Transport properties of strongly coupled gauge theories from ...
Transport in hot and dense nuclear matter: a string theory perspective Andrei Starinets Rudolf Peierls Centre for Theoretical Physics Oxford University XII Mexican Workshop on Particles and Fields Mazatlan, Mexico 9 November 2009 Over the last several years, holographic (gauge/gravity duality) methods
were used to study strongly coupled gauge theories at finite temperature and density These studies were motivated by the heavy-ion collision programs at RHIC and LHC (ALICE, ATLAS) and the necessity to understand hot and dense nuclear matter in the regime of intermediate coupling As a result, we now have a better understanding of thermodynamics and especially kinetics (transport) of strongly coupled gauge theories Of course, these calculations are done for theoretical models such as N=4 SYM and its cousins (including non-conformal theories etc). We dont know quantities such as
for QCD Heavy ion collision experiments at RHIC (2000-current) and LHC (2009-??) create hot and dense nuclear matter known as the quark-gluon plasma (note: qualitative difference between p-p and Au-Au collisions) Elliptic flow, jet quenching - focus on transport in this talk Evolution of the plasma fireball is described by relativistic fluid dynamics (relativistic Navier-Stokes equations) Need to know thermodynamics (equation of state) kinetics (first- and second-order transport coefficients)
in the regime of intermediate coupling strength: initial conditions (initial energy density profile) thermalization time (start of hydro evolution) freeze-out conditions (end of hydro evolution) Energy density vs temperature for various gauge theories Ideal gas of quarks and gluons Ideal gas of hadrons
Figure: an artistic impression from Myers and Vazquez, 0804.2423 [hep-th] Quantum field theories at finite temperature/density Equilibrium Near-equilibrium entropy equation of state .
transport coefficients emission rates perturbative non-perturbative Lattice pQCD perturbative non-perturbative ???? kinetic theory
Our understanding of gauge theories is limited Perturbation theory Lattice Conjecture: specific gauge theory in 4 dim = specific string theory in 10 dim Perturbation
theory s l y a r u o D e
th g n i tr Lattice In practice: gravity (low energy limit of string theory) in 10 dim = 4-dim gauge theory in a region of a parameter space Dual gravity Perturbation
theory Lattice Can add fundamental fermions with Hydrodynamics: fundamental d.o.f. = densities of conserved charges Need to add constitutive relations! Example: charge diffusion Conservation law Constitutive relation [Ficks law (1855)]
* Expect Einstein relations such as to hold Second-order transport (kinetic) coefficients (for theories conformal at T=0) Relaxation time Second order trasport coefficient Second order trasport coefficient Second order trasport coefficient
Second order trasport coefficient In non-conformal theories such as QCD, the total number of second-order transport coefficients is quite large What is viscosity? Friction in Newtons equation: Friction in Eulers equations Viscosity of gases and liquids Gases (Maxwell, 1867): Viscosity of a gas is independent of pressure
scales as square of temperature inversely proportional to cross-section Liquids (Frenkel, 1926): W is the activation energy In practice, A and W are chosen to fit data 10-dim gravity M,J,Q 4-dim gauge theory large N, strong coupling
Holographically dual system in thermal equilibrium M, J, Q T Gravitational fluctuations S Deviations from equilibrium ????
and B.C. Quasinormal spectrum From brane dynamics to AdS/CFT correspondence Open strings picture: dynamics of coincident D3 branes at low energy is described by Closed strings picture: dynamics of
coincident D3 branes at low energy is described by conjectured exact equivalence Maldacena (1997); Gubser, Klebanov, Polyakov (1998); Witten (1998) supersymmetric YM theory Gliozzi,Scherk,Olive77 Brink,Schwarz,Scherk77
Field content: Action: (super)conformal field theory = coupling doesnt run AdS/CFT correspondence conjectured exact equivalence Generating functional for correlation
functions of gauge-invariant operators Latest test: Janik08 String partition function In particular Classical gravity action serves as a generating functional for the gauge theory correlators Holography at finite temperature and density
Nonzero expectation values of energy and charge density translate into nontrivial background values of the metric (above extremality)=horizon and electric potential = CHARGED BLACK HOLE (with flat horizon) temperature of the dual gauge theory chemical potential of the dual theory Computing transport coefficients from first principles Fluctuation-dissipation theory (Callen, Welton, Green, Kubo)
Kubo formulae allows one to calculate transport coefficients from microscopic models In the regime described by a gravity dual the correlator can be computed using the gauge theory/gravity duality Computing transport coefficients from dual gravity Assuming validity of the gauge/gravity duality, all transport coefficients are completely determined by the lowest frequencies in quasinormal spectra of the dual gravitational background
(D.Son, A.S., hep-th/0205051, P.Kovtun, A.S., hep-th/0506184) This determines kinetics in the regime of a thermal theory where the dual gravity description is applicable Transport coefficients and quasiparticle spectra can also be obtained from thermal spectral functions Example: stress-energy tensor correlator in in the limit Zero temperature, Euclid: Finite temperature, Mink:
(in the limit ) The pole (or the lowest quasinormal freq.) Compare with hydro: In CFT: Also, (Gubser, Klebanov, Peet, 1996)
First-order transport coefficients in N = 4 SYM in the limit Shear viscosity Bulk viscosity for non-conformal theories see Buchel et al; G.D.Moore et al Gubser et al. Charge diffusion constant Supercharge diffusion constant
Thermal conductivity Electrical conductivity (G.Policastro, 2008) Shear viscosity in SYM perturbative thermal gauge theory S.Huot,S.Jeon,G.Moore, hep-ph/0608062
Strong coupling: * Charge susceptibility can be computed independently: D.T.Son, A.S., hep-th/0601157 Einstein relation holds: Is there a viscosity bound ? Minimum of
in units of P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231 A hand-waving argument Thus Gravity duals fix the coefficient: Too nave? ?
Shear viscosity - (volume) entropy density ratio from gauge-string duality In ALL theories (in the limit where dual gravity valid) : In particular, in N=4 SYM: Other higher-derivative gravity actions Y.Kats and P.Petrov: 0712.0743 [hep-th] M.Brigante, H.Liu, R.C.Myers, S.Shenker and S.Yaida: 0802.3318 [hep-th], 0712.0805 [hep-th]. R.Myers,M.Paulos, A.Sinha: 0903.2834 [hep-th] (and ref. therein many other papers) for superconformal Sp(N)
gauge theory in d=4 Also: The species problem: T.Cohen, hep-th/0702136; A. Dolbado, F.Llanes-Estrada: hep-th/0703132 Shear viscosity - (volume) entropy density ratio in QCD The value of this ratio strongly affects the elliptic flow in hydro models of QGP Chernai, Kapusta, McLerran, nucl-th/0604032 Viscosity-entropy ratio of a trapped Fermi gas T.Schafer, cond-mat/0701251
(based on experimental results by Duke U. group, J.E.Thomas et al., 2005-06) Viscosity measurements at RHIC Viscosity is ONE of the parameters used in the hydro models describing the azimuthal anisotropy of particle distribution -elliptic flow for particle species i Elliptic flow reproduced for e.g. Baier, Romatschke, nucl-th/0610108 Perturbative QCD:
Chernai, Kapusta, McLerran, nucl-th/0604032 SYM: Elliptic flow with color glass condensate initial conditions Luzum and Romatschke, 0804.4015 [nuc-th] Elliptic flow with Glauber initial conditions Luzum and Romatschke, 0804.4015 [nuc-th]
Viscosity/entropy ratio in QCD: current status Theories with gravity duals in the regime where the dual gravity description is valid [Kovtun, Son & A.S] [Buchel] [Buchel & Liu, A.S] QCD: RHIC elliptic flow analysis suggests QCD: (Indirect) LQCD simulations H.Meyer, 0805.4567 [hep-th] Trapped strongly correlated cold alkali atoms T.Schafer, 0808.0734 [nucl-th]
Liquid Helium-3 (universal limit) Holography beyond the near-equilibrium regime Other avenues of (related) research Bulk viscosity for non-conformal theories (Buchel, Benincasa, Gubser, Moore)
Non-relativistic gravity duals (Son, McGreevy, ) Gravity duals of theories with SSB, AdS/CMT (Kovtun, Herzog, Hartnoll, Horowitz) Bulk from the boundary, time evolution of QGP (Janik,) Navier-Stokes equations and their generalization from gravity (Minwalla,) Quarks moving through plasma (Chesler, Yaffe, Gubser,)
New directions S. Hartnoll Lectures on holographic methods for condensed matter physics, 0903.3246 [hep-th] C. Herzog Lectures on holographic superfluidity and superconductivity, 0904.1975 [hep-th] M. Rangamani Gravity and hydrodynamics: Lectures on the fluid-gravity correspondence, 0905.4352 [hep-th]
Hydrodynamic properties of strongly interacting hot plasmas in 4 dimensions can be related (for certain models!) to fluctuations and dynamics of 5-dimensional black holes THANK YOU AdS/CFT correspondence: the role of J For a given operator
, identify the source field , e.g. satisfies linearized supergravity e.o.m. with b.c. The recipe: To compute correlators of , one needs to solve the bulk supergravity e.o.m. for and compute the on-shell action as a functional of the b.c. Warning: e.o.m. for different bulk fields may be coupled: need self-consistent solution Then, taking functional derivatives of
gives Computing real-time correlation functions from gravity To extract transport coefficients and spectral functions from dual gravity, we need a recipe for computing Minkowski space correlators in AdS/CFT The recipe of [D.T.Son & A.S., 2001] and [C.Herzog & D.T.Son, 2002] relates real-time correlators in field theory to Penrose diagram of black hole in dual gravity Quasinormal spectrum of dual gravity = poles of the retarded correlators in 4d theory [D.T.Son & A.S., 2001] Example: R-current correlator in
in the limit Zero temperature: Finite temperature: Poles of = quasinormal spectrum of dual gravity background (D.Son, A.S., hep-th/0205051, P.Kovtun, A.S., hep-th/0506184) Analytic structure of the correlators
Strong coupling: A.S., hep-th/0207133 Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092 Spectral function and quasiparticles A B C A: scalar channel B: scalar channel - thermal part C: sound channel
Is the bound dead? Y.Kats and P.Petrov, 0712.0743 [hep-th] Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory superconformal Sp(N) gauge theory in d=4 M.~Brigante, H.~Liu, R.~C.~Myers, S.~Shenker and S.~Yaida, ``The Viscosity Bound and Causality Violation,'' 0802.3318 [hep-th], ``Viscosity Bound Violation in Higher Derivative Gravity,'' 0712.0805 [hep-th]. The species problem T.Cohen, hep-th/0702136, A.Dobado, F.Llanes-Estrada, hep-th/0703132
Universality of Theorem: For a thermal gauge theory, the ratio of shear viscosity to entropy density is equal to in the regime described by a dual gravity theory Remarks: Extended to non-zero chemical potential: Benincasa, Buchel, Naryshkin, hep-th/0610145 Extended to models with fundamental fermions in the limit
Mateos, Myers, Thomson, hep-th/0610184 String/Gravity dual to QCD is currently unknown Universality of shear viscosity in the regime described by gravity duals Gravitons component obeys equation for a minimally coupled massless scalar. But then . we get
Since the entropy (density) is Three roads to universality of The absorption argument D. Son, P. Kovtun, A.S., hep-th/0405231 Direct computation of the correlator in Kubo formula from AdS/CFT A.Buchel, hep-th/0408095 Membrane paradigm general formula for diffusion coefficient + interpretation as
lowest quasinormal frequency = pole of the shear mode correlator + Buchel-Liu theorem P. Kovtun, D.Son, A.S., hep-th/0309213, A.S., 0806.3797 [hep-th], P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175 Chernai, Kapusta, McLerran, nucl-th/0604032 QCD Chernai, Kapusta, McLerran, nucl-th/0604032 Shear viscosity at non-zero chemical potential
Reissner-Nordstrom-AdS black hole with three R charges (see e.g. Yaffe, Yamada, hep-th/0602074) We still have (Behrnd, Cvetic, Sabra, 1998) J.Mas D.Son, A.S. O.Saremi K.Maeda, M.Natsuume, T.Okamura
Photon and dilepton emission from supersymmetric Yang-Mills plasma S. Caron-Huot, P. Kovtun, G. Moore, A.S., L.G. Yaffe, hep-th/0607237 Photon emission from SYM plasma Photons interacting with matter: To leading order in Mimic by gauging global R-symmetry
Need only to compute correlators of the R-currents Now consider strongly interacting systems at finite density and LOW temperature Probing quantum liquids with holography Quantum liquid in p+1 dim Quantum Bose liquid Quantum Fermi liquid (Landau FLT)
Low-energy elementary excitations Specific heat at low T phonons fermionic quasiparticles + bosonic branch (zero sound) Departures from normal Fermi liquid occur in - 3+1 and 2+1 dimensional systems with strongly correlated electrons
- In 1+1 dimensional systems for any strength of interaction (Luttinger liquid) One can apply holography to study strongly coupled Fermi systems at low T L.D.Landau (1908-1968) The simplest candidate with a known holographic description is at finite temperature T and nonzero chemical potential associated with the baryon number density of the charge There are two dimensionless parameters: is the baryon number density
is the hypermultiplet mass The holographic dual description in the limit is given by the D3/D7 system, with D3 branes replaced by the AdSSchwarzschild geometry and D7 branes embedded in it as probes. Karch & Katz, hep-th/0205236 AdS-Schwarzschild black hole (brane) background D7 probe branes The worldvolume U(1) field
couples to the flavor current at the boundary Nontrivial background value of corresponds to nontrivial expectation value of We would like to compute - the specific heat at low - the charge density correlator
temperature The specific heat (in p+1 dimensions): (note the difference with Fermi and Bose systems) The (retarded) charge density correlator
has a pole corresponding to a propagating mode (zero sound) - even at zero temperature (note that this is NOT a superfluid phonon whose attenuation scales as New type of quantum liquid? ) Epilogue On the level of theoretical models, there exists a connection between near-equilibrium regime of certain strongly coupled
thermal field theories and fluctuations of black holes This connection allows us to compute transport coefficients for these theories At the moment, this method is the only theoretical tool available to study the near-equilibrium regime of strongly coupled thermal field theories The result for the shear viscosity turns out to be universal for all such theories in the limit of infinitely strong coupling Influences other fields (heavy ion physics, condmat) Outlook Gravity dual description of thermalization ?
Gravity duals of theories with fundamental fermions: - phase transitions - heavy quark bound states in plasma - transport properties Finite t Hooft coupling corrections to photon emission spectrum Understanding 1/N corrections Phonino Equations such as describe the low energy limit of string theory
As long as the dilaton is small, and thus the string interactions are suppressed, this limit corresponds to classical 10-dim Einstein gravity coupled to certain matter fields such as Maxwell field, p-forms, dilaton, fermions Validity conditions for the classical (super)gravity approximation - curvature invariants should be small: - quantum loop effects (string interactions = dilaton) should be small: In AdS/CFT duality, these two conditions translate into and
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