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Thermal spectral functions and holography

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.

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Thermal spectral functions and holography

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  1. 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

  2. Experimental and theoretical motivation • Heavy ion collision program at RHIC, LHC (2000-2008-2020 ??) • Studies of hot and dense nuclear matter • 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

  3. The challenge of RHIC Energy density vs temperature QCD deconfinement transition (lattice data)

  4. The challenge of RHIC (continued) Rapid thermalization ?? Large elliptic flow Jet quenching Photon/dilepton emission rates

  5. 4-dim gauge theory – large N, strong coupling 10-dim gravity M,J,Q Holographically dual system in thermal equilibrium M, J, Q T S Deviations from equilibrium Gravitational fluctuations ???? + fluctuations of other fields and B.C. Quasinormal spectrum

  6. Shear viscosity Bulk viscosity Charge diffusion constant Thermal conductivity Electrical conductivity Transport (kinetic) coefficients * Expect Einstein relations such as to hold

  7. 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 AdS-Schwarzschild black hole

  8. Computing finite-temperature correlation functions from gravity • Need to solve 5d e.o.m. of the dual fields propagating in asymptotically AdS space • Can compute Minkowski-space 4d correlators • Gravity maps into real-time finite-temperature formalism (Son and A.S., 2001; Herzog and Son, 2002)

  9. 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:

  10. 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:

  11. Predictions of hydrodynamics Hydrodynamics predicts that the retarded correlator has a “sound wave” pole at Moreover, in conformal theory

  12. Now look at the correlators obtained from gravity The correlator has poles at The speed of sound coincides with the hydro prediction!

  13. Analytic structure of the correlators Strong coupling: A.S., hep-th/0207133 Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092

  14. 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)

  15. Two-point correlation function of stress-energy tensor Field theory Zero temperature: Finite temperature: Dual gravity • Five gauge-invariant combinations • of and other fields determine • obey a system of coupled ODEs • Their (quasinormal) spectrum determines singularities of the correlator

  16. Spectral functions and quasiparticles in The slope at zero frequency determines the kinetic coefficient Peaks correspond to quasiparticles Figures show at different values of

  17. Spectral function and quasiparticles in finite-temperature “AdS + IR cutoff” model

  18. 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]

  19. Shear viscosity Bulk viscosity Charge diffusion constant Thermal conductivity Electrical conductivity Transport coefficients in N=4 SYM in the limit

  20. 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

  21. 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:

  22. 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

  23. A viscosity bound conjecture Minimum of in units of P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231

  24. Chernai, Kapusta, McLerran, nucl-th/0604032

  25. Chernai, Kapusta, McLerran, nucl-th/0604032

  26. Chernai, Kapusta, McLerran, nucl-th/0604032

  27. 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)

  28. QCD Chernai, Kapusta, McLerran, nucl-th/0604032

  29. 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:

  30. Shear viscosity at non-zero chemical potential Reissner-Nordstrom-AdS black hole with three R charges (Behrnd, Cvetic, Sabra, 1998) (see e.g. Yaffe, Yamada, hep-th/0602074) J.Mas D.Son, A.S. O.Saremi K.Maeda, M.Natsuume, T.Okamura We still have

  31. Photon and dilepton emission from supersymmetric Yang-Mills plasma S. Caron-Huot, P. Kovtun, G. Moore, A.S., L.G. Yaffe, hep-th/0607237

  32. 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

  33. Photoproduction rate in SYM (Normalized) photon production rate in SYM for various values of ‘t Hooft coupling

  34. How far is SYM from QCD? pQCD (dotted line) vs pSYM (solid line) at equal coupling (and =3) pQCD (dotted line) vs pSYM (solid line) at equal fermion thermal mass (and =3)

  35. 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

  36. THE END

  37. Some results • Shear viscosity/entropy ratio: • in the limit described by gravity duals • universal for a large class of theories • 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 • R-charge diffusion constant for N=4 SYM:

  38. Non-equilibrium regime of thermal gauge theories is of interest for RHIC and early universe physics • 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. • Lattice simulations cannot be used directly for real-time processes. • Gauge theory/gravity duality CONJECTURE provides a theoretical tool to probe non-equilibrium, non-perturbative regime of SOME thermal gauge theories

  39. Quantum field theories at finite temperature/density Equilibrium Near-equilibrium transport coefficients emission rates ……… entropy equation of state ……. perturbative non-perturbative perturbative non-perturbative ???? Lattice kinetic theory pQCD

  40. 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 • Stimulating for experimental/theoretical research in other fields

  41. 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/CFTA.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., to appear, P.Kovtun, A.S., hep-th/0506184, A.Buchel, J.Liu, hep-th/0311175

  42. Universality of shear viscosity in the regime described by gravity duals Graviton’s component obeys equation for a minimally coupled massless scalar. But then . Since theentropy (density) is we get

  43. 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:

  44. A viscosity bound conjecture P.Kovtun, D.Son, A.S., hep-th/0309213, hep-th/0405231

  45. Analytic structure of the correlators Strong coupling: A.S., hep-th/0207133 Weak coupling: S. Hartnoll and P. Kumar, hep-th/0508092

  46. 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)

  47. Spectral function and quasiparticles A B A: scalar channel C B: scalar channel - thermal part C: sound channel

  48. Pressure in perturbative QCD

  49. Quantum field theories at finite temperature/density Equilibrium Near-equilibrium transport coefficients emission rates ……… entropy equation of state ……. perturbative non-perturbative perturbative non-perturbative ???? Lattice kinetic theory pQCD

  50. 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

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