Quantum dissipation of a heavy quark from a nonlinear stochastic Schrödinger equation

Abstract

We study the open system dynamics of a heavy quark in the quark-gluon plasma using a Lindblad master equation. Applying the quantum state diffusion approach by Gisin and Percival, we derive and numerically solve a nonlinear stochastic Schrödinger equation for wave functions, which is equivalent to the Lindblad master equation for the density matrix. From our numerical analysis in one spatial dimension, it is shown that the density matrix relaxes to the Boltzmann distribution in various setups (with and without external potentials), independently of the initial conditions. We also confirm that quantum dissipation plays an essential role not only in the long-time behavior of the heavy quark but also at early times if the heavy quark initial state is localized and quantum decoherence is ineffective.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    Y. Akiba et al., The hot QCD white paper: exploring the phases of QCD at RHIC and the LHC, arXiv:1502.02730 [INSPIRE].

  2. [2]

    K. Fukushima and T. Hatsuda, The phase diagram of dense QCD, Rept. Prog. Phys. 74 (2011) 014001 [arXiv:1005.4814] [INSPIRE].

  3. [3]

    A. Adams, L.D. Carr, T. Schäfer, P. Steinberg and J.E. Thomas, Strongly correlated quantum fluids: ultracold quantum gases, quantum chromodynamic plasmas and holographic duality, New J. Phys. 14 (2012) 115009 [arXiv:1205.5180] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  4. [4]

    B.V. Jacak and B. Müller, The exploration of hot nuclear matter, Science 337 (2012) 310 [INSPIRE].

    ADS  Article  Google Scholar 

  5. [5]

    S. Borsányi, Z. Fodor, C. Hölbling, S.D. Katz, S. Krieg and K.K. Szabo, Full result for the QCD equation of state with 2 + 1 flavors, Phys. Lett. B 730 (2014) 99 [arXiv:1309.5258] [INSPIRE].

  6. [6]

    HotQCD collaboration, A. Bazavov et al., Equation of state in (2 + 1)-flavor QCD, Phys. Rev. D 90 (2014) 094503 [arXiv:1407.6387] [INSPIRE].

  7. [7]

    S. Borsányi et al., Calculation of the axion mass based on high-temperature lattice quantum chromodynamics, Nature 539 (2016) 69 [arXiv:1606.07494] [INSPIRE].

  8. [8]

    A. Bazavov, P. Petreczky and J.H. Weber, Equation of state in 2 + 1 flavor QCD at high temperatures, Phys. Rev. D 97 (2018) 014510 [arXiv:1710.05024] [INSPIRE].

  9. [9]

    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].

    ADS  Google Scholar 

  10. [10]

    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].

  11. [11]

    WHOT-QCD collaboration, Y. Maezawa et al., Heavy-quark free energy, Debye mass and spatial string tension at finite temperature in two flavor lattice QCD with Wilson quark action, Phys. Rev. D 75 (2007) 074501 [hep-lat/0702004] [INSPIRE].

  12. [12]

    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].

  13. [13]

    N. Brambilla et al., Heavy quarkonium: progress, puzzles and opportunities, Eur. Phys. J. C 71 (2011) 1534 [arXiv:1010.5827] [INSPIRE].

  14. [14]

    A. Andronic et al., Heavy-flavour and quarkonium production in the LHC era: from proton-proton to heavy-ion collisions, Eur. Phys. J. C 76 (2016) 107 [arXiv:1506.03981] [INSPIRE].

  15. [15]

    T. Matsui and H. Satz, J/ψ suppression by quark-gluon plasma formation, Phys. Lett. B 178 (1986) 416 [INSPIRE].

  16. [16]

    STAR collaboration, L. Adamczyk et al., Suppression of ϒ production in d+Au and Au+Au collisions at \( \sqrt{s_{\mathrm{NN}}}=200 \) GeV, Phys. Lett. B 735 (2014) 127 [Erratum ibid. B 743 (2015) 537] [arXiv:1312.3675] [INSPIRE].

  17. [17]

    PHENIX collaboration, A. Adare et al., Measurement of ϒ(1S + 2S + 3S) production in p+p and Au+Au collisions at \( \sqrt{{\mathrm{s}}_{\mathrm{NN}}}=200 \) GeV, Phys. Rev. C 91 (2015) 024913 [arXiv:1404.2246] [INSPIRE].

  18. [18]

    PHENIX collaboration, A. Adare et al., J/ψ production vs centrality, transverse momentum and rapidity in Au+Au collisions at \( \sqrt{s_{\mathrm{NN}}}=200 \) GeV, Phys. Rev. Lett. 98 (2007) 232301 [nucl-ex/0611020] [INSPIRE].

  19. [19]

    PHENIX collaboration, A. Adare et al., J/ψ production in \( \sqrt{{\mathrm{s}}_{\mathrm{NN}}}=200 \) GeV Cu+Cu collisions, Phys. Rev. Lett. 101 (2008) 122301 [arXiv:0801.0220] [INSPIRE].

  20. [20]

    PHENIX collaboration, A. Adare et al., J/ψ suppression at forward rapidity in Au+Au collisions at \( \sqrt{s_{\mathrm{NN}}}=200 \) GeV, Phys. Rev. C 84 (2011) 054912 [arXiv:1103.6269] [INSPIRE].

  21. [21]

    STAR collaboration, L. Adamczyk et al., J/ψ production at low p T in Au+Au and Cu+Cu collisions at \( \sqrt{s_{\mathrm{NN}}}=200 \) GeV with the STAR detector, Phys. Rev. C 90 (2014) 024906 [arXiv:1310.3563] [INSPIRE].

  22. [22]

    CMS collaboration, Indications of suppression of excited ϒ states in PbPb collisions at \( \sqrt{s_{\mathrm{NN}}}=2.76 \) TeV, Phys. Rev. Lett. 107 (2011) 052302 [arXiv:1105.4894] [INSPIRE].

  23. [23]

    CMS collaboration, Observation of sequential ϒ suppression in PbPb collisions, Phys. Rev. Lett. 109 (2012) 222301 [Erratum ibid. 120 (2018) 199903] [arXiv:1208.2826] [INSPIRE].

  24. [24]

    CMS collaboration, Suppression of ϒ(1S), ϒ(2S) and ϒ(3S) production in PbPb collisions at \( \sqrt{s_{\mathrm{NN}}}=2.76 \) TeV, Phys. Lett. B 770 (2017) 357 [arXiv:1611.01510] [INSPIRE].

  25. [25]

    ALICE collaboration, Suppression of ϒ(1S) at forward rapidity in Pb-Pb collisions at \( \sqrt{s_{\mathrm{NN}}}=2.76 \) TeV, Phys. Lett. B 738 (2014) 361 [arXiv:1405.4493] [INSPIRE].

  26. [26]

    ALICE collaboration, Centrality, rapidity and transverse momentum dependence of J/ψ suppression in Pb-Pb collisions at \( \sqrt{s_{\mathrm{NN}}}=2.76 \) TeV, Phys. Lett. B 734 (2014) 314 [arXiv:1311.0214] [INSPIRE].

  27. [27]

    ALICE collaboration, J/ψ suppression at forward rapidity in Pb-Pb collisions at \( \sqrt{s_{\mathrm{NN}}}=5.02 \) TeV, Phys. Lett. B 766 (2017) 212 [arXiv:1606.08197] [INSPIRE].

  28. [28]

    P. Braun-Munzinger and J. Stachel, (Non)thermal aspects of charmonium production and a new look at J/ψ suppression, Phys. Lett. B 490 (2000) 196 [nucl-th/0007059] [INSPIRE].

  29. [29]

    A. Andronic, P. Braun-Munzinger, K. Redlich and J. Stachel, Heavy quark(onium) at LHC: the statistical hadronization case, J. Phys. G 37 (2010) 094014 [arXiv:1002.4441] [INSPIRE].

  30. [30]

    G. Aarts et al., Heavy-flavor production and medium properties in high-energy nuclear collisionswhat next?, Eur. Phys. J. A 53 (2017) 93 [arXiv:1612.08032] [INSPIRE].

  31. [31]

    R. Rapp, D. Blaschke and P. Crochet, Charmonium and bottomonium production in heavy-ion collisions, Prog. Part. Nucl. Phys. 65 (2010) 209 [arXiv:0807.2470] [INSPIRE].

    ADS  Article  Google Scholar 

  32. [32]

    X. Zhao and R. Rapp, Charmonium in medium: from correlators to experiment, Phys. Rev. C 82 (2010) 064905 [arXiv:1008.5328] [INSPIRE].

  33. [33]

    X. Zhao and R. Rapp, Medium modifications and production of charmonia at LHC, Nucl. Phys. A 859 (2011) 114 [arXiv:1102.2194] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    K. Zhou, N. Xu, Z. Xu and P. Zhuang, Medium effects on charmonium production at ultrarelativistic energies available at the CERN Large Hadron Collider, Phys. Rev. C 89 (2014) 054911 [arXiv:1401.5845] [INSPIRE].

  35. [35]

    T. Song, K.C. Han and C.M. Ko, Bottomonia suppression in heavy-ion collisions, Phys. Rev. C 85 (2012) 014902 [arXiv:1109.6691] [INSPIRE].

  36. [36]

    A. Emerick, X. Zhao and R. Rapp, Bottomonia in the quark-gluon plasma and their production at RHIC and LHC, Eur. Phys. J. A 48 (2012) 72 [arXiv:1111.6537] [INSPIRE].

  37. [37]

    K. Zhou, N. Xu and P. Zhuang, ϒ production in heavy ion collisions at LHC, Nucl. Phys. A 931 (2014) 654 [arXiv:1408.3900] [INSPIRE].

  38. [38]

    X. Yao and B. Müller, Approach to equilibrium of quarkonium in quark-gluon plasma, Phys. Rev. C 97 (2018) 014908 [Erratum ibid. C 97 (2018) 049903] [arXiv:1709.03529] [INSPIRE].

  39. [39]

    M. Strickland, Thermal ϒ 1s and χ b1 suppression in \( \sqrt{s_{\mathrm{NN}}}=2.76 \) TeV Pb-Pb collisions at the LHC, Phys. Rev. Lett. 107 (2011) 132301 [arXiv:1106.2571] [INSPIRE].

  40. [40]

    M. Strickland and D. Bazow, Thermal bottomonium suppression at RHIC and LHC, Nucl. Phys. A 879 (2012) 25 [arXiv:1112.2761] [INSPIRE].

  41. [41]

    B. Krouppa, R. Ryblewski and M. Strickland, Bottomonia suppression in 2.76 TeV Pb-Pb collisions, Phys. Rev. C 92 (2015) 061901 [arXiv:1507.03951] [INSPIRE].

  42. [42]

    B. Krouppa and M. Strickland, Predictions for bottomonia suppression in 5.023 TeV Pb-Pb collisions, Universe 2 (2016) 16 [arXiv:1605.03561] [INSPIRE].

  43. [43]

    B. Krouppa, A. Rothkopf and M. Strickland, Bottomonium suppression using a lattice QCD vetted potential, Phys. Rev. D 97 (2018) 016017 [arXiv:1710.02319] [INSPIRE].

  44. [44]

    C. Young and K. Dusling, Quarkonium above deconfinement as an open quantum system, Phys. Rev. C 87 (2013) 065206 [arXiv:1001.0935] [INSPIRE].

  45. [45]

    N. Borghini and C. Gombeaud, Heavy quarkonia in a medium as a quantum dissipative system: master equation approach, Eur. Phys. J. C 72 (2012) 2000 [arXiv:1109.4271] [INSPIRE].

  46. [46]

    Y. Akamatsu and A. Rothkopf, Stochastic potential and quantum decoherence of heavy quarkonium in the quark-gluon plasma, Phys. Rev. D 85 (2012) 105011 [arXiv:1110.1203] [INSPIRE].

  47. [47]

    A. Rothkopf, A first look at bottomonium melting via a stochastic potential, JHEP 04 (2014) 085 [arXiv:1312.3246] [INSPIRE].

    ADS  Article  Google Scholar 

  48. [48]

    Y. Akamatsu, Heavy quark master equations in the Lindblad form at high temperatures, Phys. Rev. D 91 (2015) 056002 [arXiv:1403.5783] [INSPIRE].

  49. [49]

    Y. Akamatsu, Langevin dynamics and decoherence of heavy quarks at high temperatures, Phys. Rev. C 92 (2015) 044911 [arXiv:1503.08110] [INSPIRE].

  50. [50]

    S. Kajimoto, Y. Akamatsu, M. Asakawa and A. Rothkopf, Dynamical dissociation of quarkonia by wave function decoherence, Phys. Rev. D 97 (2018) 014003 [arXiv:1705.03365] [INSPIRE].

  51. [51]

    J.-P. Blaizot, D. De Boni, P. Faccioli and G. Garberoglio, Heavy quark bound states in a quark-gluon plasma: dissociation and recombination, Nucl. Phys. A 946 (2016) 49 [arXiv:1503.03857] [INSPIRE].

    ADS  Article  Google Scholar 

  52. [52]

    J.-P. Blaizot and M.A. Escobedo, Quantum and classical dynamics of heavy quarks in a quark-gluon plasma, JHEP 06 (2018) 034 [arXiv:1711.10812] [INSPIRE].

    ADS  Article  Google Scholar 

  53. [53]

    J.-P. Blaizot and M.A. Escobedo, The approach to equilibrium of a quarkonium in a quark-gluon plasma, arXiv:1803.07996 [INSPIRE].

  54. [54]

    N. Brambilla, M.A. Escobedo, J. Soto and A. Vairo, Quarkonium suppression in heavy-ion collisions: an open quantum system approach, Phys. Rev. D 96 (2017) 034021 [arXiv:1612.07248] [INSPIRE].

  55. [55]

    N. Brambilla, M.A. Escobedo, J. Soto and A. Vairo, Heavy quarkonium suppression in a fireball, Phys. Rev. D 97 (2018) 074009 [arXiv:1711.04515] [INSPIRE].

  56. [56]

    D. De Boni, Fate of in-medium heavy quarks via a Lindblad equation, JHEP 08 (2017) 064 [arXiv:1705.03567] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  57. [57]

    R. Katz and P.B. Gossiaux, The Schrödinger-Langevin equation with and without thermal fluctuations, Annals Phys. 368 (2016) 267 [arXiv:1504.08087] [INSPIRE].

  58. [58]

    M. Laine, O. Philipsen, P. Romatschke and M. Tassler, Real-time static potential in hot QCD, JHEP 03 (2007) 054 [hep-ph/0611300] [INSPIRE].

  59. [59]

    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].

  60. [60]

    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].

  61. [61]

    A. Rothkopf, T. Hatsuda and S. Sasaki, Complex heavy-quark potential at finite temperature from lattice QCD, Phys. Rev. Lett. 108 (2012) 162001 [arXiv:1108.1579] [INSPIRE].

    ADS  Article  Google Scholar 

  62. [62]

    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].

  63. [63]

    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].

    ADS  Google Scholar 

  64. [64]

    Y. Burnier and A. Rothkopf, Complex heavy-quark potential and Debye mass in a gluonic medium from lattice QCD, Phys. Rev. D 95 (2017) 054511 [arXiv:1607.04049] [INSPIRE].

  65. [65]

    H.P. Breuer and F. Petruccione, The theory of open quantum systems, Oxford University Press, Oxford, U.K., (2002) [INSPIRE].

  66. [66]

    N. Gisin and I.C. Percival, The quantum-state diffusion model applied to open systems, J. Phys. A 25 (1992) 5677.

  67. [67]

    I.C. Percival, Quantum state diffusion, Cambridge University Press, Cambridge, U.K., (1998).

  68. [68]

    G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48 (1976) 119 [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yukinao Akamatsu.

Additional information

ArXiv ePrint: 1805.00167

Rights and permissions

Open Access  This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.

The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Akamatsu, Y., Asakawa, M., Kajimoto, S. et al. Quantum dissipation of a heavy quark from a nonlinear stochastic Schrödinger equation. J. High Energ. Phys. 2018, 29 (2018). https://doi.org/10.1007/JHEP07(2018)029

Download citation

Keywords

  • Quantum Dissipative Systems
  • Quark-Gluon Plasma
  • Stochastic Processes