Why is theQuark-Gluon Plasma a “Perfect” Liquid ? Berndt Mueller – YITP / Duke RIKEN Workshop 8-9 July 2006 Special thanks to M. Asakawa and S.A. Bass
Lecture I • What does Lattice QCD tell us about the QGP ? • What do RHIC experiments tell us about the QGP ? • What is a “perfect” fluid ? • What are the origins of viscosity ?
What does Lattice QCD tell us about the QGP ? A: So far, a lot about thermodynamic properties and response to static probes, a little bit about spectral functions, (almost) nothing about transport properties.
RHIC Lattice - EOS F. Karsch et al. Indication of weak or strong coupling?
Phase coexistence Study of LQCD at fixed baryon density r = B/V De Forcrand & Kratochvila hep-lat/0602024
pQGP R. Gavai & S. Gupta, hep-lat/0510044 Lattice - susceptibilities
Heavy quark potentials Important for insight into medium effects on J/Y, Effective coupling as(r,T) Kaczmarek et al. Color singlet potential Kaczmarek et al.
Asakawa & Hatsuda Lattice – spectral functions Spectral functions via analytic continuation using the maximum entropy method: J/Y, etc. (At present only for quenched QCD.) Karsch et al. J/Y may survive up to 1.5 – 2 Tc !
What do RHIC experiments tell us about the QGP ? A: So far, a lot about transport properties, a little bit about thermodynamic properties, and (almost) nothing about spectral functions and response to static probes.
STAR The real road to the QGP …is the Relativistic Heavy Ion Collider
Bjorken formula teq Pre-equil. phase Liberation of saturated low-x glue fields (CGC) Space-time picture
RHIC results Important results from RHIC: • Chemical (flavor) and thermal equilibration • Jet quenching = parton energy loss, high opacity • Elliptic flow = early thermalization, low viscosity • Collective flow pattern related to valence quarks • Strong energy loss of c and b quarks • Charmonium suppression not significantly increased compared with lower (CERN) energies • Photons unaffected by medium at high pT, medium emission at low pT in agreement with models
Yield in A+A Area density of p+p coll’s in A+A Cross section in p+p coll’s p0 vs. g in Au+Au (vs. p+p) No suppression for photons Suppression of hadrons Without nuclear effects: RAA = 1.
Photons from the medium Experiment uses internal conversion of g D’Enterria, Peressounko nucl-th/0503054 t0 = 0.15 fm/c, T = 570 MeV Turbide, Rapp, Gale PRC 69 014903 (2004) t0 = 0.33 fm/c, T = 370 MeV Hard Probes 2006, June 15, 2006 – G. David, BNL
q L q q q g Scattering power of the QCD medium: Radiative energy loss Radiative energy loss: Scattering centers = color charges Density of scattering centers Range of color force
RHIC data sQGP? Eskola et al. ? RHIC QGP “Baier plot” Pion gas Cold nuclear matter q-hat at RHIC
z Reaction plane y x Collision Geometry: Elliptic Flow • Bulk evolution described by relativistic fluid dynamics, • F.D assumes that the medium is in local thermal equilibrium, • but no details of how equilibrium was reached. • Input: e(x,ti), P(e), (h,etc.). • Elliptic flow (v2): • Gradients of almond-shape surface will lead to preferential expansion in the reaction plane • Anisotropy of emission is quantified by 2nd Fourier coefficient of angular distribution: v2 • prediction of fluid dynamics
Elliptic flow is created early momentum anisotropy spatial eccentricity time evolution of the energy density: initial energy density distribution: P. Kolb, J. Sollfrank and U.Heinz, PRC 62 (2000) 054909 Model calculations suggest that flow anisotropies are generated at the earliest stages of the expansion, on a timescale of ~ 5 fm/c if a QGP EoS is assumed.
Failure of ideal hydrodynamics - tells us how hadrons form Mass splitting characteristic property of hydrodynamics v2(pT) vs. hydrodynamics
T,m,v Quark number scaling of v2 In the recombination regime, meson and baryon v2 can be obtained from the quark v2 : Emitting medium is composed of unconfined, flowing quarks.
Full 3-d Hydrodynamics • QGP evolution UrQMD Hadronization Cooper-Frye formula hadronic rescattering Monte Carlo TC TSW t fm/c From QGP to hadrons Nonaka & Bass, nucl-th/0510038 Hirano et al. nucl-th/0511046 Low (no) viscosity High viscosity Agreement with data for hQGP = 0
Boost invariant hydro with T0t0 ~ 1 requires h/s ~ 0.1. N=4 SUSY YM theory (g2Nc 1): h/s = 1/4p (Policastro, Son, Starinets). Absolute lower bound on h/s ? h/s = 1/4p implies lf ≈ 0.3 d ! Bounds onh from v2 Relativistic viscous hydrodynamics: D. Teaney QGP(T≈Tc) = sQGP ?
Ultra-cold Fermi-Gas • Li-atoms released from an optical trap (J. Thomas et al./Duke) exhibit elliptic flow analogous to that observed in relativistic heavy-ion collisions
What is an “ideal” or “perfect” liquid ?
Ideal gas vs. perfect liquid • An ideal gas is one that has strong enough interactions to reach thermal equilibrium (on a reasonable time scale), but weak enough interactions so that their effect on P(n,T) can be neglected. • This ideal can be approached arbitraily by diluting the gas and waiting very patiently (limit t first, then V ). • A perfect fluid is one that obeys the Euler equations, i.e. a fluid that has zero viscosity and infinite thermal conductivity. • There is no presumption with regard to the equation of state. • Even an imperfect fluid obeys the Euler equations in the limit of negligible velocity, density, and temperature gradients.
Ichimaru Interaction measure Viscosity of plasmas Shear viscosity of supercooled one-component plasma fluids:
Lower bound on h/s ? Argument [Kovtun, Son & Starinets, PRL 94 (2005) 111601] based on duality between thermal QFT and string theory on curved background with D-dimensional black-brane metric, e.g.: Kubo formula for shear viscosity: Dominated by absorption of (thermal) gravitons by the black hole:
Lower bound on h/s – ctd. A heuristic argument for (h/s)min is obtained using s ~ 4n : But the uncertainty relation dictates that tf (e/n) , and thus: (It is unclear whether this relation holds in the nonrelativistic domain, where s/n can be much larger than 4. But is obeyed by all known substances.) For N=4 susy SU(Nc) Yang-Mills the bound is saturated at strong coupling:
Exploring strong coupling • Ability to perform analytical strong coupling calculations in N=4 susy SU(Nc) YM and success with h have motivated other applications: • Equation of state [Gubser, Klebanov, Tseytlin, hep-th/9805156] • Spectral densities [Teaney, hep-ph/0602044] • Jet quenching parameter [Liu, Rajagopal, Wiedemann, hep-ph/0605178] • Heavy quark energy loss [Herzog, Karch, Kovtun, Kozcaz, Yaffe,hep-th/0605158] • Heavy quark diffusion [Casalderrey-Solana, Teaney, hep-ph/0605199] • Drag force on heavy quark [Gubser, hep-th/0605182 ] • …and continuing!
Some results from duality (3+1)-D world r0 horizon
Lecture II • Does “perfect” fluidity imply “strong coupling” ? • What is “anomalous” viscosity ? • Derivation of the anomalous viscosity • Manifestations of anomalous QGP transport processes
Today… …we ask the question: Is strong coupling really necessary for small h/s ?
Stellar accretion disks “A complete theory of accretion disks requires a knowledge of the viscosity. Unfortunately, viscous transport processes are not well understood. Molecular viscosity is so small that disk evolution due to this mechanism of angular momentum transport would be far too slow to be of interest. If the only source of viscosity was molecular, then n ~ h/r ~ l vT, where l is the particle mean free path and vT is the thermal velocity. Values appropriate for a disk around a newly formed star might be r ~ 1014 cm, n ~ 1015 cm-3, s ~ 10-16 cm2, so that l ~ 10 cm, and vT ~ 105 cm/s . The viscous accretion time scale would then be r2/(12n) > 1013 yr! Longer by a factor of 105 - 106 than the age conventionally ascribed to such disks. Clearly if viscous accretion explains such objects, there must be an anomalous source of viscosity. The same conclusion holds for all the other astronomical objects for which the action of accretion disks have been invoked.” (From James Graham – Astronomy 202, UC Berkeley) http://grus.berkeley.edu/~jrg/ay202/lectures.html The solution is: String theory? Unfortunately, NO.
B Anomalous viscosity Differentially rotating disc with weak magnetic field B shows an instability (Chandrasekhar) Spontaneous angular momentum transfer from inner mass to outer mass is amplified by interaction with the rotating disk and leads to instability (Balbus & Hawley – 1991). “Anomalous”, i.e. non-collisional viscosity
Anomalous viscosity: A ubiquitous phenomenon
Anomalous viscosity on the WWW Google search: Results 1 - 10 of about 322,000 for anomalousviscosity. (0.22 seconds) Chaotic Dynamics, abstract chao-dyn/9509002 Anomalous Viscosity, Resistivity, and Thermal Diffusivity of the Solar Wind Plasma Authors: Mahendra K. Verma (IIT Kanpur, India) In this paper we have estimated typical anomalous viscosity, resistivity, and thermal difffusivity of the solar wind plasma. Since the solar wind is collsionless plasma, we have assumed that the dissipation in the solar wind occurs at proton gyro radius through wave-particle interactions. Using this dissipation length-scale and the dissipation rates calculated using MHD turbulence phenomenology [Verma et al., 1995a], we estimate the viscosity and proton thermal diffusivity. The resistivity and electron's thermal diffusivity have also been estimated. We find that all our transport quantities are several orders of magnitude higher than those calculated earlier using classical transport theories of Braginskii. In this paper we have also estimated the eddy turbulent viscosity.
Anomalous viscosity - usage • Plasma physics: • A.V. = large viscosity induced in nearly collisionless plasmas by long-range fields generated by plasma instabilities. • Astrophysics - dynamics of accretion disks: • A.V. = large viscosity induced in weakly magnetized, ionized stellar accretion disks by orbital instabilities. • Biophysics: • A.V. = The viscousbehaviour of nonhomogenous fluids or suspensions, e.g., blood, in which the apparent viscosityincreases as flow or shear ratedecreases toward zero. (From: http://www.biology-online.org/dictionary)
Can the QGP viscosity be anomalous? • Can the extreme opaqueness of the QGP(seen in experiments) be explained without invoking super-strong coupling ? • Answer may lie in the peculiar properties of turbulent plasmas. • Plasma “turbulence” = random, nonthermal excitation of coherent field modes with power spectrum similar to the vorticity spectrum in a turbulent fluid [P(k) ~ 1/k2]. • Plasma turbulence arises naturally in plasmas with an anisotropic momentum distribution (Weibel-type instabilities). • Expanding plasmas (such as the QGP at RHIC) always have anisotropic momentum distributions. • Soft color fields generate anomaloustransport coefficients, which may give the medium the character of a nearly perfect fluid even at moderately weak coupling.
QGP viscosity – collisions Baym, Heiselberg, …. Danielewicz & Gyulassy, Phys. Rev. D31, 53 (85)
py px beam pz Color instabilities Spontaneous generation of color fields requires infrared instabilities. Unstable modes in plasmas occur generally when the momentum distribution of a plasma is anisotropic (Weibel instabilities – 1959). Such conditions are satisfied in HI collisions: Longitudinal expansion locally “red-shifts” the longitudinal momentum components of released small-x gluon fields (CGC) from initial state: In EM case, instabilities saturate due to effect on charged particles. In YM case, field nonlinearities lead to saturation (competition with Nielsen-Olesen instability?)
r r v v Weibel (two-stream) instability
HTL formalism “soft” “hard” Nonabelian Vlasov equations describe interaction of “hard” and “soft” color field modes and generate the “hard-thermal loop” effective theory: k ~ gT (gQs) k ~ T (Qs) Effective HTL theory permits systematic study of instabilities of “soft” color fields.
x=10 q=p/8 Mrowczynski, Strickland et al., Arnold et al. HTL instabilities Wavelength and growth rate of unstable modes can be calculated perturbatively:
Non-abelian growth Exponential growth saturates when B2 > g2 T4. Quasi-abelian growth Turbulent power spectrum From instability to “turbulence” Kolmogorov-type power spectrum of coherent field excitations [Arnold, Moore, Yaffe, hep-ph/0509226]
Color correlation length Time Non-abelian Quasi-abelian Noise Length (z) Space-time picture M. Strickland, hep-ph/0511212
Anomalous viscosity Formal derivation (Sorry – using Chapman-Enskog)