Abstract
We investigate the analytic structure of thermal energy-momentum tensor correlators at large but finite coupling in quantum field theories with gravity duals. We compute corrections to the quasinormal spectra of black branes due to the presence of higher derivative R 2 and R 4 terms in the action, focusing on the dual to \( \mathcal{N}=4 \) SYM theory and Gauss-Bonnet gravity. We observe the appearance of new poles in the complex frequency plane at finite coupling. The new poles interfere with hydrodynamic poles of the correlators leading to the breakdown of hydrodynamic description at a coupling-dependent critical value of the wave-vector. The dependence of the critical wave vector on the coupling implies that the range of validity of the hydrodynamic description increases monotonically with the coupling. The behavior of the quasinormal spectrum at large but finite coupling may be contrasted with the known properties of the hierarchy of relaxation times determined by the spectrum of a linearized kinetic operator at weak coupling. We find that the ratio of a transport coefficient such as viscosity to the relaxation time determined by the fundamental non-hydrodynamic quasinormal frequency changes rapidly in the vicinity of infinite coupling but flattens out for weaker coupling, suggesting an extrapolation from strong coupling to the kinetic theory result. We note that the behavior of the quasinormal spectrum is qualitatively different depending on whether the ratio of shear viscosity to entropy density is greater or less than the universal, infinite coupling value of ℏ/4πk B . In the former case, the density of poles increases, indicating a formation of branch cuts in the weak coupling limit, and the spectral function shows the appearance of narrow peaks. We also discuss the relation of the viscosity-entropy ratio to conjectured bounds on relaxation time in quantum systems.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Teaney, J. Lauret and E.V. Shuryak, Flow at the SPS and RHIC as a quark gluon plasma signature, Phys. Rev. Lett. 86 (2001) 4783 [nucl-th/0011058] [INSPIRE].
U.W. Heinz, Towards the little bang standard model, J. Phys. Conf. Ser. 455 (2013) 012044 [arXiv:1304.3634] [INSPIRE].
M. Luzum and P. Romatschke, Conformal relativistic viscous hydrodynamics: applications to RHIC results at \( \sqrt{s_{N\ N}} = 200 \) GeV, Phys. Rev. C 78 (2008) 034915 [Erratum ibid. C 79 (2009) 039903] [arXiv:0804.4015] [INSPIRE].
M. Luzum and P. Romatschke, Viscous hydrodynamic predictions for nuclear collisions at the LHC, Phys. Rev. Lett. 103 (2009) 262302 [arXiv:0901.4588] [INSPIRE].
B. Schenke, S. Jeon and C. Gale, Elliptic and triangular flow in event-by-event (3 + 1) D viscous hydrodynamics, Phys. Rev. Lett. 106 (2011) 042301 [arXiv:1009.3244] [INSPIRE].
H. Song, S.A. Bass, U. Heinz, T. Hirano and C. Shen, 200 A GeV Au+Au collisions serve a nearly perfect quark-gluon liquid, Phys. Rev. Lett. 106 (2011) 192301 [Erratum ibid. 109 (2012) 139904] [arXiv:1011.2783] [INSPIRE].
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].
A. Buchel, J.T. Liu and A.O. Starinets, Coupling constant dependence of the shear viscosity in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 707 (2005) 56 [hep-th/0406264] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.A. Tseytlin, Coupling constant dependence in the thermodynamics of N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 534 (1998) 202 [hep-th/9805156] [INSPIRE].
J.P. Blaizot, E. Iancu, U. Kraemmer and A. Rebhan, Hard thermal loops and the entropy of supersymmetric Yang-Mills theories, JHEP 06 (2007) 035 [hep-ph/0611393] [INSPIRE].
G. Uhlenbeck and G. Ford, Lectures in statistical mechanics, American Mathematical Society, Providence, U.S.A. (1963).
E. Gross and E. Jackson, Kinetic models and the linearized Boltzmann Equation, Phys. Fluids 2 (1959) 432.
H. Grad, Asymptotic theory of the Boltzmann equation, Phys. Fluids 6 (1963) 0147.
R. Liboff, Kinetic Theory, Springer, Germany (2003).
S. Chapman and T. Cowling, The mathematical theory of non-uniform gases, 3rd edition, Cambridge University Press, Cambridge U.K. (1970).
E. Cohen and W.E. Thirring, The Boltzmann equation, Springer, Germany (1973).
J. Casalderrey-Solana et al., Gauge/string duality, hot QCD and heavy ion collisions, Cambridge University Press, Cambridge U.K. (2014).
M. Ammon and J. Erdmenger, Gauge/gravity duality, Cambridge University Press, Cambridge, U.K. (2015).
H. Nastase, Introduction to the AdS/CFT correspondence, Cambridge University Press, Cambridge, U.K. (2015).
M. Natsuume, AdS/CFT duality user guide, Springer, Germany (2015).
J. Zaanen, Y. Liu, Y.-W. Sun and K. Schalm, Holographic duality in condensed matter physics, Cambridge University Press, Cambridge U.K. (2015).
V.P. Frolov and I.D. Novikov, Black hole physics: basic concepts and new developments, Kluwer Academic, Dordrecht Netherlands (1998).
E. Berti, V. Cardoso and A.O. Starinets, Quasinormal modes of black holes and black branes, Class. Quant. Grav. 26 (2009) 163001 [arXiv:0905.2975] [INSPIRE].
P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma, Phys. Rev. Lett. 102 (2009) 211601 [arXiv:0812.2053] [INSPIRE].
P.M. Chesler and L.G. Yaffe, Boost invariant flow, black hole formation and far-from-equilibrium dynamics in N = 4 supersymmetric Yang-Mills theory, Phys. Rev. D 82 (2010) 026006 [arXiv:0906.4426] [INSPIRE].
P.M. Chesler and L.G. Yaffe, Holography and off-center collisions of localized shock waves, JHEP 10 (2015) 070 [arXiv:1501.04644] [INSPIRE].
P.M. Chesler, N. Kilbertus and W. van der Schee, Universal hydrodynamic flow in holographic planar shock collisions, JHEP 11 (2015) 135 [arXiv:1507.02548] [INSPIRE].
J. Casalderrey-Solana, M.P. Heller, D. Mateos and W. van der Schee, From full stopping to transparency in a holographic model of heavy ion collisions, Phys. Rev. Lett. 111 (2013) 181601 [arXiv:1305.4919] [INSPIRE].
M.P. Heller, R.A. Janik and P. Witaszczyk, The characteristics of thermalization of boost-invariant plasma from holography, Phys. Rev. Lett. 108 (2012) 201602 [arXiv:1103.3452] [INSPIRE].
H. Bantilan and P. Romatschke, Simulation of black hole collisions in asymptotically Anti-de Sitter spacetimes, Phys. Rev. Lett. 114 (2015) 081601 [arXiv:1410.4799] [INSPIRE].
A. Buchel, M.P. Heller and R.C. Myers, Equilibration rates in a strongly coupled nonconformal quark-gluon plasma, Phys. Rev. Lett. 114 (2015) 251601 [arXiv:1503.07114] [INSPIRE].
J. Jankowski, G. Plewa and M. Spalinski, Statistics of thermalization in Bjorken Flow, JHEP 12 (2014) 105 [arXiv:1411.1969] [INSPIRE].
V. Keranen and P. Kleinert, Thermalization of Wightman functions in AdS/CFT and quasinormal modes, arXiv:1511.08187 [INSPIRE].
R.A. Janik, G. Plewa, H. Soltanpanahi and M. Spalinski, Linearized nonequilibrium dynamics in nonconformal plasma, Phys. Rev. D 91 (2015) 126013 [arXiv:1503.07149] [INSPIRE].
R.A. Janik, J. Jankowski and H. Soltanpanahi, Non-equilibrium dynamics and phase transitions, arXiv:1512.06871 [INSPIRE].
M. Attems et al., Thermodynamics, transport and relaxation in non-conformal theories, arXiv:1603.01254 [INSPIRE].
R.A. Janik, J. Jankowski and H. Soltanpanahi, Quasinormal modes and the phase structure of strongly coupled matter, JHEP 06 (2016) 047 [arXiv:1603.05950] [INSPIRE].
U. Gürsoy, A. Jansen and W. van der Schee, A new dynamical instability in Anti-de-Sitter spacetime, arXiv:1603.07724 [INSPIRE].
P. Romatschke, Retarded correlators in kinetic theory: branch cuts, poles and hydrodynamic onset transitions, Eur. Phys. J. C 76 (2016) 352 [arXiv:1512.02641] [INSPIRE].
S.A. Stricker, Holographic thermalization in N = 4 super Yang-Mills theory at finite coupling, Eur. Phys. J. C 74 (2014) 2727 [arXiv:1307.2736] [INSPIRE].
S. Waeber, A. Schäfer, A. Vuorinen and L.G. Yaffe, Finite coupling corrections to holographic predictions for hot QCD, JHEP 11 (2015) 087 [arXiv:1509.02983] [INSPIRE].
X.O. Camanho, J.D. Edelstein, J. Maldacena and A. Zhiboedov, Causality constraints on corrections to the graviton three-point coupling, JHEP 02 (2016) 020 [arXiv:1407.5597] [INSPIRE].
H. Reall, N. Tanahashi and B. Way, Causality and hyperbolicity of lovelock theories, Class. Quant. Grav. 31 (2014) 205005 [arXiv:1406.3379] [INSPIRE].
J. Ferziger and H. Kaper, Mathematical theory of transport processes in gases, North-Holland Publishing Company, Amsterdam Netherlands (1972).
V. Silin, Introduction to kinetic theory of gases (in Russian), 3rd edition, Nauka, Moscow, Russia (1971).
I.A. Kvasnikov, Thermodynamics and statistical physics: a theory of non-equilibrium systems (in Russian), Moscow University Press, Moscow, Russia (1987).
P.L. Bhatnagar, E.P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev. 94 (1954) 511.
S. de Groot, W. van Leeuwen and C. van Weert, Relativistic kinetic theory, North-Holland Publishing Company, Amsterdam Netherlands (1980).
L. Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Springer, Germany (2009).
P.B. Arnold, G.D. Moore and L.G. Yaffe, Effective kinetic theory for high temperature gauge theories, JHEP 01 (2003) 030 [hep-ph/0209353] [INSPIRE].
P.B. Arnold, G.D. Moore and L.G. Yaffe, Transport coefficients in high temperature gauge theories. 2. Beyond leading log, JHEP 05 (2003) 051 [hep-ph/0302165] [INSPIRE].
P. Kovtun, D.T. Son and A.O. Starinets, Holography and hydrodynamics: Diffusion on stretched horizons, JHEP 10 (2003) 064 [hep-th/0309213] [INSPIRE].
S. Sachdev, Quantum phase transitions, 2nd edition, Cambridge University Press, Cambridge U.K. (2011).
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
D. Birmingham, I. Sachs and S.N. Solodukhin, Conformal field theory interpretation of black hole quasinormal modes, Phys. Rev. Lett. 88 (2002) 151301 [hep-th/0112055] [INSPIRE].
S. Kalyana Rama and B. Sathiapalan, On the role of chaos in the AdS/CFT connection, Mod. Phys. Lett. A 14 (1999) 2635 [hep-th/9905219] [INSPIRE].
G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].
U.H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Black hole formation in AdS and thermalization on the boundary, JHEP 02 (2000) 039 [hep-th/9912209] [INSPIRE].
P.K. Kovtun and A.O. Starinets, Quasinormal modes and holography, Phys. Rev. D 72 (2005) 086009 [hep-th/0506184] [INSPIRE].
A.O. Starinets, Quasinormal modes of near extremal black branes, Phys. Rev. D 66 (2002) 124013 [hep-th/0207133] [INSPIRE].
S.A. Hartnoll and S.P. Kumar, AdS black holes and thermal Yang-Mills correlators, JHEP 12 (2005) 036 [hep-th/0508092] [INSPIRE].
A. Núñez and A.O. Starinets, AdS/CFT correspondence, quasinormal modes and thermal correlators in N = 4 SYM, Phys. Rev. D 67 (2003) 124013 [hep-th/0302026] [INSPIRE].
V. Cardoso, J. Natario and R. Schiappa, Asymptotic quasinormal frequencies for black holes in nonasymptotically flat space-times, J. Math. Phys. 45 (2004) 4698 [hep-th/0403132] [INSPIRE].
J. Natario and R. Schiappa, On the classification of asymptotic quasinormal frequencies for d-dimensional black holes and quantum gravity, Adv. Theor. Math. Phys. 8 (2004) 1001 [hep-th/0411267] [INSPIRE].
G. Festuccia and H. Liu, A Bohr-Sommerfeld quantization formula for quasinormal frequencies of AdS black holes, Adv. Sci. Lett. 2 (2009) 221 [arXiv:0811.1033] [INSPIRE].
S. Hod, Universal bound on dynamical relaxation times and black-hole quasinormal ringing, Phys. Rev. D 75 (2007) 064013 [gr-qc/0611004] [INSPIRE].
S. Grozdanov and A. O. Starinets, Second order transport and quasinormal modes in holographic Gauss-Bonnet liquid, to appear (2016).
J. Pawelczyk and S. Theisen, AdS 5 × S 5 black hole metric at O(α ′3), JHEP 09 (1998) 010 [hep-th/9808126] [INSPIRE].
A. Buchel, Resolving disagreement for η/s in a CFT plasma at finite coupling, Nucl. Phys. B 803 (2008) 166 [arXiv:0805.2683] [INSPIRE].
P. Benincasa and A. Buchel, Transport properties of N = 4 supersymmetric Yang-Mills theory at finite coupling, JHEP 01 (2006) 103 [hep-th/0510041] [INSPIRE].
A. Buchel, Shear viscosity of boost invariant plasma at finite coupling, Nucl. Phys. B 802 (2008) 281 [arXiv:0801.4421] [INSPIRE].
A. Buchel and M. Paulos, Relaxation time of a CFT plasma at finite coupling, Nucl. Phys. B 805 (2008) 59 [arXiv:0806.0788] [INSPIRE].
A. Buchel and M. Paulos, Second order hydrodynamics of a CFT plasma from boost invariant expansion, Nucl. Phys. B 810 (2009) 40 [arXiv:0808.1601] [INSPIRE].
O. Saremi and K.A. Sohrabi, Causal three-point functions and nonlinear second-order hydrodynamic coefficients in AdS/CFT, JHEP 11 (2011) 147 [arXiv:1105.4870] [INSPIRE].
S. Grozdanov and A.O. Starinets, On the universal identity in second order hydrodynamics, JHEP 03 (2015) 007 [arXiv:1412.5685] [INSPIRE].
M.T. Grisaru, A.E.M. van de Ven and D. Zanon, Four Loop β-function for the N = 1 and N =2 supersymmetric nonlinear σ-model in two-dimensions, Phys. Lett. B 173(1986) 423 [INSPIRE].
D.J. Gross and E. Witten, Superstring modifications of Einstein’s equations, Nucl. Phys. B 277 (1986) 1 [INSPIRE].
M.B. Green and C. Stahn, D3-branes on the Coulomb branch and instantons, JHEP 09 (2003) 052 [hep-th/0308061] [INSPIRE].
S. de Haro, A. Sinkovics and K. Skenderis, On α ′ corrections to D-brane solutions, Phys. Rev. D 68 (2003) 066001 [hep-th/0302136] [INSPIRE].
M.B. Green, K. Peeters and C. Stahn, Superfield integrals in high dimensions, JHEP 08 (2005) 093 [hep-th/0506161] [INSPIRE].
M.F. Paulos, Higher derivative terms including the Ramond-Ramond five-form, JHEP 10 (2008) 047 [arXiv:0804.0763] [INSPIRE].
R.C. Myers, M.F. Paulos and A. Sinha, Quantum corrections to η/s, Phys. Rev. D 79 (2009) 041901 [arXiv:0806.2156] [INSPIRE].
A. Buchel, R.C. Myers, M.F. Paulos and A. Sinha, Universal holographic hydrodynamics at finite coupling, Phys. Lett. B 669 (2008) 364 [arXiv:0808.1837] [INSPIRE].
G. Policastro, D.T. Son and A.O. Starinets, From AdS/CFT correspondence to hydrodynamics, JHEP 09 (2002) 043 [hep-th/0205052] [INSPIRE].
R. Baier, P. Romatschke, D.T. Son, A.O. Starinets and M.A. Stephanov, Relativistic viscous hydrodynamics, conformal invariance and holography, JHEP 04 (2008) 100 [arXiv:0712.2451] [INSPIRE].
S. Grozdanov and N. Kaplis, Constructing higher-order hydrodynamics: the third order, Phys. Rev. D 93 (2016) 066012 [arXiv:1507.02461] [INSPIRE].
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].
S. Bhattacharyya, V.E. Hubeny, S. Minwalla and M. Rangamani, Nonlinear Fluid Dynamics from Gravity, JHEP 02 (2008) 045 [arXiv:0712.2456] [INSPIRE].
M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, Viscosity Bound Violation in Higher Derivative Gravity, Phys. Rev. D 77 (2008) 126006 [arXiv:0712.0805] [INSPIRE].
M. Brigante, H. Liu, R.C. Myers, S. Shenker and S. Yaida, The viscosity bound and causality violation, Phys. Rev. Lett. 100 (2008) 191601 [arXiv:0802.3318] [INSPIRE].
A. Buchel and R.C. Myers, Causality of holographic hydrodynamics, JHEP 08 (2009) 016 [arXiv:0906.2922] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, AdS 7 /CF T 6 , Gauss-Bonnet gravity and viscosity bound, JHEP 03 (2010) 087 [arXiv:0910.5347] [INSPIRE].
X.O. Camanho and J.D. Edelstein, Causality constraints in AdS/CFT from conformal collider physics and Gauss-Bonnet gravity, JHEP 04 (2010) 007 [arXiv:0911.3160] [INSPIRE].
A. Buchel, J. Escobedo, R.C. Myers, M.F. Paulos, A. Sinha and M. Smolkin, Holographic GB gravity in arbitrary dimensions, JHEP 03 (2010) 111 [arXiv:0911.4257] [INSPIRE].
J. de Boer, M. Kulaxizi and A. Parnachev, Holographic Lovelock gravities and black holes, JHEP 06 (2010) 008 [arXiv:0912.1877] [INSPIRE].
X.O. Camanho and J.D. Edelstein, Causality in AdS/CFT and Lovelock theory, JHEP 06 (2010) 099 [arXiv:0912.1944] [INSPIRE].
X.O. Camanho, J.D. Edelstein and M.F. Paulos, Lovelock theories, holography and the fate of the viscosity bound, JHEP 05 (2011) 127 [arXiv:1010.1682] [INSPIRE].
G. Papallo and H.S. Reall, Graviton time delay and a speed limit for small black holes in Einstein-Gauss-Bonnet theory, JHEP 11 (2015) 109 [arXiv:1508.05303] [INSPIRE].
R.-G. Cai, Gauss-Bonnet black holes in AdS spaces, Phys. Rev. D 65 (2002) 084014 [hep-th/0109133] [INSPIRE].
S. Nojiri and S.D. Odintsov, Anti-de Sitter black hole thermodynamics in higher derivative gravity and new confining deconfining phases in dual CFT, Phys. Lett. B 521 (2001) 87 [Erratum ibid. B 542 (2002) 301] [hep-th/0109122] [INSPIRE].
Y.M. Cho and I.P. Neupane, Anti-de Sitter black holes, thermal phase transition and holography in higher curvature gravity, Phys. Rev. D 66 (2002) 024044 [hep-th/0202140] [INSPIRE].
I.P. Neupane, Black hole entropy in string generated gravity models, Phys. Rev. D 67 (2003) 061501 [hep-th/0212092] [INSPIRE].
I.P. Neupane, Thermodynamic and gravitational instability on hyperbolic spaces, Phys. Rev. D 69 (2004) 084011 [hep-th/0302132] [INSPIRE].
S. Grozdanov and A.O. Starinets, Zero-viscosity limit in a holographic Gauss-Bonnet liquid, Theor. Math. Phys. 182 (2015) 61 [Teor. Mat. Fiz. 182 (2014) 76].
P. Kovtun and A. Starinets, Thermal spectral functions of strongly coupled N = 4 supersymmetric Yang-Mills theory, Phys. Rev. Lett. 96 (2006) 131601 [hep-th/0602059] [INSPIRE].
M.P. Heller, R.A. Janik and P. Witaszczyk, Hydrodynamic gradient expansion in gauge theory plasmas, Phys. Rev. Lett. 110 (2013) 211602 [arXiv:1302.0697] [INSPIRE].
M.P. Heller and M. Spalinski, Hydrodynamics beyond the gradient expansion: resurgence and resummation, Phys. Rev. Lett. 115 (2015) 072501 [arXiv:1503.07514] [INSPIRE].
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].
S. Cremonini, The shear viscosity to entropy ratio: a status report, Mod. Phys. Lett. B 25 (2011) 1867 [arXiv:1108.0677] [INSPIRE].
V.E. Fortov and V. Mintsev, Quantum Bound of the Shear Viscosity of a strongly coupled plasma, Phys. Rev. Lett. 111 (2013) 125004.
V.E. Fortov et al., Viscosity of a strongly coupled dust component in a weakly ionized plasma, Phys. Rev. Lett. 109 (2012) 055002.
U. Hohm, On the ratio of the shear viscosity to the density of entropy of the rare gases and H 2 , N 2 , CH 4 , and CF 4, Chem. Phys. 444 (2014) 39.
E.W. Leaver, Quasinormal modes of Reissner-Nordstrom black holes, Phys. Rev. D 41 (1990) 2986 [INSPIRE].
F. Denef, S.A. Hartnoll and S. Sachdev, Quantum oscillations and black hole ringing, Phys. Rev. D 80 (2009) 126016 [arXiv:0908.1788] [INSPIRE].
M. Edalati, J.I. Jottar and R.G. Leigh, Shear modes, criticality and extremal black holes, JHEP 04 (2010) 075 [arXiv:1001.0779] [INSPIRE].
M. Edalati, J.I. Jottar and R.G. Leigh, Holography and the sound of criticality, JHEP 10 (2010) 058 [arXiv:1005.4075] [INSPIRE].
R.A. Davison and N.K. Kaplis, Bosonic excitations of the AdS 4 Reissner-Nordstrom black hole, JHEP 12 (2011) 037 [arXiv:1111.0660] [INSPIRE].
M. Kaminski, K. Landsteiner, J. Mas, J.P. Shock and J. Tarrio, Holographic operator mixing and quasinormal modes on the brane, JHEP 02 (2010) 021 [arXiv:0911.3610] [INSPIRE].
M. Kaminski, K. Landsteiner, F. Pena-Benitez, J. Erdmenger, C. Greubel and P. Kerner, Quasinormal modes of massive charged flavor branes, JHEP 03 (2010) 117 [arXiv:0911.3544] [INSPIRE].
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
Corresponding author
Additional information
ArXiv ePrint: 1605.02173
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Grozdanov, S., Kaplis, N. & Starinets, A.O. From strong to weak coupling in holographic models of thermalization. J. High Energ. Phys. 2016, 151 (2016). https://doi.org/10.1007/JHEP07(2016)151
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2016)151