Journal of High Energy Physics

, 2017:206 | Cite as

Thermal quarkonium physics in the pseudoscalar channel

  • Y. Burnier
  • H.-T. Ding
  • O. Kaczmarek
  • A.-L. Kruse
  • M. LaineEmail author
  • H. Ohno
  • H. Sandmeyer
Open Access
Regular Article - Theoretical Physics


The pseudoscalar correlator is an ideal lattice probe for thermal modifications to quarkonium spectra, given that it is not compromised by a contribution from a large transport peak. We construct a perturbative spectral function incorporating resummed thermal effects around the threshold and vacuum asymptotics above the threshold, and compare the corresponding imaginary-time correlators with continuum-extrapolated lattice data for quenched SU(3) at several temperatures. Modest differences are observed, which may originate from non-perturbative mass shifts or renormalization factors, however no resonance peaks are needed for describing the quenched lattice data for charmonium at and above T ∼ 1.1Tc ∼ 350 MeV. For comparison, in the bottomonium case a good description of the lattice data is obtained with a spectral function containing a single thermally broadened resonance peak.


Quark-Gluon Plasma Thermal Field Theory Lattice QCD Perturbative QCD 


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]
    A. Mócsy, P. Petreczky and M. Strickland, Quarkonia in the Quark Gluon Plasma, Int. J. Mod. Phys. A 28 (2013) 1340012 [arXiv:1302.2180] [INSPIRE].CrossRefGoogle Scholar
  2. [2]
    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].ADSCrossRefGoogle Scholar
  3. [3]
    K. Morita and S.H. Lee, Heavy quarkonium correlators at finite temperature: QCD sum rule approach, Phys. Rev. D 82 (2010) 054008 [arXiv:0908.2856] [INSPIRE].
  4. [4]
    A. Ayala, C.A. Dominguez and M. Loewe, Finite Temperature QCD Sum Rules: a Review, Adv. High Energy Phys. 2017 (2017) 9291623 [arXiv:1608.04284] [INSPIRE].CrossRefGoogle Scholar
  5. [5]
    P. Petreczky and D. Teaney, Heavy quark diffusion from the lattice, Phys. Rev. D 73 (2006) 014508 [hep-ph/0507318] [INSPIRE].
  6. [6]
    T. Umeda, Constant contribution in meson correlators at finite temperature, Phys. Rev. D 75 (2007) 094502 [hep-lat/0701005] [INSPIRE].
  7. [7]
    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].
  8. [8]
    A. Francis, O. Kaczmarek, M. Laine, T. Neuhaus and H. Ohno, Nonperturbative estimate of the heavy quark momentum diffusion coefficient, Phys. Rev. D 92 (2015) 116003 [arXiv:1508.04543] [INSPIRE].ADSGoogle Scholar
  9. [9]
    C. Allton, T. Harris, S. Kim, M.P. Lombardo, S.M. Ryan and J.I. Skullerud, The bottomonium spectrum at finite temperature from N f = 2 + 1 lattice QCD, JHEP 07 (2014) 097 [arXiv:1402.6210] [INSPIRE].ADSGoogle Scholar
  10. [10]
    S. Kim, P. Petreczky and A. Rothkopf, Lattice NRQCD study of S- and P-wave bottomonium states in a thermal medium with N f = 2 + 1 light flavors, Phys. Rev. D 91 (2015) 054511 [arXiv:1409.3630] [INSPIRE].
  11. [11]
    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
  12. [12]
    Y. Burnier, O. Kaczmarek and A. Rothkopf, In-medium P-wave quarkonium from the complex lattice QCD potential, JHEP 10 (2016) 032 [arXiv:1606.06211] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    S. Kim, Heavy Flavours in Quark-Gluon Plasma, PoS(LATTICE2016)007 [arXiv:1702.02297] [INSPIRE].
  14. [14]
    F. Karsch, E. Laermann, P. Petreczky and S. Stickan, Infinite temperature limit of meson spectral functions calculated on the lattice, Phys. Rev. D 68 (2003) 014504 [hep-lat/0303017] [INSPIRE].
  15. [15]
    G. Aarts and J.M. Martínez Resco, Continuum and lattice meson spectral functions at nonzero momentum and high temperature, Nucl. Phys. B 726 (2005) 93 [hep-lat/0507004] [INSPIRE].
  16. [16]
    S. Borsányi et al., Charmonium spectral functions from 2 + 1 flavour lattice QCD, JHEP 04 (2014) 132 [arXiv:1401.5940] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    A. Ikeda, M. Asakawa and M. Kitazawa, In-medium dispersion relations of charmonia studied by maximum entropy method, Phys. Rev. D 95 (2017) 014504 [arXiv:1610.07787] [INSPIRE].
  18. [18]
    H.T. Ding, A. Francis, O. Kaczmarek, F. Karsch, H. Satz and W. Soeldner, Charmonium properties in hot quenched lattice QCD, Phys. Rev. D 86 (2012) 014509 [arXiv:1204.4945] [INSPIRE].
  19. [19]
    Y. Burnier and M. Laine, Massive vector current correlator in thermal QCD, JHEP 11 (2012) 086 [arXiv:1210.1064] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  20. [20]
    Y. Burnier and M. Laine, Charm mass effects in bulk channel correlations, JHEP 11 (2013) 012 [arXiv:1309.1573] [INSPIRE].ADSCrossRefGoogle Scholar
  21. [21]
    Y. Burnier and M. Laine, Temporal mesonic correlators at NLO for any quark mass, PoS(LATTICE 2013)218 [arXiv:1310.6124] [INSPIRE].
  22. [22]
    S. Caron-Huot, Asymptotics of thermal spectral functions, Phys. Rev. D 79 (2009) 125009 [arXiv:0903.3958] [INSPIRE].ADSGoogle Scholar
  23. [23]
    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
  24. [24]
    T. Matsui and H. Satz, J/ψ Suppression by Quark-Gluon Plasma Formation, Phys. Lett. B 178 (1986) 416 [INSPIRE].
  25. [25]
    M. Laine, O. Philipsen, P. Romatschke and M. Tassler, Real-time static potential in hot QCD, JHEP 03 (2007) 054 [hep-ph/0611300] [INSPIRE].
  26. [26]
    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].
  27. [27]
    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].
  28. [28]
    Y. Burnier, M. Laine and M. Vepsäläinen, Heavy quarkonium in any channel in resummed hot QCD, JHEP 01 (2008) 043 [arXiv:0711.1743] [INSPIRE].ADSCrossRefGoogle Scholar
  29. [29]
    M.A. Escobedo and J. Soto, Non-relativistic bound states at finite temperature (I): The Hydrogen atom, Phys. Rev. A 78 (2008) 032520 [arXiv:0804.0691] [INSPIRE].
  30. [30]
    F. Dominguez and B. Wu, On dissociation of heavy mesons in a hot quark-gluon plasma, Nucl. Phys. A 818 (2009) 246 [arXiv:0811.1058] [INSPIRE].ADSCrossRefGoogle Scholar
  31. [31]
    G. ’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
  32. [32]
    S.A. Larin, The Renormalization of the axial anomaly in dimensional regularization, Phys. Lett. B 303 (1993) 113 [hep-ph/9302240] [INSPIRE].
  33. [33]
    N. Gray, D.J. Broadhurst, W. Grafe and K. Schilcher, Three Loop Relation of Quark \( \overline{M\kern0.1em S} \) and Pole Masses, Z. Phys. C 48 (1990) 673 [INSPIRE].
  34. [34]
    K. Melnikov and T.v. Ritbergen, The Three loop relation between the \( \overline{M\kern0.1em S} \) and the pole quark masses, Phys. Lett. B 482 (2000) 99 [hep-ph/9912391] [INSPIRE].
  35. [35]
    P. Marquard, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and D. Wellmann, \( \overline{\mathrm{MS}} \) -on-shell quark mass relation up to four loops in QCD and a general SU(N) gauge group, Phys. Rev. D 94 (2016) 074025 [arXiv:1606.06754] [INSPIRE].
  36. [36]
    K.G. Chetyrkin, J.H. Kühn and M. Steinhauser, Heavy quark current correlators to O(α s2), Nucl. Phys. B 505 (1997) 40 [hep-ph/9705254] [INSPIRE].
  37. [37]
    A.H. Hoang and T. Teubner, Analytic calculation of two loop corrections to heavy quark pair production vertices induced by light quarks, Nucl. Phys. B 519 (1998) 285 [hep-ph/9707496] [INSPIRE].
  38. [38]
    A.H. Hoang, V. Mateu and S. Mohammad Zebarjad, Heavy Quark Vacuum Polarization Function at O(α s2) and O(α s3), Nucl. Phys. B 813 (2009) 349 [arXiv:0807.4173] [INSPIRE].
  39. [39]
    Y. Kiyo, A. Maier, P. Maierhofer and P. Marquard, Reconstruction of heavy quark current correlators at O(α s3), Nucl. Phys. B 823 (2009) 269 [arXiv:0907.2120] [INSPIRE].
  40. [40]
    A. Maier and P. Marquard, Life of Π, arXiv:1710.03724 [INSPIRE].
  41. [41]
    B.A. Kniehl, A. Onishchenko, J.H. Piclum and M. Steinhauser, Two-loop matching coefficients for heavy quark currents, Phys. Lett. B 638 (2006) 209 [hep-ph/0604072] [INSPIRE].
  42. [42]
    M. Beneke, A. Maier, J. Piclum and T. Rauh, The bottom-quark mass from non-relativistic sum rules at NNNLO, Nucl. Phys. B 891 (2015) 42 [arXiv:1411.3132] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  43. [43]
    R. Harlander and M. Steinhauser, Higgs decay to top quarks at O(α s2), Phys. Rev. D 56 (1997) 3980 [hep-ph/9704436] [INSPIRE].
  44. [44]
    A. Maier and P. Marquard, Low- and High-Energy Expansion of Heavy-Quark Correlators at Next-To-Next-To-Leading Order, Nucl. Phys. B 859 (2012) 1 [arXiv:1110.5581] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  45. [45]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Quark Mass and Field Anomalous Dimensions to \( \mathcal{O}\left({\alpha}_s^5\right) \), JHEP 10 (2014) 076 [arXiv:1402.6611] [INSPIRE].
  46. [46]
    T. Luthe, A. Maier, P. Marquard and Y. Schröder, Five-loop quark mass and field anomalous dimensions for a general gauge group, JHEP 01 (2017) 081 [arXiv:1612.05512] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  47. [47]
    P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Five-Loop Running of the QCD coupling constant, Phys. Rev. Lett. 118 (2017) 082002 [arXiv:1606.08659] [INSPIRE].ADSCrossRefGoogle Scholar
  48. [48]
    F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    S. Capitani, M. Lüscher, R. Sommer and H. Wittig, Non-perturbative quark mass renormalization in quenched lattice QCD, Nucl. Phys. B 544 (1999) 669 [Erratum ibid. B 582 (2000) 762] [hep-lat/9810063] [INSPIRE].
  50. [50]
    R. Sommer, Scale setting in lattice QCD, PoS(LATTICE 2013)015 [arXiv:1401.3270] [INSPIRE].
  51. [51]
    K.-I. Ishikawa, I. Kanamori, Y. Murakami, A. Nakamura, M. Okawa and R. Ueno, Non-perturbative determination of the Λ-parameter in the pure SU(3) gauge theory from the twisted gradient flow coupling, arXiv:1702.06289 [INSPIRE].
  52. [52]
    M. Laine and Y. Schröder, Two-loop QCD gauge coupling at high temperatures, JHEP 03 (2005) 067 [hep-ph/0503061] [INSPIRE].
  53. [53]
    I. Ghisoiu, J. Möller and Y. Schröder, Debye screening mass of hot Yang-Mills theory to three-loop order, JHEP 11 (2015) 121 [arXiv:1509.08727] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  54. [54]
    Y. Schröder, The Static potential in QCD to two loops, Phys. Lett. B 447 (1999) 321 [hep-ph/9812205] [INSPIRE].
  55. [55]
    R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, Analytic three-loop static potential, Phys. Rev. D 94 (2016) 054029 [arXiv:1608.02603] [INSPIRE].
  56. [56]
    ALPHA collaboration, S. Schaefer, R. Sommer and F. Virotta, Critical slowing down and error analysis in lattice QCD simulations, Nucl. Phys. B 845 (2011) 93 [arXiv:1009.5228] [INSPIRE].
  57. [57]
    A. Francis, O. Kaczmarek, M. Laine, T. Neuhaus and H. Ohno, Critical point and scale setting in SU(3) plasma: An update, Phys. Rev. D 91 (2015) 096002 [arXiv:1503.05652] [INSPIRE].
  58. [58]
    ALPHA collaboration, M. Guagnelli, R. Sommer and H. Wittig, Precision computation of a low-energy reference scale in quenched lattice QCD, Nucl. Phys. B 535 (1998) 389 [hep-lat/9806005] [INSPIRE].
  59. [59]
    S. Necco and R. Sommer, The N f = 0 heavy quark potential from short to intermediate distances, Nucl. Phys. B 622 (2002) 328 [hep-lat/0108008] [INSPIRE].
  60. [60]
    B. Sheikholeslami and R. Wohlert, Improved Continuum Limit Lattice Action for QCD with Wilson Fermions, Nucl. Phys. B 259 (1985) 572 [INSPIRE].ADSCrossRefGoogle Scholar
  61. [61]
    M. Lüscher, S. Sint, R. Sommer, P. Weisz and U. Wolff, Nonperturbative O(a) improvement of lattice QCD, Nucl. Phys. B 491 (1997) 323 [hep-lat/9609035] [INSPIRE].
  62. [62]
    R. Sommer, A New way to set the energy scale in lattice gauge theories and its applications to the static force and α s in SU(2) Yang-Mills theory, Nucl. Phys. B 411 (1994) 839 [hep-lat/9310022] [INSPIRE].
  63. [63]
    S. Capitani et al., Renormalization and off-shell improvement in lattice perturbation theory, Nucl. Phys. B 593 (2001) 183 [hep-lat/0007004] [INSPIRE].
  64. [64]
    A. Skouroupathis and H. Panagopoulos, Two-loop renormalization of scalar and pseudoscalar fermion bilinears on the lattice, Phys. Rev. D 76 (2007) 094514 [Erratum ibid. D 78 (2008) 119901] [arXiv:0707.2906] [INSPIRE].
  65. [65]
    M. Lüscher, S. Sint, R. Sommer and P. Weisz, Chiral symmetry and O(a) improvement in lattice QCD, Nucl. Phys. B 478 (1996) 365 [hep-lat/9605038] [INSPIRE].
  66. [66]
    S. Sint and P. Weisz, Further results on O(a) improved lattice QCD to one loop order of perturbation theory, Nucl. Phys. B 502 (1997) 251 [hep-lat/9704001] [INSPIRE].
  67. [67]
    M. Göckeler et al., Perturbative and Nonperturbative Renormalization in Lattice QCD, Phys. Rev. D 82 (2010) 114511 [Erratum ibid. D 86 (2012) 099903] [arXiv:1003.5756] [INSPIRE].
  68. [68]
    S. Gupta, K. Hüebner and O. Kaczmarek, Renormalized Polyakov loops in many representations, Phys. Rev. D 77 (2008) 034503 [arXiv:0711.2251] [INSPIRE].
  69. [69]
    P. Korcyl and G.S. Bali, Non-perturbative determination of improvement coefficients using coordinate space correlators in N f = 2 + 1 lattice QCD, Phys. Rev. D 95 (2017) 014505 [arXiv:1607.07090] [INSPIRE].
  70. [70]
    M. Lüscher, Stochastic locality and master-field simulations of very large lattices, arXiv:1707.09758 [INSPIRE].
  71. [71]
    Y. Burnier, Quarkonium spectral function in medium at next-to-leading order for any quark mass, Eur. Phys. J. C 75 (2015) 529 [arXiv:1410.1304] [INSPIRE].ADSCrossRefGoogle Scholar
  72. [72]
    Y. Burnier, M. Laine and M. Vepsäläinen, Heavy quark medium polarization at next-to-leading order, JHEP 02 (2009) 008 [arXiv:0812.2105] [INSPIRE].ADSCrossRefGoogle Scholar
  73. [73]
    D.J. Broadhurst, J. Fleischer and O.V. Tarasov, Two loop two point functions with masses: Asymptotic expansions and Taylor series, in any dimension, Z. Phys. C 60 (1993) 287 [hep-ph/9304303] [INSPIRE].
  74. [74]
    J.F. Donoghue, B.R. Holstein and R.W. Robinett, Quantum Electrodynamics at Finite Temperature, Annals Phys. 164 (1985) 233 [Erratum ibid. 172 (1986) 483] [INSPIRE].

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Y. Burnier
    • 1
  • H.-T. Ding
    • 2
  • O. Kaczmarek
    • 2
    • 3
  • A.-L. Kruse
    • 3
  • M. Laine
    • 4
    Email author
  • H. Ohno
    • 5
    • 6
  • H. Sandmeyer
    • 3
  1. 1.Gymnase de RenensRenensSwitzerland
  2. 2.Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle PhysicsCentral China Normal UniversityWuhanChina
  3. 3.Fakultät für PhysikUniversität BielefeldBielefeldGermany
  4. 4.AEC, ITP, University of BernBernSwitzerland
  5. 5.Center for Computational SciencesUniversity of TsukubaIbarakiJapan
  6. 6.Physics Department, Brookhaven National LaboratoryUptonU.S.A.

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