Status of the viscosity bound

Transcription

1 Status of the viscosity bound Aninda Sinha Indian Institute of Science, Bangalore & Perimeter Institute, Canada 1

2 Muller, 2010 Accreting PBH 2

3 Preliminaries Wikipedia KSS conjecture η: Shear viscosity s: Entropy density Kinetic theory, quasiparticles >>1 Not true in strongly coupled plasmas 3

4 Preliminaries Molecular theory tells us that the viscosity at high temperatures of a dilute gas increases with temperature At low temperature, a gas condenses to a liquid. As temperature is lowered, it can become a solid. So viscosity increases with decreasing temperature. Thus may expect ratio to be minimum near criticality. Same behaviour expected in QCD. High temperature (asymptotic freedom) is perturbative while low temperature is dominated by hadrons (cross section decreases).csernai, Kapusta, McLerran, 2006 So expect a minimum 4

5 IS ħ??? k B NOTE THAT WE DO NOT KNOW OF ANY FLUID IN NATURE THAT VIOLATES THIS BOUND ALTHOUGH WE WILL FIND THEORETICAL CONSTRUCTS WHERE THIS IS VIOLATED 6

6 Kovtun, Son, Starinets Phys.Rev.Lett. 94 (2005) Source: ADS 7

7 VISCOSITY BOUND EXPERIMENT THEORY RHIC LHC OTHERS STRINGS QCD OTHERS Graphene Cold atoms Fluid/Gravity Lattice pqcd Accretion disk Violation with N f Violation with anisotropy? I will mainly consider QFTs in 3+1 dimensions. 8

8 Quark Gluon Plasma RHIC, LHC, Early universe High temperature High density Early universe, neutron stars 9

9 938 MeV S.Gupta et al : T c =175 MeV. [Paper selected as one of the top 10 breakthroughs in 2011 by Physics World!!] 10

10 at RHIC and LHC φ dn 1 = (1 + 2v1 cosϕ + 2v2 cos 2ϕ +...) dϕ 2π y x z Hydrod initial s to mom y x p y p x 11

11 Heinz et al RHIC: 0.16 LHC: Conservatively < : A 20% uncertainty in eccentricity can lead to a 100% error in the ratio. To remove uncertainties, consider triangular flow v 3. Charged hadron elliptic flow Coupled hydro with realistic model of late hadronic stage Viscous Israel Stewart Hydro used Initial conditions (energy density) model dependent. Heinz et al claim that once model dependent uncertainties in LHC data is removed, it could yield same as RHIC. In other words, expected enhancement due to weaker coupling may be absent. 12

12 VISCOSITY BOUND EXPERIMENT THEORY RHIC LHC OTHERS STRINGS QCD OTHERS Graphene Cold atoms Fluid/Gravity Lattice pqcd λφ 4 Accretion disk Violation with N f Violation with anisotropy? 13

13 Reynold s number increases G R A P H E N E Muller et al Allows opportunity to study properties of ultrarelativistic plasma at moderate temperatures. Massless dispersion relation for quasiparticles. Use Boltzmann transport relation to compute shear viscosity. Reynold s number (QGP Re 1). May exhibit turbulent behaviour at low temperatures. May be relevant for nano-applications. Turbulence may enhance or diminish shear viscosity 14

14 in cold atoms Schafer Li T 10-9 K Dilute fermi gas of Lithium-6 or Potassium-40 Elliptic flow observed Release Fermi gas from optical trap and study evolution of aspect ratio Similar problem as QGP need better understanding of transition from hydro to kinetic theory. May be one can learn from the other. 15

15 VISCOSITY BOUND EXPERIMENT THEORY RHIC LHC OTHERS STRINGS QCD OTHERS Graphene Cold atoms Fluid/Gravity Lattice pqcd λφ 4 Accretion disk Violation with N f Violation with anisotropy? 16

16 in black hole accretion B. Mukhopadhyay, AS: Black holes accrete matter, give off radiation. We can observe this radiation. X-ray spectrum can be studied. Eg. Supermassive black hole Sgr A* ( 10 6 x Mass of Sun). Spiralling matter falling into black hole is modeled as a viscous fluid. Shear viscosity is key for the matter to lose angular momentum. Solve relativistic hydro in Kerr background. Assume radiatively inefficient flow (ADAF R.Narayan et al). We used lattice QCD eos (low density leads to low chemical potential). Temperature close to black hole is as big as K. Model viscosity using famous Shakura-Sunyaev (>5000cits) prescription: α : typically to explain spectrum Origin is said to be due to turbulence 17

17 Ratio is independent of rate of infalling matter, black hole spin (artifact of ADAF). Similar result even if we use ideal gas eos that is used usually. For astrophysical black holes this is HUGE. I do not think existing accretion disk models do proper justice to subleading terms in the hydro expansion. A naïve estimate leads to viscous term being 25% leading term. When is this small? For primordial black holes. Curiously for surviving primordial black holes Open question to explore other scenarios So that where radiation effects are considered. How justified is hydro in these models? Important since they seem to explain observation. 18

18 May be can ask what fluid (experimentally!) has the LARGEST ratio! 19

19 VISCOSITY BOUND EXPERIMENT THEORY RHIC LHC OTHERS STRINGS QCD OTHERS Graphene Cold atoms Fluid/Gravity Lattice pqcd λφ 4 Accretion disk Violation with N f Violation with anisotropy? 20

20 from lattice QCD 2+1 flavour Close to RHIC/LHC fits Borsanyi et al Meyer State of the art BUT Results only for SU(3) pure glue Analytic continuation of Euclidean correlators A very hard problem with little progress over the last 2-3 years 21

21 η and η/s for hadron gas in equlibrium Viscosity increases with Temperature. Viscosity decreases with finite baryon number density. Demir, Bass 2007

22 from QCD Greiner et al Trace anomaly Including Hagedorn states Excluding Hagedorn states, Hadron Resonance Gas. M<2GeV. Trace anomaly matches better with lattice KEY FEATURE: Including Hagedorn states LOWERS the ratio even in HADRONIC phase. CAVEAT: Linear extrapolation was used at high temperatures. 23

23 VISCOSITY BOUND EXPERIMENT THEORY RHIC LHC OTHERS STRINGS QCD OTHERS Graphene Cold atoms Fluid/Gravity Lattice pqcd Accretion disk Violation with N f Violation with anisotropy? 24

24 Strongly coupled QGP seems to be conformal, just above T c RHIC 1 scale energy density by free result ε/ε 0 T/T c

25 AdS/CFT does not give identical physics to QCD, but may be useful to study strongly coupled QGP RHIC 1 scale energy density by free result 0.75 ε/ε 0 N=4 SYM T/T c

26 Shear viscosity: defined by Kubo formula AdS/CFT evaluates as correlator of gravitons: [Policastro, Son & Starinets; Son & Starinets; Herzog & Son] Einstein gravity horizon entropy: result: universal result for all known theories with Einstein gravity dual: (in units of ) [Kovtun, Son & Starinets; Buchel & Liu; Benincasa, Buchel &Naryshkin; Iqbal & Liu;... ] Isotropy assumed. Wait for later! 27

27 Long wavelength limit of linear response of boundary theory fluid is completely captured by horizon fluid. Connection with membrane paradigm. [Damour; Starinets; KSS; Iqbal, Liu; Brustein, Medved;Strominger et al.] Certain transport coefficients could be evaluated by computing geometric quantities at the black hole horizon. Away from the long wavelength (zero frequency) limit, the full geometry of spacetime would play a role. In linear response we have for an operator O and source φ 0 Transport coefficient 28

28 In AdS/CFT our starting point is a metric of the form: ISOTROPY Consider a minimally coupled scalar Plug solution into action. Surface term Throw away horizon contr. Effective coupling We will soon use regularity at horizon to replace r-derivative with a t-derivative NOTE: Here everything is at boundary at infinity. 29

29 Using the equations of motion we find trivial RG flow This triviality only works in the low frequency limit This means that we can calculate at horizon rather than at boundary! If we demand regularity of field at horizon, we get from the use of Eddington-Finkelstein coordinates 30

30 This immediately leads to the beautifully simple result For shear viscosity we turn on a metric perturbation h xy. In Einstein gravity this works out to be a minimally coupled scalar with effective coupling 16πG N. Immediately get the universal result At finite coupling, this story generalizes. The effective coupling now changes which corrects the ratio. 31

31 Finite coupling Kubo Formula Effective action For fun set α s =0.5, get ratio 0.11 t Hooft coupling Fixed by conformal anomalies. Needs additional matter. 1/N c 32

32 Sadly, after so many years of string theory, the leading order higher derivative corrections in IIB are not completely known. Thankfully, not all of the terms are needed to compute the leading correction to the ratio. Typical argument relies on the fact that most fields in the classical solution are set to zero or constant. e.g., The only fields turned on are the metric, 5-form flux and a constant dilaton. So a field with quadratic kinetic terms that is sourced by the classical correction at order ε will only contribute at order ε 2. Need to worry about the 5-form flux though. So the question is are all the terms involving the 5-form flux and metric known? 33

33 Green, Stahn propose the following to contain all flux and metric terms at α 3 in IIB ~g R+ DF+F 2 [Myers, Paulos, AS, 2008] Somewhat surprisingly the above flux terms do not change the existing result. Generic flux terms would have! [Buchel, Liu, Starinets, 2004; Buchel, 2008; Myers, Paulos, AS, 2008] TRUE FOR ALL THEORIES WITH UNIVERSAL GRAVITY DUAL!! AND ISOTROPIC PLASMA >0, is this true always? NO!!. [Kats & Petrov; Buchel, Myers, AS 2008] 34

34 Superconformal gauge theories with marginal gauge coupling [Buchel, Myers, AS]: for example: scft s with SU(N c ) gauge group matter fields N=4 SYM * Hoyos,Karch shear viscosity: Open problem: [Buchel, Find a model Myers with,as] δ 11/Nc ~ but <0 These theories violate KSS bound!! *: String construction known 35

35 Known String constructions: N=2 Superconformal fixed points from F- theory [Aharony, Tachikawa] N c D3-branes probing F-theory singularities generated by n 7 (p,q) 7-branes Considered by Kats and Petrov. Violates bound if >0 so bound is violated For susy theories, c-a=1/96(n s -6N v ). I do not know of any reason why N s >6 N v. 36

36 Adding fundamental matter seems to LOWER the ratio. In the hadron gas calculation of Demir and Bass, increasing chemical potential LOWERED the ratio. Seems to be a prediction for lattice, if the techniques ever allow us to compute at finite chemical potential. 37

37 How low can you get? Brigante et al considered Gauss-Bonnet gravity. Speed of graviton on constant r hypersurface can exceed c if λ is too big Same constraint demanding positive energy flux! Hofman, Maldacena 38

38 Another toy model: Quasi-topological gravity [Myers, Robinson; Myers, Paulos, AS] No causality violation Constrain λ,μ demanding positive energy fluxes, c>0. Cannot get arbitrarily low values CAVEAT: Buchel and Cremonini find lower values in presence of phase transition. However stability 39 may be an issue.

39 Until recently, the only known violations of the bound needed higher order corrections in the gravity side. A purist would say the violation is tiny in order to trust the gravity description. Anisotropy seems to change all this. Claims have appeared that 1. Anistropy introduces additional viscous parameters which do not obey universality. 2. Even at 2-derivative level, the bound is violated for these additional parameters. WHY CARE? Because QGP is generally anisotropic. 40

40 Anisotropy Erdmenger et al, Consider transversely isotropic (yz direction) fluid. p-wave superfluid. SU(2) gauge field. Still universal Controls anisotropy Breaks SO(3) to SO(2). Introduces current density. Metric ansatz: Asymptotically AdS. At horizon f(r) >1. That is why no violation. Wait for later. 41

41 Solve system numerically. Use Kubo formula to compute shear viscosity. Find non-universality. CAVEAT. This model describes a physical system that is intrinsically anistropic. QCD is NOT such a system. Rather to model an anistropic plasma, the anistropy should be in the initial conditions. Non-universal ratios in this model are still ABOVE bound! 42

42 Bound violation using anistropy Rebhan, Steineder The results were published in Physical Review Letters and highlighted as an "editors' selection. Disclaimer: I was not the referee! Mateos, Trancanelli Model describes static plasma in presence of anisotropic external source D7 branes (a) dissolved in geometry. Nonsingular metric. RRscalar Dilaton Conformal anomaly RG flow between AdS and Lifshitz. 43

43 D7 clumps (nucleation) UNSTABLE STABLE a: anistropy Does it asymptote? Complicated phase diagram due to anisotropy Anisotropic shear viscosity can violate the bound. ~e -φ >1 44

44 Although QCD is not anisotropic, in heavy ion collisions, there are claims that metastable phases with non-zero Θ can exist. So this model does seem like a good starting point to investigate holographic duals. Rebhan-Steineder result is interesting and should be thoroughly examined. Are there hidden instabilities in the system when the bound is violated? Thermodynamically the claim is NO. But sound waves have not been studied in any detail. May be bound violation is accompanied by instability. If correct the Rebhan-Steineder result points at the fact that the viscosity bound (if one exists!) only applies to isotropic plasmas. 45

45 Open problems in fluid/gravity Beautiful connection between GR and fluid mechanics [Bhattacharrya, Hubeny, Minwalla, Rangamani ]. One key equation Gibbons- Hawking Counterterm Can this be extended to include classical higher derivative corrections? Generalized Gibbons-Hawking seem really hard except for Lovelock theories, generalized counterterms seem more tractable. Can one take into account the effect of fluctuations in fluid mechanics? Cf. Kovtun et al Formulate the correspondence in an anisotropic set-up. 46

46 Summary Search for perfect fluid still on. LHC data still being analysed. Ratio from LHC/RHIC are somewhere between Uncertainty in initial conditions still exist although people have come up with better models to reduce them. Cold atoms seem to give results that are at least twice as big. Black hole accretion disk gave a HUGE ratio for astrophysical black holes. Makes hydro models used to computing spectrum questionable. 47

47 Lattice methods have made little progress. Understanding analytic continuation seems to be an issue. QCD methods have led to some interesting results. Increasing chemical potential seems to lower the ratio. AdS/CFT: Fundamental matter leads to 1/N violation of the bound. Anistropy seems to ruin universality. A model has appeared where even in a 2-derivative theory anisotropic shear viscosity violates the bound. 48

48 Acknowledgments Rob Myers, Alex Buchel, Miguel Paulos, Banibrata Mukhopadhyay for collaboration Pavel Kovtun, David Mateos, Hansjoerg Zeller for correspondence Perimeter Institute and Hebrew University for hospitality. 49