/0612168] [INSPIRE].
  59. [59]
    H. Liu, K. Rajagopal and Y. Shi, Robustness and Infrared Sensitivity of Various Observables in the Application of AdS/CFT to Heavy Ion Collisions, JHEP 08 (2008) 048 [arXiv:0803.3214] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  60. [60]
    S. He, M. Huang and Q.-S. Yan, Logarithmic correction in the deformed AdS 5 model to produce the heavy quark potential and QCD β-function, Phys. Rev. D 83 (2011) 045034 [arXiv:1004.1880] [INSPIRE].
  61. [61]
    K. Bitaghsir Fadafan, Heavy quarks in the presence of higher derivative corrections from AdS/CFT, Eur. Phys. J. C 71 (2011) 1799 [arXiv:1102.2289] [INSPIRE].
  62. [62]
    C. Ewerz and K. Schade, Applications of Holography to Strongly Coupled Plasmas, PoS(Confinement X)270 [arXiv:1307.6161] [INSPIRE].
  63. [63]
    D. Giataganas, Probing strongly coupled anisotropic plasma, JHEP 07 (2012) 031 [arXiv:1202.4436] [INSPIRE].ADSCrossRefGoogle Scholar
  64. [64]
    A. Rebhan and D. Steineder, Probing Two Holographic Models of Strongly Coupled Anisotropic Plasma, JHEP 08 (2012) 020 [arXiv:1205.4684] [INSPIRE].ADSCrossRefGoogle Scholar
  65. [65]
    E. Caceres, M. Natsuume and T. Okamura, Screening length in plasma winds, JHEP 10 (2006) 011 [hep-th/0607233] [INSPIRE].ADSCrossRefGoogle Scholar
  66. [66]
    A. Samberg and C. Ewerz, Heavy Probes in Strongly Coupled Plasmas with Chemical Potential, Springer Proc. Phys. 170 (2016) 401 [arXiv:1312.5999] [INSPIRE].CrossRefGoogle Scholar
  67. [67]
    S.D. Avramis, K. Sfetsos and D. Zoakos, On the velocity and chemical-potential dependence of the heavy-quark interaction in N = 4 SYM plasmas, Phys. Rev. D 75 (2007) 025009 [hep-th/0609079] [INSPIRE].
  68. [68]
    O. Kaczmarek, Screening at finite temperature and density, PoS(CPOD07)043 [arXiv:0710.0498] [INSPIRE].
  69. [69]
    M. Cheng et al., The QCD equation of state with almost physical quark masses, Phys. Rev. D 77 (2008) 014511 [arXiv:0710.0354] [INSPIRE].
  70. [70]
    S. Gupta, K. Hübner and O. Kaczmarek, Renormalized Polyakov loops in many representations, Phys. Rev. D 77 (2008) 034503 [arXiv:0711.2251] [INSPIRE].
  71. [71]
    K. Schade, Applications of Holography to Strongly Coupled Plasmas, Ph.D. Thesis, Universität Heidelberg, Heidelberg, Germany (2012).Google Scholar
  72. [72]
    E. Nakano, S. Teraguchi and W.-Y. Wen, Drag force, jet quenching and AdS/QCD, Phys. Rev. D 75 (2007) 085016 [hep-ph/0608274] [INSPIRE].
  73. [73]
    O. DeWolfe and C. Rosen, Robustness of Sound Speed and Jet Quenching for Gauge/Gravity Models of Hot QCD, JHEP 07 (2009) 022 [arXiv:0903.1458] [INSPIRE].ADSCrossRefGoogle Scholar
  74. [74]
    O. Kaczmarek and F. Zantow, Static quark anti-quark interactions at zero and finite temperature QCD. II. Quark anti-quark internal energy and entropy, hep-lat/0506019 [INSPIRE].
  75. [75]
    P. Petreczky, Heavy quark potentials and quarkonia binding, Eur. Phys. J. C 43 (2005) 51 [hep-lat/0502008] [INSPIRE].
  76. [76]
    I. Iatrakis and D.E. Kharzeev, Holographic entropy and real-time dynamics of quarkonium dissociation in non-Abelian plasma, Phys. Rev. D 93 (2016) 086009 [arXiv:1509.08286] [INSPIRE].
  77. [77]
    K. Bitaghsir Fadafan and S.K. Tabatabaei, Entropic destruction of a moving heavy quarkonium, Phys. Rev. D 94 (2016) 026007 [arXiv:1512.08254] [INSPIRE].
  78. [78]
    J. Noronha, The Heavy Quark Free Energy in QCD and in Gauge Theories with Gravity Duals, Phys. Rev. D 82 (2010) 065016 [arXiv:1003.0914] [INSPIRE].
  79. [79]
    J. Noronha, Connecting Polyakov Loops to the Thermodynamics of SU(N c) Gauge Theories Using the Gauge-String Duality, Phys. Rev. D 81 (2010) 045011 [arXiv:0910.1261] [INSPIRE].
  80. [80]
    M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Mineola, U.S.A. (1964).Google Scholar
  81. [81]
    O. Andreev, Renormalized Polyakov Loop in the Deconfined Phase of SU(N ) Gauge Theory and Gauge/String Duality, Phys. Rev. Lett. 102 (2009) 212001 [arXiv:0903.4375] [INSPIRE].
  82. [82]
    M. Panero, Thermodynamics of the QCD plasma and the large-N limit, Phys. Rev. Lett. 103 (2009) 232001 [arXiv:0907.3719] [INSPIRE].ADSCrossRefGoogle Scholar
  83. [83]
    U. Gürsoy, E. Kiritsis, L. Mazzanti, G. Michalogiorgakis and F. Nitti, Improved Holographic QCD, Lect. Notes Phys. 828 (2011) 79 [arXiv:1006.5461] [INSPIRE].ADSCrossRefMATHGoogle Scholar
  84. [84]
    A. Bazavov et al., Equation of state and QCD transition at finite temperature, Phys. Rev. D 80 (2009) 014504 [arXiv:0903.4379] [INSPIRE].
  85. [85]
    S.S. Gubser, Comparing the drag force on heavy quarks in N = 4 super-Yang-Mills theory and QCD, Phys. Rev. D 76 (2007) 126003 [hep-th/0611272] [INSPIRE].
  86. [86]
    A. Bazavov and P. Petreczky, Polyakov loop in 2 + 1 flavor QCD, Phys. Rev. D 87 (2013) 094505 [arXiv:1301.3943] [INSPIRE].
  87. [87]
    A. Bazavov et al., Polyakov loop in 2 + 1 flavor QCD from low to high temperatures, Phys. Rev. D 93 (2016) 114502 [arXiv:1603.06637] [INSPIRE].
  88. [88]
    D.E. Kharzeev, Deconfinement as an entropic self-destruction: a solution for the quarkonium suppression puzzle?, Phys. Rev. D 90 (2014) 074007 [arXiv:1409.2496] [INSPIRE].
  89. [89]
    H. Satz, Quarkonium Binding and Entropic Force, Eur. Phys. J. C 75 (2015) 193 [arXiv:1501.03940] [INSPIRE].

Copyright information

© The Author(s) 2018

Authors and Affiliations

  • Carlo Ewerz
    • 1
    • 2
    • 3
  • Olaf Kaczmarek
    • 4
  • Andreas Samberg
    • 1
    • 2
  1. 1.Institut für Theoretische Physik, Ruprecht-Karls-Universität HeidelbergHeidelbergGermany
  2. 2.ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum für SchwerionenforschungDarmstadtGermany
  3. 3.Frankfurt Institute for Advanced StudiesFrankfurtGermany
  4. 4.Fakultät für Physik, Universität BielefeldBielefeldGermany

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