Holographic recipes for hot and dense strongly coupled matter

1. Holographic recipes for hot and dense strongly coupled matter SEWM-2008 Amsterdam David Teniers the Younger (1610-1690) “The Alchemist”

2. Andrei Starinets University of Southampton

3. Outline Introductory review of gauge/string duality From gauge/string to gauge/gravity duality Finite temperature, holography and black hole physics Transport in strongly coupled gauge theories from black hole physics First- and second-order hydrodynamics and dual gravity Photon/dilepton emission rates from dual gravity Quantum liquids and holography Other approaches

4. 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 detector) 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 don’t know for QCD

5. What is string theory?

6. 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

7. 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)

8. AdS/CFT correspondence conjectured exact equivalence Generating functional for correlation functions of gauge-invariant operators String partition function In particular Classical gravity action serves as a generating functional for the gauge theory correlators

9. The bulk and the boundary in AdS/CFT correspondence UV/IR: the AdS metric is invariant under z plays a role of inverse energy scale in 4D theory z 5D bulk (+5 internal dimensions) strings or supergravity fields 0 gauge fields 4D boundary

10. 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

11. supersymmetric Yang-Mills is the harmonic oscillator of the XXI century!

12. Field content: supersymmetric YM theory Gliozzi,Scherk,Olive’77 Brink,Schwarz,Scherk’77 • Action: (super)conformal field theory = coupling doesn’t run

13. AdS/CFT correspondence is the simplest example of the gauge/string (gauge/gravity) duality

14. Gauge/gravity duality and QCD coupling energy

15. Gauge/gravity duality and QCD coupling energy

16. Gauge/gravity duality and QCD coupling energy

17. 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

18. 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

19. 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

20. 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

21. 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

22. First-order transport (kinetic) coefficients Shear viscosity Bulk viscosity Charge diffusion constant Supercharge diffusion constant Thermal conductivity Electrical conductivity * Expect Einstein relations such as to hold

23. 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 (G.Policastro, 2008) Thermal conductivity Electrical conductivity

24. 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:

25. Shear viscosity in SYM perturbative thermal gauge theory S.Huot,S.Jeon,G.Moore, hep-ph/0608062 Correction to : Buchel, Liu, A.S., hep-th/0406264 Buchel, 0805.2683 [hep-th]; Myers, Paulos, Sinha, 0806.2156 [hep-th]

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

27. 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 But see A.Buchel, 0804.3161 [hep-th] • 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

28. Second-order transport coefficients in N = 4 SYM in the limit Relaxation time Here we also used results from: S.Bhattacharyya,V.Hubeny,S.Minwalla,M.Rangamani, 0712.2456 [hep-th] Generalized to CFT in D dim: Haack & Yarom, 0806.4602 [hep-th]; Finite ‘t Hooft coupling correction computed: Buchel and Paulos, 0806.0788 [hep-th] Question: does this affect RHIC numerics? Luzum and Romatschke, 0804.4015 [nuc-th]

29. 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:

30. Viscosity/entropy ratio in QCD: current status Theories with gravity duals in the regime where the dual gravity description is valid (universal limit) 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

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

32. Supersymmetric sound mode (“phonino”) in Hydrodynamic mode (infinitely slowly relaxing fluctuation of the charge density) Hydro pole in the retarded correlator of the charge density Conserved charge Sound wave pole: Supersound wave pole: Lebedev & Smilga, 1988 (see also Kovtun & Yaffe, 2003)

33. Sound and supersymmetric sound in In 4d CFT Sound mode: Supersound mode: Quasinormal modes in dual gravity Graviton: Gravitino:

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

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

36. Now consider strongly interacting systems at finite density and LOW temperature

37. This part of the talk - “Holographic recipes at finite density and low temperature” - is based on the paper “Zero Sound from Holography” arXiv: 0806.3796 [hep-th] by Andreas Karch Dam Thanh Son A.S.

38. Probing quantum liquids with holography Low-energy elementary excitations Specific heat at low T Quantum liquid in p+1 dim Quantum Bose liquid phonons Quantum Fermi liquid (Landau FLT) 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

39. L.D.Landau (1908-1968) Fermi Liquid Theory: 1956-58

40. 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 AdS- Schwarzschild geometry and D7 branes embedded in it as probes. Karch & Katz, hep-th/0205236

41. 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 temperature - the charge density correlator

42. 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?

43. Other avenues of (related) research Bulk viscosity for non-conformal theories (Buchel, Gubser,…) Non-relativistic gravity duals (Son, McGreevy,… ) Gravity duals of theories with SSB (Kovtun, Herzog,…) Bulk from the boundary (Janik,…) Navier-Stokes equations and their generalization from gravity (Minwalla,…) Quarks moving through plasma (Chesler, Yaffe, Gubser,…)

45. THANK YOU

46. 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]

47. Conclusions The program of computing first- and second-order transport coefficients in N=4 SYM at strong coupling and large number of colors is essentially complete These calculations are helpful in numerical simulations of non-equilibrium QCD and stimulated developments in other fields such as LQCD, pQCD, condmat Need better understanding of - thermalization, isotropisation, and time evolution [Janik et al.] - corrections from higher-derivative terms [Kats & Petrov; Brigante,Myers,Shenker,Yaida] [ Buchel, Myers, Paulos, Sinha, 0808.1837 [hep-th] ] LHC (ALICE) will provide more data at higher T At low temperature and finite density, does holography identify a new type of quantum liquid? (perturbative study of SUSY models at finite temperature and density would be very helpful)

48. Elliptic flow with Glauber initial conditions Luzum and Romatschke, 0804.4015 [nuc-th]

49. Elliptic flow with color glass condensate initial conditions Luzum and Romatschke, 0804.4015 [nuc-th]

50. 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 But see A.Buchel, 0804.3161 [hep-th] • 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