Heavy Ion Physics in AdS/CFT

Transcription

1 Heavy Ion Physics in AdS/CFT Urs Achim Wiedemann CERN TH Porto, 9 Sept 2009

2 Viscosity: Bounds from theory Quantum field theory ; AdS/CFT η s > 1 4π Heavy Ion Phenomenology Arnold, Moore, Yaffe, JHEP 11 (2000) 001 Strong coupling limit of N=4 SYM Kovtun, Son, Starinets, hep-th/ ς(3) 1+ 4π 8 (2λ) / 2 λ g 2 N c

3 Elliptic Flow: Hallmark of a collective phenomenon bounce dn dφ 1+ 2v 2 [ cos( 2φ) ] squeeze

4 Particle production w.r.t. reaction plane Single 2->2 process Maximal asymmetry NOT correlated to the reaction plane Many 2->2 or 2-> n processes Reduced asymmetry ~ 1 N NOT correlated to the reaction plane final state interactions asymmetry caused not only by multiplicity fluctuations collective component is correlated to the reaction plane

5 Particle production w.r.t. reaction plane Want to measure particle production as function of angle w.r.t. reaction plane φ v n ( D) = e i n φ D But reaction plane is unknown... Have to measure particle correlations: e i n ( φ 1 φ 2 ) But this requires signals = v n D 1 D 1 D 2 ( ) v n ( D 2 ) + e i n ( φ 1 φ 2 ) v n > 1 N corr D 1 D 2 Non-flow effects ~ O(1 N) Improve measurement with higher cumulants: Borghini, Dinh, Ollitrault, PRC (2001) e i n ( φ 1 +φ 2 φ 3 φ 4 ) e i n ( φ 1 φ 3) e i n ( φ 2 φ 4 ) e i n ( φ 1 φ 4 ) e i n ( φ 2 φ 3 ) = v 4 n + O( 1 N 3 ) This requires signals v n > 1 N 3 4

6 Elliptic flow: v 2 Momentum space: E dn d 3 p = 1 dn 2π p T dp T dη 1+ 2v 2( p T )cos 2φ [ ( )] Reaction plane 'Non-flow' effect for 2 nd order cumulants N ~ N ~ O(v 2 ) for 4 th order cumulants STAR Coll, Phys. Rev. C66 (2002) N 3 4 ~ 0.03 << v 2 strong unambiguous signal of collectivity

7 Elliptic flow vs. hydrodynamic simulations Assumptions: - perfect (non-dissipative) liquid T µν = ε + p ( ) u µ u ν p g µν - Bjorken boost invariance - realistic equation of state - realistic initial conditions - realistic decoupling (freeze-out) Results: - initial transverse pressure gradient - dependence of flow field elliptic flow v 2 ( p T ) φ u µ - size and pt-dependence of data accounted for by hydro ( maximal ) - characteristic mass dependence, since all particle species emerge from common flow field u µ v 2 φ Equal energy density lines PRC 72 (05) GeV Au+Au min-bias Reaction plane Kolb, Heinz; Teaney, Shuryak; Hirano, Nara; Huovinen

8 Hydrodynamic simulations are consistent with minimal shear viscosity at RHIC. M. Luzum, P. Romatschke, PRC 78 (2008)

9 Open issues in the interpretation of v2 Can the generic multiplicity dependence of v2/epsilon (spatial anisotropy) result from hydrodynamics? Can hydro reproduce the generic trends in the energy- and rapidity dependence of v2?

10 Dissipative Hydrodynamics Do not neglect non-ideal components N µ i = n i u µ + n i T µν = ε u µ u ν pδ µν + q µ u ν + q ν u µ + Π µν Region of validity? Spatio-temporal variations of macroscopic fluid should be small if compared to microscopic reaction rates Γ nσ >> θ = µ u µ Gradient expansion 2nd order viscous Israel-Stewart hydrodynamics (shear viscosity only, neglect vorticity) Entropy increase determined by viscosity: T µ S µ Π µν Π µν 2η Equations of motion involve relaxation time and viscosity τ π Δ µ α Δ ν β DΠ αβ + Π µν = η µ u ν 2τ π Π α(µ ν ) ω α

11 2 nd order transport coefficients conformal case (6 transport coefficients) η,τ π,κ,λ 1,λ 2,λ 3 Result for N=4 SYM κ λ<<1 ~ λ 1 λ<<1 τ π λ>>1 τ π λ<<1 ~ 5.9 η ε + p 375 ~ 4 2ln2 + 8 ς(3)λ 3 / 2 η ε + p 0.2 T 5s 8π 2 T κ λ>>1 ~ ς(3)λ 3 / 2 s 4π 2 T ~ 5.2 η2 ε + p λ ~ λ>>1 4 ς(3)λ 3 / 2 η2 ε + p Relaxation time is very short λ 2 λ<<1 ~ 2ητ π Remarkable curiosity: all modes propagate causal λ 3 = 0? λ 2 λ>>1 ~ ln2 η πt Bhattacharyya, Hubeny, Minwalla,Rangamani 2008 Kanitschneider, Skenderis (2009) Buchel, Myers (2009) Romatschke (2009) non-conformal case: - (9 additional transport coefficients + velocity of sound) - coefficients given in powers of 1 3 c 2 s

12 The sonic booma more differential measure of collectivity If a jet deposits energy in a nondissipative perfect fluid, then this energy must propagate in a conical shock wave.

13 Dragging a quark through N=4 SYM medium Apply force to maintain momentum p of quark p = µp + F = 0 Different from QCD quenching: since strongly coupled on all scales (divergence for v-> 1) de q dx = π 2 λt v 2 1 v 2 Lost energy dissipates in hydrodynamic modes (Mach cone) Fries, Gubser, Michalogiorgakis, Pufu, hep-th/ Herzog, Karch, Kovtun, Kozcaz, Yaffe hep-th/ S. Gubser, hep-th/ Casalderrey-Solana, Teaney, hep-ph/ Range of validity of this picture set by Schwinger mechanism F crit M 2 λ coshη < Λ

14 AdS/CFT calculation of Mach cone Chesler, Yaffe arxiv: Energy density (specific heat is ~ N c 2 is, which amplifies the effect of heat over motion. So this plot shows where the heat, i.e. compression, i.e. sound goes.) Momentum flow: is in Mach cone and wake

15

16

17 Messages on viscous hydro Elliptic flow is an unambiguously established experimental signal of collectivity, measured on top of a quantitatively controlled background. The dependence of elliptic flow on transverse momentum and particle species are consistent with a fluid dynamic interpretation. Hydrodynamic simulations reproduce the measured v2 at mid-rapidity under the assumption of minimal shear viscosity and short thermalization times (< 1 fm/c). This supports the picture of a strongly coupled plasma. AdS/CFT has provided qualitatively novel insight to elliptic flow studies, mainly by showing that extremely small shear viscosity and short relaxation times can emerge in non-abelian finite temperature quantum field theories. There are generic trends in elliptic flow data, which persist over many order of magnitude and which (so far) have not found a natural explanation in hydrodynamics. The observed Mach-cone-like structures are not (yet?) unambiguous signals of collective excitations.

18 Jet quenching Heavy Ion Collisions produce auto-generated probes at high s NN Q >> T 150 MeV Q: How sensitive are such hard probes?

19 High p T Hadron Spectra R AA ( p T,η) = dn AA dp T dη n coll dn NN dp T dη Centrality dependence: 0-5% 70-90% L large L small

20 Strong suppression persists to highest p T Centrality dependence: 0-5% 70-90% L large L small

21 Parton Propagation in an External Color Field High-energy scattering determined by eikonal Wilson line Ψ in = ψ( α i,x i ) α i, x i Ψ out = ψ α i, x i { α i,x i } ( ) = P exp i dz T a A + a ( x i,z ) W x i [ ] During scattering, transverse coordinates are frozen, color rotates { } α i,x i r ( ) Π i W i αi β i ( x i ) ( ) β i, x i Example: gluon radiation in q+a scattering Incoming free quark wave function dressed to O(g) Outgoing from target Ψ in = α + dx f (x) T b αβ β,b(x) f (x) g b(x) = α α + α β Ψ out = W F αγ ( 0) γ + dx f (x) W F βγ ( 0)W A bc ( x)t b αβ γ,c(x) T αβ b x x 2

22 Example: gluon production in q+a Component of outgoing function orthogonal to incoming one [ Ψ in ] Ψ out δψ = 1 Ψ in Number of produced gluons N qa N qa ( k T ) = 1 δψ α a + d k T N c α ( k T ) g 2 dx dy x. y x 2 y 2 eik T Perturbative x y ( ) ( )a d ( k T )δψ α T arg et [ N(x,0) + N(y,0) N(x,y) ] Non-perturbative Kovchegov Mueller 1998 This illustrates that, in general, QCD radiation involves perturbative and nonperturbative scales. Target average defined in terms of adjoint light-like Wilson loop [ ] ( N 2 c 1) exp 1 4 ˆ N(x, y) 1 Tr W A + (x)w A (y) This quantity appears in QCD jet quenching calculations, and has been calculated in AdS/CFT q L 2 L2

23 Jet quenching parameter at RHIC PHENIX, PRC77 (2008) One parameter fit to RHIC data accounts for nuclear modification factors. (For model dependence of parameter value, see also PHENIX publication.)

24 Results for quenching parameter [ ] = exp 1 4 ˆ W A (C) exp i( S(C) S 0 ) ( ) ( ) q ˆ SYM = π 3 / 2 Γ 3 4 Γ 5 4 q L 2 L2 λt α SYM N c T 3 Liu, Rajagopal, Wiedemann, hep-ph/ Range of validity of this picture Our (3+1)-dim world Wilson loop C in our world r = Λ γ > Λ S(C) : area of string world sheet with boundary C. horizon r = r 0 Numerology: If we relate N=4 SYM to QCD by fixing N c = 3 α SYM =1 2 q ˆ SYM = 4.4 GeV 2 for T = 300 MeV fm q ˆ SYM =10.6 GeV 2 fm for T = 400 MeV

25 Speculation: what sets in when drag breaks down? Opposite ranges of validity: γ > Λ γ < Λ Drag without balancing force leads to deceleration. Estimate, when this can be neglected: quenching drag de drag dt = π 2 λt 2 v 2 1 v 2 de accel dt = λ 2π ( dp dt) 2 m 2 Assume: de drag dt 1 1 π p >> de accel dt dp dt 2 >> T ( 1 v ) 1/ 4 Λ m λt >> γ (use lin. drag on LHS again) coincides with the range of validity of linear drag Motivated us to consider drag calculation including acceleration

26 Stirring string: formulation of the problem Consider heavy test quark, moved by external force on - Circle of radius - with constant angular frequency L ω Two-dimensional world-sheet of corresponding rotating string: X µ ( τ,σ) = t = τ,ρ = ρ(u),ϕ = ωτ + θ(u), x 3 = 0,u = σ ( ) Motivation: - this quark radiates in the vacuum (synchrotron radiation) due to acceleration a = ω 2 L, L = ρ(u = ) v = ω L - propagation in medium with constant velocity implies energy loss due to drag. What is the interplay between linear drag and energy loss due to acceleration? Which effect dominates in which kinematical regime? Also: rotating quark is a perfectly localized energy perturbation. How do these perturbations dissipate?

27 A model to study interplay between linear drag and acceleration A rotating string drag solution K. Bitaghsir, H. Liu, K. Rajagopal, U. Wiedemann, 2008 ω = 5.0 Π =100. v = ω ρ 0, a T = ω 2 ρ 0 A constant of motion z L ( z θ) = 0, Π = L ( z θ) de q dx = π 2 λt 2 v 1 v 2

28 BACK-UP: A special point on the stirring string θ' 2 = Π 2 ( u 4 ρ 2 ω 2 u 4 1) ρ' u 4 1 ρ 2 ( u 4 1) ρ 2 ( u 4 1) Π 2 [ ] Numerator vanishes if ρ c ω = u c, ρ c = ρ(u c ) Velocity of light in the bulk at u c Denominator must vanish at same point since u c = ω ρ c = θ' Πω Π2 ω 2 2Πω Πω Π 2 ω 2 real ρ'(u c ) can be determined exactly, too. This is the initial condition for Solving 2 nd order differential equation for radial e.o.m.

29 Energy loss of the stirring string The conjugate momenta Π µ σ = L ( ) σ X µ Characterize de dt = Π u t = π 2 λt 2 ω Π dl angular dt = Π ϕ u = Π t 1 ω Energy loss is known if we know functional dependence of Limiting cases: ω Π ω Π linear drag = v 2 1 v 2, ω Π ω Π vac rad = v 2 ω 2 ( 1 v 2 ) 2 Π v = ρ( )ω Mikhailov s result: bremsstrahlung in QED and N=4 SYM differ by pre-factor only.

30 Energy loss of the stirring string l = ρ( )

31 1. Compare: Observations from stirring string 2. Destructive interference: ω Π vac rad >> ω Π linear drag ω >> ( 1 v 2 ) 3 / 4 That s why vac rad always dominates for ω >1 Π <Π vac rad + Π linear drag holds everywhere For small frequencies, the minimum of well-described by ( ) Π / Π vac rad + Π linear drag ω Π vac rad = ω Π linear drag ω = ( 1 v 2 ) 3 / 4 l = 1 ω 4 / 3 /w 3. Why is vacuum radiation relevant? No complete answer, but Critical synchrotron energy is If ω >> ( 1 v 2 ) 3 / 4 then ω c = 3γ 3 ω ω c >> π T

32 Messages on jet quenching Jet quenching is a robust and generic (unambiguously established) experimental signal on the level of single hadron spectra and di-hadron correlations. It is expected to persist for internal jet structures (which will be accessible at LHC) Jet physics in QCD involves perturbative and non-perturbative scales. Phenomenology requires to account for baseline of jet fragmentation in vacuum before establishing medium effects on top of it. AdS/CFT has provided qualitatively novel insight to jet quenching studies, e.g. by showing that quenching parameter can be very large or how hard probes could excite collective modes in the medium. (One may imagine further qualitative insights e.g. from understanding which energy loss mechanisms set in at velocities where linear drag seizes to be valid.)

33 AdS/CFT vs lattice QCD What is the evidence that lattice QCD is strongly coupled? At which temperatures could weak coupling techniques apply to QCD thermodynamics?

34 Strong coupling close to Tc RBC-Bielefeld, PoS LAT2007:217,2007. Strong coupling approximation ( α s = 0.5, N c = 3 λ 6π λ >>1 ) seems satisfied What about perturbative description in terms of quasi-particles

35 The renormalized Polyakov loop in QCD Trace of Wilson line TrΩ x ( ) Tr P exp i dτ A 0 ( τ, x ) - transforms under Z(3) TrΩ x - interpretation TrΩ x β ( ) z TrΩ( x ) 0 [ ] ( ) = exp β ΔF q ( x ) Good order parameter for pure SU(3) TrΩ x ( ) Limit reached above 3-5 T c RBC-Bielefeld, PoS LAT2007:217,2007. A.Kurkela

36 Relation to the validity of quasi-particle pictures The effective potential at T>>T c - height of potential barrier V eff [ ( )] TrΩ x V eff MAX T 4 - fluctuations between minima negligible only above 3-5 T c Re[ A 0 ] Gross, Pisarski, Yaffe Rev.Mod.Phys.53:43,1981 [ ] Im A 0 Quasi-particle models - generally rely on expanding V eff around A 0 =(0,0) - this breaks Z(3) invariance - the complex minima lie outside domain of validity of description - no sound basis for T< 2-3 T c

37 SPS RHIC Deviations from conformality LHC Karsch, Laermann, Peikert, NPB605 (2001) 579 At the peak of the interaction measure, one finds 1 3 c 2 s 1 6

38 Quark susceptibilities Appear to agree with (NLA) perturbative calculations

39 Finite temperature QCD vs. AdS/CFT chiral condensate N=4 SYM in vacuum NO QCD in vacuum YES N=4 SYM finite T NO QCD at finite T melted asymptotic freedom NO YES NO less important confinement NO YES NO deconfined superymmetric YES NO broken NO conformal YES NO YES approx at T >>T c degrees of freedom Physics near vacuum and at high energy is very different Very different, but may be taken care of by normalization EXP: Conceptual questions in reach of LHC TH: Which field theoretical tools are best suited? quasi-particles ε 3p NO Zero Conceivable for T>2-3 Tc Small for T>2-3 Tc only