Skip to main content

Drag force to all orders in gradients

A preprint version of the article is available at arXiv.

Abstract

We study the energy loss of a heavy quark slowly moving through an evolving strongly coupled plasma. We use the linearized fluid/gravity correspondence to describe small perturbations of the medium flow with general spacetime dependence. This all order linearized hydrodynamics results in a drag force exerted on a heavy quark even when it is at rest with the fluid element. We show how the general contribution to the drag force can be derived order by order in the medium velocity gradients and provide explicit results valid up to the third order. We then obtain an approximate semi-analytic result for the drag force to all orders in the gradient expansion but linearized in the medium velocity. Thus, the effects of a class of hydrodynamic gradients on the drag force are re-summed, giving further insight into the dissipative properties of strongly coupled plasmas. The all order result allows us to study the drag force in the non-hydrodynamic regime of linear medium perturbations that vary rapidly in space and time.

References

  1. W. Busza, K. Rajagopal and W. van der Schee, Heavy ion collisions: the big picture and the big questions, Ann. Rev . Nucl. Part. Sci. 68 (2018) 339 [arXiv:1802. 04801] [INSPIRE].

  2. Y.L. Dokshitzer and D.E. Kharzeev, Heavy quark colorimetry of QCD matter, Phys. Lett. B 519 (2001) 199 [hep-ph/0106202] [INSPIRE].

  3. R. Baier, Y.L. Dokshitzer, A.H. Mueller and D. Schiff , Quenching of hadron spectra in media, JHEP 09 (2001) 033 [hep-ph/0106347] [INSPIRE].

  4. S. Jeon and G.D. Moore, Energy loss of leading partons in a thermal QCD medium, Phys. Rev. C 71 (2005) 034901 [hep-ph/0309332] [INSPIRE].

  5. M.G. Mustafa and M.H. Thoma, Quenching of hadron spectra due to the collisional energy loss of partons in the quark gluon plasma, Acta Phys. Hung. A 22 (2005) 93 [hep-ph/0311168] [INSPIRE].

  6. M. Djordjevic and M. Gyulassy, Where is the charm quark energy loss at RHIC?, Phys. Lett. B 560 (2003) 37 [nucl-th/0302069] [INSPIRE].

  7. M. Djordjevic and M. Gyulassy, Heavy quark radiative energy loss in QCD matter, Nucl. Phys. A 733 (2004) 265 [nucl-th/0310076] [INSPIRE].

  8. B.-W. Zhang, E. Wang and X.-N. Wang, Heavy quark energy loss in nuclear medium, Phys. Rev. Lett. 93 (2004) 072301 [nucl-th/0309040] [INSPIRE].

  9. 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].

  10. S. Wicks, W. Horowitz, M. Djordjevic and M. Gyulassy, Elastic, inelastic and path length fluctuations in jet tomography, Nucl. Phys. A 784 (2007) 426 [nucl-th/0512076] [INSPIRE].

  11. I. Vitev, Non-Abelian energy loss in cold nuclear matter, Phys. Rev. C 75 (2007) 064906 [hep-ph/0703002] [INSPIRE].

  12. Z.-B. Kang, F. Ringer and I. Vitev, Effective field theory approach to open heavy flavor production in heavy-ion collisions, JHEP 03 (2017) 146 [arXiv:1610.02043] [INSPIRE].

  13. B. Blagojevic, M. Djordjevic and M. Djordjevic, Calculating hard probe radiative energy loss beyond the soft-gluon approximation: examining the approximation validity, Phys. Rev. C 99 (2019) 024901 [arXiv:1804.07593] [INSPIRE].

  14. PHENIX collaboration, 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].

  15. BRAHMS collaboration, 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].

  16. PHOBOS collaboration, The PHOBOS perspective on discoveries at RHIC, Nucl. Phys. A 757 (2005) 28 [nucl-ex/0410022] [INSPIRE].

  17. STAR collaboration, 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].

  18. ALICE collaboration, Elliptic flow of charged particles in Pb-Pb collisions at 2.76 TeV, Phys. Rev. Lett. 105 (2010) 252302 [arXiv:1011.3914] [INSPIRE].

  19. ATLAS collaboration, Measurement of the pseudorapidity and transverse momentum dependence of the elliptic flow of charged particles in lead-lead collisions at \( \sqrt{s_{\mathrm{NN}}} \) = 2.76 TeV with the ATLAS detector, Phys. Lett. B 707 (2012) 330 [arXiv:1108.6018] [INSPIRE].

  20. CMS collaboration, Measurement of the elliptic anisotropy of charged particles produced in Pb-Pb collisions at \( \sqrt{s_{\mathrm{NN}}} \) = 2.76 TeV, Phys. Rev. C 87 (2013) 014902 [arXiv:1204.1409] [INSPIRE].

  21. P. Romatschke and U. Romatschke, Relativistic fluid dynamics in and out of equilibrium, Cambridge University Press, Cambridge, U.K. (2019) [arXiv:1712.05815] [INSPIRE].

  22. J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].

  23. 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].

  24. E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].

  25. 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) [arXiv:1101.0618] [INSPIRE].

  26. 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].

  27. C.P. Herzog, A. Karch, P. Kovtun, C. Kozcaz and L.G. Yaffe, Energy loss of a heavy quark moving through N = 4 supersymmetric Yang-Mills plasma, JHEP 07 (2006) 013 [hep-th/0605158] [INSPIRE].

  28. S.S. Gubser, Drag force in AdS/CFT, Phys. Rev. D 74 (2006) 126005 [hep-th/0605182] [INSPIRE].

  29. J. Casalderrey-Solana and D. Teaney, Heavy quark diffusion in strongly coupled N = 4 Yang-Mills, Phys. Rev. D 74 (2006) 085012 [hep-ph/0605199] [INSPIRE].

  30. S.S. Gubser, Momentum fluctuations of heavy quarks in the gauge-string duality, Nucl. Phys. B 790 (2008) 175 [hep-th/0612143] [INSPIRE].

  31. 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].

  32. H. Liu, K. Rajagopal and U.A. Wiedemann, Calculating the jet quenching parameter from AdS/CFT, Phys. Rev. Lett. 97 (2006) 182301 [hep-ph/0605178] [INSPIRE].

  33. H. Liu, K. Rajagopal and U.A. Wiedemann, An AdS/CFT calculation of screening in a hot wind, Phys. Rev. Lett. 98 (2007) 182301 [hep-ph/0607062] [INSPIRE].

  34. E. Caceres, M. Natsuume and T. Okamura, Screening length in plasma winds, JHEP 10 (2006) 011 [hep-th/0607233] [INSPIRE].

  35. S.S. Gubser, D.R. Gulotta, S.S. Pufu and F.D. Rocha, Gluon energy loss in the gauge-string duality, JHEP 10 (2008) 052 [arXiv:0803.1470] [INSPIRE].

  36. P.M. Chesler, K. Jensen, A. Karch and L.G. Yaffe, Light quark energy loss in strongly-coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. D 79 (2009) 125015 [arXiv:0810.1985] [INSPIRE].

  37. Y. Hatta, E. Iancu and A.H. Mueller, Jet evolution in the N = 4 SYM plasma at strong coupling, JHEP 05 (2008) 037 [arXiv:0803.2481] [INSPIRE].

  38. K. Bitaghsir Fadafan, H. Liu, K. Rajagopal and U.A. Wiedemann, Stirring strongly coupled plasma, Eur. Phys. J. C 61 (2009) 553 [arXiv:0809.2869] [INSPIRE].

  39. P. Arnold and D. Vaman, Jet quenching in hot strongly coupled gauge theories revisited: 3-point correlators with gauge-gravity duality, JHEP 10 (2010) 099 [arXiv:1008.4023] [INSPIRE].

  40. P.M. Chesler, Y.-Y. Ho and K. Rajagopal, Shining a gluon beam through quark-gluon plasma, Phys. Rev. D 85 (2012) 126006 [arXiv:1111.1691] [INSPIRE].

  41. P. Arnold, P. Szepietowski, D. Vaman and G. Wong, Tidal stretching of gravitons into classical strings: application to jet quenching with AdS/CFT, JHEP 02 (2013) 130 [arXiv:1212.3321] [INSPIRE].

  42. P. Arnold, P. Szepietowski and D. Vaman, Coupling dependence of jet quenching in hot strongly-coupled gauge theories, JHEP 07 (2012) 024 [arXiv:1203.6658] [INSPIRE].

  43. M. Chernicoff, D. Fernandez, D. Mateos and D. Trancanelli, Quarkonium dissociation by anisotropy, JHEP 01 (2013) 170 [arXiv:1208.2672] [INSPIRE].

  44. D. Giataganas, Probing strongly coupled anisotropic plasma, JHEP 07 (2012) 031 [arXiv:1202.4436] [INSPIRE].

  45. P.M. Chesler, M. Lekaveckas and K. Rajagopal, Heavy quark energy loss far from equilibrium in a strongly coupled collision, JHEP 10 (2013) 013 [arXiv:1306.0564] [INSPIRE].

  46. M. Lekaveckas and K. Rajagopal, Effects of fluid velocity gradients on heavy quark energy loss, JHEP 02 (2014) 068 [arXiv:1311.5577] [INSPIRE].

  47. A. Ficnar, S.S. Gubser and M. Gyulassy, Shooting string holography of jet quenching at RHIC and LHC, Phys. Lett. B 738 (2014) 464 [arXiv:1311.6160] [INSPIRE].

  48. D. Dudal and T.G. Mertens, Melting of charmonium in a magnetic field from an effective AdS/QCD model, Phys. Rev. D 91 (2015) 086002 [arXiv:1410.3297] [INSPIRE].

  49. A.V. Sadofyev and Y. Yin, The charmonium dissociation in an “anomalous wind”, JHEP 01 (2016) 052 [arXiv:1510.06760] [INSPIRE].

  50. K. Rajagopal and A.V. Sadofyev, Chiral drag force, JHEP 10 (2015) 018 [arXiv:1505.07379] [INSPIRE].

  51. P.M. Chesler and K. Rajagopal, On the evolution of jet energy and opening angle in strongly coupled plasma, JHEP 05 (2016) 098 [arXiv:1511.07567] [INSPIRE].

  52. K.A. Mamo, Energy loss of a nonaccelerating quark moving through a strongly coupled N = 4 super Yang-Mills vacuum or plasma in strong magnetic field, Phys. Rev. D 94 (2016) 041901 [arXiv:1606.01598] [INSPIRE].

  53. S.I. Finazzo, R. Critelli, R. Rougemont and J. Noronha, Momentum transport in strongly coupled anisotropic plasmas in the presence of strong magnetic fields, Phys. Rev. D 94 (2016) 054020 [Erratum ibid. 96 (2017) 019903] [arXiv:1605.06061] [INSPIRE].

  54. D. Dudal and S. Mahapatra, Confining gauge theories and holographic entanglement entropy with a magnetic field, JHEP 04 (2017) 031 [arXiv:1612.06248] [INSPIRE].

  55. S. Li, K.A. Mamo and H.-U. Yee, Jet quenching parameter of the quark-gluon plasma in a strong magnetic field: perturbative QCD and AdS/CFT correspondence, Phys. Rev. D 94 (2016) 085016 [arXiv:1605.00188] [INSPIRE].

  56. B. Singh, L. Thakur and H. Mishra, Heavy quark complex potential in a strongly magnetized hot QGP medium, Phys. Rev. D 97 (2018) 096011 [arXiv:1711.03071] [INSPIRE].

  57. D. Dudal and T.G. Mertens, Holographic estimate of heavy quark diffusion in a magnetic field, Phys. Rev. D 97 (2018) 054035 [arXiv:1802.02805] [INSPIRE].

  58. H. Bohra, D. Dudal, A. Hajilou and S. Mahapatra, Anisotropic string tensions and inversely magnetic catalyzed deconfinement from a dynamical AdS/QCD model, Phys. Lett. B 801 (2020) 135184 [arXiv:1907.01852] [INSPIRE].

  59. J. Casalderrey-Solana, D.C. Gulhan, J.G. Milhano, D. Pablos and K. Rajagopal, A hybrid strong/weak coupling approach to jet quenching, JHEP 10 (2014) 019 [Erratum ibid. 09 (2015) 175] [arXiv:1405.3864] [INSPIRE].

  60. J. Casalderrey-Solana, D.C. Gulhan, J.G. Milhano, D. Pablos and K. Rajagopal, Predictions for boson-jet observables and fragmentation function ratios from a hybrid strong/weak coupling model for jet quenching, JHEP 03 (2016) 053 [arXiv:1508.00815] [INSPIRE].

  61. K. Rajagopal, A.V. Sadofyev and W. van der Schee, Evolution of the jet opening angle distribution in holographic plasma, Phys. Rev. Lett. 116 (2016) 211603 [arXiv:1602.04187] [INSPIRE].

  62. J. Casalderrey-Solana, D. Gulhan, G. Milhano, D. Pablos and K. Rajagopal, Angular structure of jet quenching within a hybrid strong/weak coupling model, JHEP 03 (2017) 135 [arXiv:1609.05842] [INSPIRE].

  63. Z. Hulcher, D. Pablos and K. Rajagopal, Resolution effects in the hybrid strong/weak coupling model, JHEP 03 (2018) 010 [arXiv:1707.05245] [INSPIRE].

  64. J. Brewer, K. Rajagopal, A. Sadofyev and W. van der Schee, Holographic jet shapes and their evolution in strongly coupled plasma, Nucl. Phys. A 967 (2017) 508 [arXiv:1704.05455] [INSPIRE].

  65. J. Brewer, K. Rajagopal, A. Sadofyev and W. Van Der Schee, Evolution of the mean jet shape and dijet asymmetry distribution of an ensemble of holographic jets in strongly coupled plasma, JHEP 02 (2018) 015 [arXiv:1710.03237] [INSPIRE].

  66. J. Brewer, A. Sadofyev and W. van der Schee, Jet shape modifications in holographic dijet systems, arXiv:1809.10695 [INSPIRE].

  67. J. Casalderrey-Solana, G. Milhano, D. Pablos and K. Rajagopal, Modification of jet substructure in heavy ion collisions as a probe of the resolution length of quark-gluon plasma, JHEP 01 (2020) 044 [arXiv:1907.11248] [INSPIRE].

  68. S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear fluid dynamics from gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].

  69. J. Erdmenger, M. Haack, M. Kaminski and A. Yarom, Fluid dynamics of R-charged black holes, JHEP 01 (2009) 055 [arXiv:0809.2488] [INSPIRE].

  70. N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam and P. Surowka, Hydrodynamics from charged black branes, JHEP 01 (2011) 094 [arXiv:0809.2596] [INSPIRE].

  71. D.E. Kharzeev, J. Liao, S.A. Voloshin and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions — a status report, Prog. Part. Nucl. Phys. 88 (2016) 1 [arXiv:1511.04050] [INSPIRE].

  72. Y. Bu and M. Lublinsky, All order linearized hydrodynamics from fluid-gravity correspondence, Phys. Rev. D 90 (2014) 086003 [arXiv:1406.7222] [INSPIRE].

  73. Y. Bu and M. Lublinsky, Linearized fluid/gravity correspondence: from shear viscosity to all order hydrodynamics, JHEP 11 (2014) 064 [arXiv:1409.3095] [INSPIRE].

  74. Y. Bu and M. Lublinsky, Linearly resummed hydrodynamics in a weakly curved spacetime, JHEP 04 (2015) 136 [arXiv:1502.08044] [INSPIRE].

  75. Y. Bu, M. Lublinsky and A. Sharon, Hydrodynamics dual to Einstein-Gauss-Bonnet gravity: all-order gradient resummation, JHEP 06 (2015) 162 [arXiv:1504.01370] [INSPIRE].

  76. M. Lublinsky and E. Shuryak, How much entropy is produced in strongly coupled Quark-Gluon Plasma (sQGP) by dissipative effects?, Phys. Rev. C 76 (2007) 021901 [arXiv:0704.1647] [INSPIRE].

  77. M. Lublinsky and E. Shuryak, Improved hydrodynamics from the AdS/CFT, Phys. Rev. D 80 (2009) 065026 [arXiv:0905.4069] [INSPIRE].

  78. L.D. Landau and E.M. Lifshitz, Fluid mechanics, Course of theoretical physics 6, Pergamon Press, New York, NY, U.S.A. (1959).

  79. G. Policastro, D.T. Son and A.O. Starinets, The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 87 (2001) 081601 [hep-th/0104066] [INSPIRE].

  80. G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].

  81. P. Kovtun, D.T. Son and A.O. Starinets, Viscosity in strongly interacting quantum field theories from black hole physics, Phys. Rev. Lett. 94 (2005) 111601 [hep-th/0405231] [INSPIRE].

  82. R. Gregory, S. Kanno and J. Soda, Holographic superconductors with higher curvature corrections, JHEP 10 (2009) 010 [arXiv:0907.3203] [INSPIRE].

  83. J.P. Boyd, Chebyshev and Fourier spectral methods, Dover Publications, New York, NY, U.S.A. (2001).

  84. W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical recipes: the art of scientific computing, Cambridge University Press, New York, NY, U.S.A. (2007).

  85. S. Chapman, Y. Neiman and Y. Oz, Fluid/gravity correspondence, local Wald entropy current and gravitational anomaly, JHEP 07 (2012) 128 [arXiv:1202.2469] [INSPIRE].

  86. C. Eling, A. Meyer and Y. Oz, Local entropy current in higher curvature gravity and Rindler hydrodynamics, JHEP 08 (2012) 088 [arXiv:1205.4249] [INSPIRE].

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

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jared Reiten.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

ArXiv ePrint: 1912.08816

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Reiten, J., Sadofyev, A.V. Drag force to all orders in gradients. J. High Energ. Phys. 2020, 146 (2020). https://doi.org/10.1007/JHEP07(2020)146

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP07(2020)146

Keywords

  • Holography and quark-gluon plasmas
  • Gauge-gravity correspondence