Thermal spectral functions and holography Andrei Starinets Perimeter
- Slides: 60
Thermal spectral functions and holography Andrei Starinets (Perimeter Institute) “Strong Fields, Integrability and Strings” program Isaac Newton Institute for Mathematical Sciences Cambridge, 31. VII. 2007
Experimental and theoretical motivation Ø Heavy ion collision program at RHIC, LHC (2000 -2008 -2020 ? ? ) Ø Studies of hot and dense nuclear matter Ø Abundance of experimental results, poor theoretical understanding: - the collision apparently creates a fireball of “quark-gluon fluid” - need to understand both thermodynamics and kinetics -in particular, need theoretical predictions for parameters entering equations of relativistic hydrodynamics – viscosity etc – computed from the underlying microscopic theory (thermal QCD) -this is difficult since the fireball is a strongly interacting nuclear fluid, not a dilute gas
The challenge of RHIC Energy density vs temperature QCD deconfinement transition (lattice data)
The challenge of RHIC (continued) Rapid thermalization ? ? Large elliptic flow Jet quenching Photon/dilepton emission rates
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 ? ? + fluctuations of other fields and B. C. Quasinormal spectrum
Transport (kinetic) coefficients • Shear viscosity • Bulk viscosity • Charge diffusion constant • Thermal conductivity • Electrical conductivity * Expect Einstein relations such as to hold
Gauge/gravity dictionary determines correlators of gauge-invariant operators from gravity (in the regime where gravity description is valid!) Maldacena; Gubser, Klebanov, Polyakov; Witten For example, one can compute the correlators such as by solving the equations describing fluctuations of the 10 -dim gravity background involving Ad. S-Schwarzschild black hole
Computing finite-temperature correlation functions from gravity § Need to solve 5 d e. o. m. of the dual fields propagating in asymptotically Ad. S space § Can compute Minkowski-space 4 d correlators § Gravity maps into real-time finite-temperature formalism (Son and A. S. , 2001; Herzog and Son, 2002)
Hydrodynamics: fundamental d. o. f. = densities of conserved charges Need to add constitutive relations! Example: charge diffusion Conservation law Constitutive relation [Fick’s law (1855)] Diffusion equation Dispersion relation Expansion parameters:
Similarly, one can analyze another conserved quantity – energy-momentum tensor: This is equivalent to analyzing fluctuations of energy and pressure We obtain a dispersion relation for the sound wave:
Predictions of hydrodynamics Hydrodynamics predicts that the retarded correlator has a “sound wave” pole at Moreover, in conformal theory
Now look at the correlators obtained from gravity The correlator has poles at The speed of sound coincides with the hydro prediction!
Analytic structure of the correlators Strong coupling: A. S. , hep-th/0207133 Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092
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)
Two-point correlation function of stress-energy tensor Field theory Zero temperature: Finite temperature: Dual gravity Ø Five gauge-invariant combinations of and other fields determine Ø obey a system of coupled ODEs Ø Their (quasinormal) spectrum determines singularities of the correlator
Spectral functions and quasiparticles in The slope at zero frequency determines the kinetic coefficient Figures show Peaks correspond to quasiparticles at different values of
Spectral function and quasiparticles in finite-temperature “Ad. S + IR cutoff” model
Holographic models with fundamental fermions Thermal spectral functions of flavor currents Additional parameter makes life more interesting… R. Myers, A. S. , R. Thomson, 0706. 0162 [hep-th]
Transport coefficients in N=4 SYM in the limit • Shear viscosity • Bulk viscosity • Charge diffusion constant • Thermal conductivity • Electrical conductivity
Shear viscosity in SYM perturbative thermal gauge theory S. Huot, S. Jeon, G. Moore, hep-ph/0608062 Correction to : A. Buchel, J. Liu, A. S. , hep-th/0406264
Electrical conductivity in SYM Weak coupling: Strong coupling: * Charge susceptibility can be computed independently: D. T. Son, A. S. , hep-th/0601157 Einstein relation holds:
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
A viscosity bound conjecture Minimum of in units of P. Kovtun, D. Son, A. S. , hep-th/0309213, hep-th/0405231
Chernai, Kapusta, Mc. Lerran, nucl-th/0604032
Chernai, Kapusta, Mc. Lerran, nucl-th/0604032
Chernai, Kapusta, Mc. Lerran, 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)
Chernai, Kapusta, Mc. Lerran, nucl-th/0604032 QCD
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, Mc. Lerran, nucl-th/0604032 SYM:
Shear viscosity at non-zero chemical potential Reissner-Nordstrom-Ad. S 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
Photoproduction rate in SYM (Normalized) photon production rate in SYM for various values of ‘t Hooft coupling
How far is SYM from QCD? p. QCD (dotted line) vs p. SYM (solid line) at equal coupling (and =3) p. QCD (dotted line) vs p. SYM (solid line) at equal fermion thermal mass (and =3)
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
THE END
Some results ü Shear viscosity/entropy ratio: • in the limit described by gravity duals • universal for a large class of theories ü Bulk viscosity for non-conformal theories • in the limit described by gravity duals • in the high-T regime (but see Buchel et al, to appear…) • model-dependent ü R-charge diffusion constant for N=4 SYM:
v Non-equilibrium regime of thermal gauge theories is of interest for RHIC and early universe physics v This regime can be studied in perturbation theory, assuming the system is a weakly interacting one. However, this is often NOT the case. Nonperturbative approaches are needed. v Lattice simulations cannot be used directly for real-time processes. v Gauge theory/gravity duality CONJECTURE provides a theoretical tool to probe non-equilibrium, non-perturbative regime of SOME thermal gauge theories
Quantum field theories at finite temperature/density Equilibrium Near-equilibrium entropy equation of state ……. transport coefficients emission rates ……… perturbative non-perturbative Lattice p. QCD perturbative non-perturbative ? ? kinetic theory
Epilogue Ø 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 Ø This connection allows us to compute transport coefficients for these theories Ø At the moment, this method is the only theoretical tool available to study the near-equilibrium regime of strongly coupled thermal field theories Ø The result for the shear viscosity turns out to be universal for all such theories in the limit of infinitely strong coupling Ø Stimulating for experimental/theoretical research in other fields
Three roads to universality of Ø The absorption argument D. Son, P. Kovtun, A. S. , hep-th/0405231 Ø Direct computation of the correlator in Kubo formula from Ad. S/CFT A. Buchel, hep-th/0408095 Ø “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. , to appear, P. Kovtun, A. S. , hep-th/0506184, A. Buchel, J. Liu, hep-th/0311175
Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then. we get Since the entropy (density) is
Example 2 (continued): 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: )
A viscosity bound conjecture P. Kovtun, D. Son, A. S. , hep-th/0309213, hep-th/0405231
Analytic structure of the correlators Strong coupling: A. S. , hep-th/0207133 Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092
Example 2: 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)
Spectral function and quasiparticles A B C A: scalar channel B: scalar channel - thermal part C: sound channel
Pressure in perturbative QCD
Quantum field theories at finite temperature/density Equilibrium Near-equilibrium entropy equation of state ……. transport coefficients emission rates ……… perturbative non-perturbative Lattice p. QCD perturbative non-perturbative ? ? kinetic theory
Thermal spectral functions and holography Andrei Starinets Perimeter Institute for Theoretical Physics “Strong Fields, Integrability and Strings” program Isaac Newton Institute for Mathematical Sciences Cambridge July 31, 2007
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, Mc. Lerran, nucl-th/0604032 SYM:
A hand-waving argument Thus Gravity duals fix the coefficient:
Thermal conductivity Non-relativistic theory: Relativistic theory: Kubo formula: In SYM with non-zero chemical potential One can compare this with the Wiedemann-Franz law for the ratio of thermal to electric conductivity:
Classification of fluctuations and universality O(2) symmetry in x-y plane Shear channel: Sound channel: Scalar channel: Other fluctuations (e. g. But not the shear channel ) may affect sound channel universality of
Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then. we get Since the entropy (density) is
Three roads to universality of Ø The absorption argument D. Son, P. Kovtun, A. S. , hep-th/0405231 Ø Direct computation of the correlator in Kubo formula from Ad. S/CFT A. Buchel, hep-th/0408095 Ø “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. , to appear, P. Kovtun, A. S. , hep-th/0506184, A. Buchel, J. Liu, hep-th/0311175
Effect of viscosity on elliptic flow
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
Sound wave pole Compare: In CFT:
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