The Gauge-String Duality and QCD at Finite Temperature

The gauge-string duality and QCD at ﬁnite temperature

Steve Gubser Princeton University

Strings 2008, CERN

August 21, 2008

ptContents

1 Near-extremal D3-branes 3

2 Shear viscosity 5

3 Equation of state and bulk viscosity 9

4 The trailing string 14 4.1 Stochastic forces on heavy quarks ...... 19

5 Jet-splitting? 21

6 Jet quenching 25

7 Falling strings 28

8 Total multiplicity 32

9 Outlook 36 The gauge-string duality and QCD at ﬁnite temperature

Steve Gubser Princeton University

Strings 2008, CERN

August 21, 2008

ptContents

1 Near-extremal D3-branes 3

2 Shear viscosity 5

3 Equation of state and bulk viscosity 9

4 The trailing string 14 4.1 Stochastic forces on heavy quarks ...... 19

5 Jet-splitting? 21

6 Jet quenching 25

7 Falling strings 28

8 Total multiplicity 32

9 Outlook 36 Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 3 1 Near-extremal D3-branes 1. Near-extremal D3-branes The near-extremal D3-brane metric describes =4gauge theory at ﬁnite temper- N ature [Gubser et al. 1996] (also unpublished work of Strominger):

2 2 1/2 2 2 1/2 dr 2 2 ds = H− hdt + dx + H + r dΩ − h 5 (1) L4 r4 H =1+ h =1 0 . r4 − r4 In the now-familiar strong coupling limit of AdS/CFT [Maldacena 1998; Gubser et al. 1998a; Witten 1998] L8 2N 2 L4 2 (2) = 4 1 2 = λ gYMN 1 G10 π α ≡ One ﬁnds free energy density [Gubser et al. 1998b] F 3 15ζ(3) f(λ) = = + + ... f (3) V 4 8λ3/2 free 2 where f = π (N 2 1)T 4 for SU(N) super-Yang-Mills. free − 6 −

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 4 1 Near-extremal D3-branes At weak coupling [Fotopoulos and Taylor 1999; Vazquez-Mozo 1999; Kim and Rey 2000; Nieto and Tytgat 1999],

3 √2+3 3/2 f(λ) = 1 2 λ + 3 λ + ... ffree (4) − 2π π The most modern treatment I know of is by [Blaizot et al. 2006]: (3) and (4) uniquely ﬁx a (4,4) Pade´ estimate, f 1+αλ1/2 + βλ + γλ3/2 = (5) 1/2 3/2 ffree 1+¯αλ + βλ¯ +¯γλ

/ = 4 super-Yang-Mills Comparison with a hard thermal loop S S0 1 N weak-coupling to order λ3/2 calculation of s/sfree (roughly, two-loop 0.95 −3/2 perturbation theory supplemented by a strong-coupling to order λ self-consistent gap equation for thermal 0.9 0.85 masses) does pretty well out to λ cΛ = 0 ∼ 1 0.8 λ N 4. L A Pad´e 0.75 λ → ∞ 1 cΛ = 2 1 2 HTL (green) calculations of entropy in =4 0.7 N 0 2 4 6 8 10 12 14 [Blaizot et al. 2006]. λ g2N ≡ Figure 2: Weak and strong coupling results for the entropy density of = 4 SYM theory together N with the NLA results obtained in analogy to QCD (cp. Fig. 1), but as a function of λ, which here is

a free parameter. The dashed and full heavy gray lines represent the Pad´e approximants R[1,1] and R[4,4] which interpolate between weak and strong coupling results to leading and next-to-leading orders, respectively.

uniquely all the coeﬃcients in the R[4,4] approximant

1 + αλ1/2 + βλ + γλ3/2 + δλ2 R = . (3.1) [4,4] 1 + α¯λ1/2 + β¯λ + γ¯λ3/2 + δ¯λ2

The fact that the weak coupling result has no term of order λ1/2, while the strong- coupling result approaches 3/4 at large couplings, where it has no terms of order λ−1/2 or λ−1, completely constrains the coefficients in the denominator of (3.1) in terms of those in the numerator: 4 4 4 α¯ = α, β¯ = β, γ¯ = γ, δ¯ = δ. (3.2) 3 3 3 The remaining four independent coefficients are then uniquely fixed by the results quoted above for the weak-coupling expansion to order λ3/2 and, respectively, the strong-coupling expansion to order λ−3/2, together with the expectation that the subsequent term in the latter should be suppressed by at least a factor of λ−1. One thus finds

2(9 + 3√2 + γπ3) 9 α = , β = , 9π 2π2 2 2 γ = , δ = α . (3.3) 15ζ(3) 15ζ(3)

Gratifyingly, the resulting rational function has no pole at positive values of λ and it interpolates monotonically between zero and infinite coupling, as commonly expected [3]. In Fig. 2, the Padé approximant (3.1) to the weak- and strong-coupling results is represented by the heavy gray line marked “Padé”. In order to get an idea of the convergence of

successive Pad´e approximants, the simpler Pad´e approximant R[1,1] which only reproduces

– 8 – Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 3 1 Near-extremal D3-branes 1. Near-extremal D3-branes The near-extremal D3-brane metric describes =4gauge theory at ﬁnite temper- N ature [Gubser et al. 1996] (also unpublished work of Strominger):

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 4 1 Near-extremal D3-branes At weak coupling [Fotopoulos and Taylor 1999; Vazquez-Mozo 1999; Kim and Rey 2000; Nieto and Tytgat 1999],

uniquely all the coeﬃcients in the R[4,4] approximant

1 + αλ1/2 + βλ + γλ3/2 + δλ2 R = . (3.1) [4,4] 1 + α¯λ1/2 + β¯λ + γ¯λ3/2 + δ¯λ2

2(9 + 3√2 + γπ3) 9 α = , β = , 9π 2π2 2 2 γ = , δ = α . (3.3) 15ζ(3) 15ζ(3)

successive Pad´e approximants, the simpler Pad´e approximant R[1,1] which only reproduces

– 8 – Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 5 2 Shear viscosity 2. Shear viscosity Neglecting loop and stringy corrections to two-derivative gravity, a broad set of black branes have [Policastro et al. 2001; Buchel and Liu 2004; Kovtun et al. 2005] η 1 = ; (6) s 4π and D3-branes in particular have [Buchel et al. 2005] η 1 135ζ(3) = 1+ + ... . (7) s 4π 8λ3/2 Loop corrections may lead to violations [Kats and Petrov 2007; Brigante et al. 2008] of the conjectured bound η/s 1/4π. ≥ η is a key input for relativistic hydrodynamics:

µν µ ν µν µα νβ 2 λ T =( + p)u u + pg P P η αuβ + βuα gαβ λu − ∇ ∇ − 3 ∇ (8) λ µν µν µ ν + ζgαβ λu where P = g + u u . ∇

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 6 2 Shear viscosity Lattice simulations of pure glue [Meyer 2007] indicate η 5/3 η =0.134 < 1 @ 90% CL (9) s best ≈ 4π s ∼ This is hard work for the lattice because viscosities arise from real-time correlators:

2 1 R η δikδjl + δilδjk δijδkl + ζδijδkl = lim Im Gij,kl(ω) − 3 − ω 0 ω → (10) GR (ω) i d3x dt eiωtθ(t) [T (t, x),T (0, 0)] , ij,kl ≡− ij kl whereas lattice provides direct access only to Euclidean correlators: β 2πn GE(ω ) = dτ d3xeiωnτ T (τ, x) (0) ω = n E O O n β 0 (11) R ∞ ρ(ω) = G (iωn)= dω for n>0. − ω iωn −∞ − To get GR(ω) for real ω starting from lattice data, some assumptions about spectral density ρ(ω) have to be made. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 5 2 Shear viscosity 2. Shear viscosity Neglecting loop and stringy corrections to two-derivative gravity, a broad set of black branes have [Policastro et al. 2001; Buchel and Liu 2004; Kovtun et al. 2005] η 1 = ; (6) s 4π and D3-branes in particular have [Buchel et al. 2005] η 1 135ζ(3) = 1+ + ... . (7) s 4π 8λ3/2 Loop corrections may lead to violations [Kats and Petrov 2007; Brigante et al. 2008] of the conjectured bound η/s 1/4π. ≥ η is a key input for relativistic hydrodynamics:

µν µ ν µν µα νβ 2 λ T =( + p)u u + pg P P η αuβ + βuα gαβ λu − ∇ ∇ − 3 ∇ (8) λ µν µν µ ν + ζgαβ λu where P = g + u u . ∇

Side view of an off-center gold-gold collision. 2R/! x The reaction plane is the plane of the page b as a vector is approximately determined for each z event. b γ 100 at RHIC, 2800 at LHC. 2R = 14 fm ≈ Au Au Elliptic Flow: A collective effect Beam’s eye view of a Particles prefer to be “in plane”: non-central collision: ! y

Cartoon of elliptic ﬂow.dN/d( From! "# [BakerR ) =2001 N0 (].1 Uneven+ 2V1c pressureos (!"# gradientsR) + 2V lead2cos to (2 anisotropic(!"#R)) + ex- ... ) pansion.

Elliptic flow !

Gubser, AdS/CFT and QCD at finite T , 8-21-08 8 2 Shear viscosity $ Experimental measure of elliptic flow is dM-waveark D. B coefficientaker in an expansion of az- 1 imuthal distribution of particles (here y = tanh− pz/E is rapidity): dN dN = [1 + 2v2 cos 2φ + ...] (12) pT dpT dydφ pT dpT dy

) 0.2 T Viscosity dependence of v2 was studied e.g. in (p 2

v 0.18 b ! 6.8 fm (16-24% Central) [Teaney 2003] in terms of Γs/τo, where 0.16 STAR Data 4 η 0.14 Γ = sound attenuation length # /" = 0 s 0.12 s o 3T s 0.1 #s/"o = 0.1 τoT 1 characteristic expansion (13) 0.08 ≈ 0.06 Γ η 1 s =0.1 0.04 #s/"o = 0.2 τ0 ←→ s ≈ 4π 0.02 But... Ideal hydro, Γs =0, was “designed” to 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 agree with data in this study. pT(GeV) Effect of shear viscosity on predictions of Upshot: data favors the range FIG. 3: Elliptic flow v2 avs2a(pfuTn)c.t Fromion of p [TTeaneyfor diff2003erent ].val Dataues of pointsΓs/τo. The data points are four η 5/2 particle cummulant data arefrom pions,the ST fromAR co STARllabora [tAdlerion [3]. etO al.nly2002statis].tical errors are shown0. The < 0.2 . (14) ≤ s ∼ ≈ 4π difference between the ideal and viscous curves is linearly proportional to Γs/τo.

8 8 (fm) (fm) L 7 L 7 , R , R

S Ideal S

, R 6 , R 6 O O R R 5 5

4 4

3 3 "o = 7.0 fm RO RO 2 2 Ro = 10 fm RS RS r uo = 0.55 c 1 RL 1 RL To = 160 MeV 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

pT (GeV) pT(GeV)

FIG. 4: (a) Ideal blast wave ﬁt to the experimental HBT radii RO, RS, and RL shown in (b) as a function of transverse momentum KT . The solid symbols are from the STAR collaboration [11] and the open symbols are from the PHENIX collaboration [12]. For clarity, the experimental points have been slightly shifted horizontally.

14 Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 7 2 Shear viscosity Elliptic ﬂow in heavy ion collisions puts bounds on η. Here’s the relevant geometry:

Cartoon of elliptic ﬂow.dN/d( From! "# [BakerR ) =2001 N0 (].1 Uneven+ 2V1c pressureos (!"# gradientsR) + 2V lead2cos to (2 anisotropic(!"#R)) + ex- ... ) pansion.

Elliptic flow !

) 0.2 T Viscosity dependence of v2 was studied e.g. in (p 2

8 8 (fm) (fm) L 7 L 7 , R , R

S Ideal S

, R 6 , R 6 O O R R 5 5

4 4

3 3 "o = 7.0 fm RO RO 2 2 Ro = 10 fm RS RS r uo = 0.55 c 1 RL 1 RL To = 160 MeV 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

pT (GeV) pT(GeV)

14 Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 9 3 Equation of state and bulk viscosity 3. Equation of state and bulk viscosity

RHIC RESULTS 39 QCD is signiﬁcantly non-conformal near Tc, and conﬁnement is a smooth cross- over, not a phase transition.

Lattice results for the equa- 16.0 ! /T4 tion of state of QCD. From SB 14.0 [Karsch 2002]. SB is the energy density for free 12.0 /T4 quarks and gluons. The 20% ! 3 10.0 (3/4)s/T 4 deﬁcit in /SB is suggestive 3p/T of strong coupling. 8.0 6.0 T 170 MeV. 3 flavor, N"=4, p4 staggered c 4.0 • ≈ m =770 MeV RHIC operates at # • 2.0 T 280 MeV. T/Tc ≈ 0.0 LHC will operate at 1.0 1.5 2.0 2.5 3.0 3.5 4.0 • T 600 MeV. Figure 1: Figure of ε(T )/T 4, P (T )/T 4, and s(T )/T 3 for three light ﬂavors of ≈≈ quarks on the lattice.

Table 1: Table of RHIC Performance. Run Species Particle Energy Total Delivered Average Store [GeV/n] Luminosity Polarization Run-1 2000 Au + Au 27.9 < 0.001µb−1 - −1 Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 Au +10Au 65.2 32 Equation0 µb of state and bulk- viscosity Run-2 2001-2 Au + Au 100.0 258 µb−1 - In a bottom-up approach [Gubser andA Nelloreu + Au 20089.8], we can0.4 reproduceµb−1 the- lattice pol. p + p 100.0 1.4 µb−1 14% eos using Run-3 2002-3 d + Au 100.0 1.4 pb−1 - pol. p + p 100.0 5.5 pb−1 34% Run-14 2003-4 Au1+ Au 2 100.0 3740 µb−1 - = 2 R Au +(A∂φu ) 31V.2 (φ) . 67 µb−1 - (15) L 2κ5 −pol.2p + p −100.0 7.1 pb−1 45% − Run-5 2004-5 Cu + Cu 100.0 42.1 nb 1 - V (φ) can be adjusted to match dependenceCu + Cu of 31.2 67 µb−1 - Cu + Cu 11.2 0.02 nb−1 - pol. p + p 100.dp0 29.5 pb 1 46% speed of sound:pol. p + p c2 204.9 0.1 pb−1 30% (16) s ≡ d 2 4 on T . Then adjust κ5 to get desired /T at some high scale (say 3 GeV). Here’s a quasi-realistic choice: 12 cosh γφ + bφ2 V (φ)=− γ =0.606 ,b=2.057 . (17) L2

Authors of [Gursoy and Kiritsis 2008; Gursoy et al. 2008ab] took same starting 2 φ point (15) further: an appropriate V (φ), with V φ2e√ 3 , gives a Hawking- ∼− Page transition to confinement, logarithmic RG in UV, and glueball with m2 n, ∼ as in linear confinement. Gubser, AdS/CFT and QCD at finite T , 8-21-08 9 3 Equation of state and bulk viscosity 3. Equation of state and bulk viscosity

RHIC RESULTS 39 QCD is signiﬁcantly non-conformal near Tc, and conﬁnement is a smooth cross- over, not a phase transition.

Authors of [Gursoy and Kiritsis 2008; Gursoy et al. 2008ab] took same starting 2 φ point (15) further: an appropriate V (φ), with V φ2e√ 3 , gives a Hawking- ∼− Page transition to confinement, logarithmic RG in UV, and glueball with m2 n, ∼ as in linear confinement. Gubser, AdS/CFT and QCD at finite T , 8-21-08 11 3 Equation of state and bulk viscosity Once conformal invariance is broken, we can investigate bulk viscosity [Gubser et al. 2008cb], following a number of earlier works, e.g. [Parnachev and Starinets 2005; Buchel 2005 2007]:

1 1 3 iωt µ ν ζ = lim Im d x dt e θ(t) [T µ(t, x),T ν(0, 0)] . (18) 9 ω 0 ω → R3,1 t,x Shear viscosity relates to absorption probability for h h an h12 graviton. Bulk vis- 12 ii / $ z cosity relates to absorption of a mixture of the hii graviton and the scalar φ. p12 pii / $ ! " absorb horizon # " absorb z = z H dr2 ds2 = e2A(r) h(r)dt2 + dx2 + e2B(r) φ = φ(r) . (19) − h(r) 2A 2A 2A In a gauge where δφ =0 , let’s set h11 =e− δg11 = e− δg22 = e− δg33. Then 2A+2B 1 h e− 2 h hB h11 = 4A +3B h11 + 2 ω + h11 −3A − − h − h 6hA − h (20)

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 12 3 Equation of state and bulk viscosity

2 Type I: smooth cross- cs

• " " " " " " " " " " " " " "!"! ! ! ! ! ! ! ! ! ! ! ! ! ! over, like (17). " " "! ! ! ! " "! ! 0.30 " ! " ! "! Type II: nearly second " 0.25 ! " • order, c2 0 at T . " QPM s c !" ! ! → 0.20 lattice, 2 1 flavors " lattice, pure glue Type III: No BH below ! • 0.15 " Type I BH T , like [Gursoy et al. " c " " Type II BH 0.10 " " 2008b]. " Type III BH " 0.05

0.00 T!Tc Ζ!s 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 # Type I BH 2 " Type II BH Sharper behavior of cs gives 1.000 • sharper ζ/s. 0.500! ! Type III BH ! ! lattice, pure glue Large ζ at Tc is hard to ar- ! 0.100 • ""!"" " #"##!##"#" sum rule, 2 1 flavors range with a reasonably re- ###"" #"#"## 0.050######"""! "## ########"#"#""""" "# ##"""""" ! ""## ! "# alistic EOS. "## ! " # ! " # ! " # 0.010 ! # ! " # ! " # Poses a challenge for “soft ! " # 0.005 ! " # ! " • ! # statistical hadronization” ! " # ! " # ! ! " ! proposal of [Karsch et al. ! T!T 1.0 1.5 2.0 2.5 3.0 3.5 4.0 c 2007]. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 11 3 Equation of state and bulk viscosity Once conformal invariance is broken, we can investigate bulk viscosity [Gubser et al. 2008cb], following a number of earlier works, e.g. [Parnachev and Starinets 2005; Buchel 2005 2007]:

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 12 3 Equation of state and bulk viscosity

2 Type I: smooth cross- cs

0.00 T!Tc Ζ!s 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 # Type I BH 2 " Type II BH Sharper behavior of cs gives 1.000 • sharper ζ/s. 0.500! ! Type III BH ! ! lattice, pure glue Large ζ at Tc is hard to ar- ! 0.100 • ""!"" " #"##!##"#" sum rule, 2 1 flavors range with a reasonably re- ###"" #"#"## 0.050######"""! "## ########"#"#""""" "# ##"""""" ! ""## ! "# alistic EOS. "## ! " # ! " # ! " # 0.010 ! # ! " # ! " # Poses a challenge for “soft ! " # 0.005 ! " # ! " • ! # statistical hadronization” ! " # ! " # ! ! " ! proposal of [Karsch et al. ! T!T 1.0 1.5 2.0 2.5 3.0 3.5 4.0 c 2007]. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 13 3 Equation of state and bulk viscosity Is bulk viscosity experimentally relevant? Interesting proposal of Kharzeev and collaborators [Kharzeev and Tuchin 2007; Karsch et al. 2007]: bulk viscosity is a strong correction to hydro at T = Tc leading to last-instant entropy production accompanying freezeout:

expansion

If ζ is large, much entropy / many soft particles are produced as thermal medium expands. This depiction is in imitation of a ﬁgure in [Kharzeev]. Bottom-up calculations in AdS suggest that it’s hard to get ζ/s > 0.1 with quasi- realistic eos. If that’s right, then expansion-induced entropy is probably not so sig- niﬁcant.

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 14 4 The trailing string 4. The trailing string A heavy external quark moving at speed v experiences a drag force [Herzog et al. 2006; Gubser 2006a] (see also [Casalderrey-Solana and Teaney 2006]): dp π√λ v = T 2 . (21) dt − 2 √1 v2 − (21) arises in a simple way: a fundamental string trails out behind the quark into AdS5-Schwarzschild, pulling back upon it. q r q q v R3,1 x R3,1 AdS !Schwarzschild y 5 AdS5!Schwarzschild fundamental horizon string horizon momentum flow !(y) static drag Static force versus drag force. In both cases, the classical shape of the string is known analytically. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 13 3 Equation of state and bulk viscosity Is bulk viscosity experimentally relevant? Interesting proposal of Kharzeev and collaborators [Kharzeev and Tuchin 2007; Karsch et al. 2007]: bulk viscosity is a strong correction to hydro at T = Tc leading to last-instant entropy production accompanying freezeout:

expansion

Gubser, AdS/CFT and QCD at finite T , 8-21-08 14 4 The trailing string 4. The trailing string A heavy external quark moving at speed v experiences a drag force [Herzog et al. 2006; Gubser 2006a] (see also [Casalderrey-Solana and Teaney 2006]): dp π√λ v = T 2 . (21) dt − 2 √1 v2 − (21) arises in a simple way: a fundamental string trails out behind the quark into AdS5-Schwarzschild, pulling back upon it. q r q q v R3,1 x R3,1 AdS !Schwarzschild y 5 AdS5!Schwarzschild fundamental horizon string horizon momentum flow !(y) static drag Static force versus drag force. In both cases, the classical shape of the string is known analytically. Gubser, AdS/CFT and QCD at finite T , 8-21-08 15 4 The trailing string Mass is formally infinite, but if we use instead a finite heavy quark mass M, find dp p 2 M = where τQ = 2 , (22) dt −τQ π√λ T

So characteristic stopping length / time is τQ. To get a numerical value for τ , I favor comparing =4SYM to QCD at ﬁxed Q N energy density rather than temperature. SU(3) SYM has about 3 the number of × degrees of freedom as QCD, and I expect τQ to decrease with number of dof’s.

To ﬁx λ, I favor [Gubser 2006c] using the following effective measure of αs:

3 ∂Fqq¯ α (r, T ) r2 F is excess free energy from heavy q-q¯ pair. (23) qq¯ ≡ 4 ∂r qq¯ q r q q r q R3,1 x R3,1 y AdS5!Schwarzschild AdS5!Schwarzschild fundamental massless exchange horizon string horizon

Two string theory conﬁgurations contributing to Fqq¯. Only U-shape is fully understood. But see [Bak et al. 2007] for recent work on exchange diagram.

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 16 4 The trailing string Simplest approximation to U-curve contribution is zero temperature result: 2 3 ∂Vqq¯ 3π α (T =0) r2 = √λ . (24) SYM ≡ 4 ∂r Γ(1/4)4 To ﬁx λ 5.5, compare to lattice at largest r where U-shape dominates. ≈ " " a TSYM 190 MeV Static quark force forb =4TSYM (yellow250 MeV N 0.1 0.25 0.5 1 band) versus Nf =20.1lattice results0.25 from0.5 1 0.6 0.6 [Kaczmarek and0.6 Zantow 2005]. 0.6

0.4 ! 0.4 SYM =0.4QCD !means 0.4 • 1/4 0.2 0.2 TSYM 0.2= TQCD/3 . I took 0.2 Α Α T 250 MeV here. 0 0 QCD ≈ 0 0 An alternative perspective can be -0.2 -0.2 • -0.2 -0.2 0.1 0.25 0.5 1 found in [Sin0.1 and Zahed0.2520070.5]. 1 r r The match is conspicuously imperfect! At least we fix λ from a leading-order effect. Matching Debye length in large r tail gives even smaller λ [Bak et al. 2007]. 2 A sensible alternative is T = T with λ 6π from setting gYM = α 0.5. QCD SYM ≈ 4π s ≈ Always, N =3. Gubser, AdS/CFT and QCD at finite T , 8-21-08 15 4 The trailing string Mass is formally infinite, but if we use instead a finite heavy quark mass M, find dp p 2 M = where τQ = 2 , (22) dt −τQ π√λ T

To ﬁx λ, I favor [Gubser 2006c] using the following effective measure of αs:

Two string theory conﬁgurations contributing to Fqq¯. Only U-shape is fully understood. But see [Bak et al. 2007] for recent work on exchange diagram.

Using my preferred comparison scheme, τc 2 fm/c for charm at RHIC; also ≈ τb/τc = mb/mc. So charm equilibrates, and b does so only partially. 5

RAA and v2 for heavy quarks. pT is for 1.8 AA AA 1.4 Au+Au @ s = 200 GeV 0-10% central R (a) a non-photonicNN electron. From [Adare R 1.6 Armesto et al. (I) 1.2 et al. 2006]. 1.4 van Hees et al. (II) 1.2 3/(2!T) Moore & 1 "12/(2!T) Teaney (III) Crudely, RAA(pT ) is the % of 1 0.8 • 0.8 charm quarks escaping at a given 0.6 0.6 transverse momentum. 0.4 0.4 e± : p > 0.3 ButGeV/c p shown is for e± decay 0.2 T T Au+Au @ sNN = 200 GeV e± : p > •3.0 GeV/c 0.2 0.2 T product, so roughly double it to HF 2 0 (b) 0 ! : p > 4.0 GeV/c v ! R , p > 4 GeV/c T AA T 0 get pT of c. 0.15 minimum bias !0 v , p > 2 GeV/c 0 50 100 150 200 250 300 350 2 T e± R , e± vHF Npart AA 2 Smaller RAA and bigger v2 go 0.1 FIG. 2: RAA of hea•vy-ﬂavor electrons with pT above 0.3 and 0 3 GeV/c and of π wtogether.ith prmT > (4 GeV/c as function of 0.05 centrality given by Npart. Error bars (brackets) depict statistical (point-by-poinvant sys Heestematic curves) uncerta haveinties.τcThe r4ig.h5t fm. 0 (left) box at RAA =•1 shows the relative uncertainty fr≈om the 0 1 2 3 4 5 6 7 8 9 p [GeV/c] p+p reference common to all points for prmT > 0.3(3) GeV/c. T

Upshot: Data favors larger τc, butFI notG. 3 much: (a) RA larger,A of hea thanvy-flav stringor elect theoryrons in 0 analysis.-10% central collisions compared with π0 data [6] and model calculations measurement is addFored fo anr m alternativeinimum bias e viewpoint,vents. see e.g.(curv [eTeaneys I [30], II [200831], an];d I also,II [32]) beware. The boxbatcontribution.RAA = 1 shows HF the uncertainty in TAA. (b) v of heavy-flavor electrons in Figure 1 shows the invariant pT spectra of electrons 2 π0 from heavy-flavor decay for minimum bias events and in minimum bias collisions compared with data [29] and the same models. Errors are shown as in Fig. 2. 3 five centrality classes. The curves overlayed are the fit to the corresponding data from p+p collisions [18] with the spectral shape taken from a FONLL calculation [1case7] of a geometric path average over a static, finite, uni- medium parameters. In particular, high value extrapola- Gubser, AdS/CFT and QCD at finite T , 8-21-08 18 4 The0 trailing string and scaled by the nuclear overlap integral TAA for eacformh wplasmaith centofralthicity, knessand it Lis; lthenarger than RAA of π with tions of the qˆ parameter proposed in [26] could simulate ! " centrality class [6]. The insert in Fig. 1 shows the ratio prmT > 4 GeV/c [6]. Since above 3 GeV/c electrons Tagging b’s and c’s should be possible after detector upgrades at RHIC, and at LHC. the flat pT independent prediction from AdS/CFT. of electrons from heavy-flavor decays to background. It from charm decays originanteQmµQaLinly from D mesons with Q 1 e 1 We propose to use the double ratio of charm to bot- increases rapidly wiAth distinctiveprmT , reach differenceing one for p [rHorowitzmT p andT ab GyulassyRovAAe 4(pGTeV) /=2007c th−is]c betweenomparison pQCDindicate ands ,a s AdS/CFTlightly(2) ≈ smaller high p supprenssQioµnQoLf heav≈y-flnaQvµorQmLesons than tom RAA to amplify the observable difference between 1.5 GeV/c, reflectingpredictionsthe small am fromount of RHICmateria tol in LHCthe energies mayT come from detector acceptance. It is this large signal to background observed for light mesons. the mass and pT dependencies of the AdS/CFT drag where the pT dependence is carried entirelyHFby the spec- and pQCD-inspired energy loss models. One can see in ratio which makes the accurate measurement of heavy-t Figurem3 shows the measured RAA and v2 of heavy- HF b tralbottomindex nQ(pcharmT ). RAA can be interpreted for L Q flavor electron spectra and v2 possible. flavor electrons in 0-1for0% c AdS/CFTentral and minimum bias col- Fig. 2 that not only are most overall normalization dif- cb RAA 0 ≡ For all centralities, the Au+AuRspectra agree well wit1h/t(charmnQliµsiQon≈)s,asamnthedbottomoufractionr corresponQdi/Lng πof dthatea Q[6,jets29]. thatThe laescaptter(25)e ferences canceled, but also that the curves remarkably AA c 0 the p+p reference at low pT but a ≡supRpreAAssio∼n witunstopph areedrestfromrictedtheto psTtrronglyanges wcoupledhere RAAplasmaand v22withinof π dothe 1 pcb/pT for pQCD, pcb qLˆ bunch to either AdS/CFT-like or pQCD-like generic re-  not depend strongly on pT such that a comparison of respect to p+p develops towards high prmT . This AdS/CFTis −approximation. ∝ sults regardless of the input parameters used. quantified by the nuclear modification factor R = heavy-flavor electrons and π0 is not obscured by decay AA The numerical value of Rcb shown in Fig. 2 for dNAu+Au/( TAA dσpQCDp+p), whe predictionsre dNAu+Au is forthe differ- kinematics. The data indicate strong coupling of heavy ! " ential yield in Au+Aucband dσp+p is the differential cross quarks to the medium. The suppression is large and sim- AdS/CFT can be roughly understood analytically from RAA separate cleanly 0 section in p+p in a given p bin. For p < 1.6 GeV/c, ilar to that of π for prmT > 4 GeV/c where a significant Eq. (2) as, fromT AdS/CFTrm becauseT dσp+p, is taken bin-by-bin from [18], whereas a fit to contribution from bottom decays is expected. The large the same data (curvassumptionses in Fig. 1) is aboutused a initialt higher p , vHF shows that the charm relaxation time is compara- b rmT 2 cb Mc n (pT ) Mc taking the normalizconditionsation uncer canceltainty i out.nto a Butccount. Sys- ble to the short time scale of flow development in the RAdS c 0.26, (3) ≈ Mb n (pT ) ≈ Mb ≈ tematic uncertaintiebewares in dσp+p uncertaintyand TAA are i onncluded. produced medium. Figure 2 shows RAA for electrons from heavy-flavor More quantitative statements require theoretical guid- the limits of validity of where in this approximation all λ, T ∗, L, and nc(pT ) decays for two different pT ranges as a function of the ance. Figure 3 compares the RAA and v2 of heavy-flavor ≈ AdS/CFT. nb(pT ) dependences drop out. number of participant nucleons, Npart. For prmT > electrons with models calculating both quantities simul- 0.3 GeV/c, which contains more than half of the heavy- taneously. A perturbative QCD calculation with radia- The pQCD trend in Fig. 2 can be understood qualita- flavor decay electrons [18], RAA is close to unity for tive energy loss (curves I) [30] can describe the measured tively from the expected behavior of ¯pQCD noted above all Npart in accordance with the binary scaling of the RAA reasonably well using a large transport coefficient giving (with nc nb = n) 2 total heavy-flavor yield [19]. For prmT > 3 GeV/c, qˆ = 14 GeV /fm, which leads to a consistent descrip- ≈ the heavy flavor electron RAA decreases systematically tion of light hadron suppression as well. This value of qˆ cb pcb FIG. 2: The double ratio of Rc (p ) to Rb (p ) predictions RpQCD 1 , (4) Related studies by Brasoveanu and d’Enterria are in progress.AA T AA T ≈ − pT for LHC using Eq. (1) for AdS/CFT and WHDG [25] for pQCD with a wide range of input parameters. The generic 2 where pcb = κn(pT )L log(Mb/Mc)qˆ sets the relevant mo- difference between the pQCD results tending to unity con- mentum scale. Thus Rcb 1 more slowly for higher trasted to the much smaller and nearly pT -independent results → from AdS/CFT can be easily distinguished at LHC. opacity. One can see this behavior reflected in the full numerical results shown in Fig. 2 for moderate suppression, but that the extreme opacity qˆ = 100 case deviates Two implementations of pQCD energy loss are used in from Eq. (4). this paper. The first is the full WHDG model convolving The maximum momentum for which string theoretic fluctuating elastic and inelastic loss with fluctuating path predictions for Rcb can be trusted is not well understood. geometry [25]. The second restricts WHDG to include Eq. (1) was derived assuming a constant heavy quark only radiative loss in order to facilitate comparison to velocity. Supposing this is maintained by the presence [30]. Note that when realistic nuclear geometries with of an electromagnetic field, the Born-Infeld action gives Bjorken expansion are used, the “fragility” of R for AA a “speed limit” of γ = M 2/λ(T )2 [37]. The work of large qˆ reported in [36] is absent in both implementations c ∗ [19] relaxed the assumptions of infinite quark mass and of WHDG. constant velocity; nevertheless Eq. (1) well approximates Unlike the AdS/CFT dynamics, pQCD predicts the full results. Requiring a time-like endpoint on the [23, 24, 25] that the average energy loss fraction probe brane for a constant velocity string representing a in a static uniform plasma is approximately ¯ 2 ≈ finite mass quark leads to [21] a parametrically similar κL qˆlog(pT /MQ)/pT , with κ a proportionality constant 2 cutoff, and qˆ = µD/λg. The most important feature in pQCD relative to AdS/CFT is that ¯ 0 asymptotically 2 pQCD → 2M 4M 2 at high-pT while ¯AdS remains constant. nQ(pT ) is a γc = 1 + 2 . (5) pQCD √λT ∗ ≈ λ(T ∗) slowly increasing function of momentum; thus RAA AdS increases with pT whereas RAA decreases. This generic There is no known limit yet for the dynamic velocity difference can be observed in Fig. 1, which shows repre- case. To get a sense of the pT scale where the AdS/CFT sentative predictions from the full numerical calculations approximation may break down, we plot the momentum of charm and bottom RAA(pT ) at LHC. cutoffs from Eq. (5) for the given SYM input parameters Q Double Ratio of charm to bottom RAA A disadvantage corresponding to T ∗(τ0) and Tc∗. These are depicted by of the RQ (p ) observable alone is that its normaliza- “O” and “ ” in the figures, respectively. AA T | tion and slow pT dependence can be fit with different Conclusions Possible strong coupling deviations from model assumptions compensated by using very different pQCD in nuclear collisions were studied based on a recent Gubser, AdS/CFT and QCD at finite T , 8-21-08 17 4 The trailing string

Using my preferred comparison scheme, τc 2 fm/c for charm at RHIC; also ≈ τb/τc = mb/mc. So charm equilibrates, and b does so only partially. 5

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 20 4 The trailing string q v x R3,1

y timelike AdS5!Schwarzschild signals go yv this way spacelike Stochastic ﬂuctuations are controlled horizon by causal horizon on the worldsheet. The horizon on the worldsheet is at y = yv. AdS5-Schwarzschild geometry is L2π2T 2 1 dy2 ds2 = (1 y4)dt2 + dx2 + . (29) 5 y2 − − π2T 2 1 y4 − Consider observers who stay at ﬁxed y while holding onto the trailing string:

dτ 2 > 0 if y>y √4 1 v2: “outside” the worldsheet black hole. • v ≡ − dτ 2 < 0 if y

Something roughly like Hawking radiation must emanate from the worldsheet horizon, leading to stochastic F (t). Actual computations directly access F (t )F (t ) . i 1 j 2 Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 19 4 The trailing string 4.1. Stochastic forces on heavy quarks Drag force is not the whole story: in a Langevin description [Casalderrey-Solana and Teaney 2006; Gubser 2006b; Casalderrey-Solana and Teaney 2007] dp π√λT 2 = ηp + F (t) η = (26) dt − 2m where F is a stochastic force: if p is in the 1ˆ direction, then T 3 F (t )F (t ) κ δ(t t ) , κ = π√λ 1 1 1 2 ≈ L 1 − 2 L (1 v2)5/4 − (27) T 3 Fi(t1)Fj(t2) κT δijδ(t1 t2) , κT = π√λ ≈ − √4 1 v2 − String theory value for κL exceeds Einstein relation except near v =0: 1 κL = 2TEη , (28) (1 v2)3/4 − hinting that Langevin description doesn’t capture all the physics. Also: correlation time in F (t) diverges as 1/√4 1 v2. −

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 20 4 The trailing string q v x R3,1

dτ 2 > 0 if y>y √4 1 v2: “outside” the worldsheet black hole. • v ≡ − dτ 2 < 0 if y

Something roughly like Hawking radiation must emanate from the worldsheet horizon, leading to stochastic F (t). Actual computations directly access F (t )F (t ) . i 1 j 2 2 Jiangyong Jia for the PHENIX Collaboration

malization factor that is fixed by the experimentally measured RAA, which varies with energy loss models. But single hadron yield can’t constrain the shape of ! P (∆E) = dφd−→r P (∆E, φ, −→r ), hence the dynamics of energy loss models, very well (the fragility [ 5]). ! look for the jet on the other side To regain sensitivity to the energy loss mechanisms, one inevitably need to Gubser, AdS/CFT and QCD at finitesTtu, 8-21-08dy the distribution o21f the energy lost by the partons via two5- or Jet-splitting?multi-particle STAR PRL 90, 082302 (2003) Peripheral Au + Au correlations. For intermediate pT charged hadron pairs, the away-side jet was observed to peak at ∆φ π 1.1 [ 6, 7], suggesting that the energy lost by high 5. Jet-splitting? ∼ ± pT partons is transported to lower pT hadrons at angles away from ∆φ π. The proposed mechanisms for such energy transport include medium deflection∼of hard [ 8] or showAer hardparto processns [ 9], occurringlarge-angle neargluo then ra edgediatio ofn the[ 10, medium11], Cherenkov gluon radiation [produces12], and “ aM near-sideach Shock “trigger”” medium jetex (red).citatio Thens [ 1 away-side3]. In thispartonprocee interactsding, we stronglydiscuss so withme a thespec medium.ts of the d Fromi-hadr [oJacakn correlation results from PHE2006NIX,].with an eye to constrain the possible energy loss mechanism as well the response of the partonic matter to the energy loss.

2. AwayJet-sid reconstructione pT scan: Me isdiu impractical,m-induced socom makepone histogramsnt and Punch- Central Au + Au through component of azimuthal separation between two energetic hadrons. 0.4 3-4 ! 0.4-1 GeV/c 3-4 ! 1-2 GeV/c 3-4 ! 2-3 GeV/c 0.3 Au-Au 0-20% ! 1.5 ! 6.0 p - p PHENIX Preliminary 0.2 WithM appropriateedium ipsT ocuts,paq ob-ue! S H S

" 0.1 serve_ h ai double-humpgh densi structurety # on away-side: “jet-splitting.” 0 dN/d From [ Jia la2007rge]. !interaction A 3-4 ! 3-4 GeV/c 4-5 ! 4-5 GeV/c 5-10 ! 5-10 GeV/c ! 2.5 ! 2.5 More inclusive cuts ﬁll in 1/N 0.02 the region around ∆φ = π: 0.01 “jet-broadening” [Adams et al. 2005]. 0 0 2 4 0 2 4 0 2 4 #" (rad)

A B Fig. 1. Per-trigger yield vs. ∆φ for various trigger and partner pT (pT pT ) in p+p and 0-20% Au+Au collisions. The Data in some panels are scaled as ind⊗icated. Solid lines (shaded bands) indicate elliptic ﬂow (ZYAM method) uncertainties. Arrows Gubser, AdS/CFT and QCD at ﬁniteiTn,F 8-21-08ig. 1c depict the “H22ead Region” (H) and the “Shoulder Reg5ion Jet-splitting?” (S). A string theory calculation has been done for heavy quarks:[Gubser et al. 2007; Chesler and Yaffe 2007] andFig.1 refsshow therein.s the per-trigger yield distributions for p + p and central Au+Au

A heavy quark trails a string mn v behind it. The string cou- T q ples to gravitons dual to 3,1 T in the gauge theory. R mn hmn AdS5 !Schwarzschild Calculate hmn using lin- earized Einstein equations. fundamental string One big calculation gives horizon T 0m over a broad range of scales; high k asymptotics pioneered in [Yarom 2007] turn out to be especially interesting.

Render all quantities dimensionless: 2 √1 v 0i 0i X = πTxSi(X ) − T (0, x) T (0, x) . (30) ≡ (πT )4√λ − Coulomb 2 Jiangyong Jia for the PHENIX Collaboration

Render all quantities dimensionless: 2 √1 v 0i 0i X = πTxSi(X ) − T (0, x) T (0, x) . (30) ≡ (πT )4√λ − Coulomb E for v!0.75 E for v!0.75 0.2 14 0.00296125 0.285598 12 0.15 10 p 8 p 0.1 X 6 X 4 0.05 2 " " 0 0.00273632 0 0.092848 -15 -10 -5 0 5 10 15 -0.2 -0.1 0 0.1 0.2 X1 X1 S1 for v!0.75 S1 for v!0.75 0.5 14 0.00221565 0.161937 12 0.4 10 0.3 p 8 p X 6 X 0.2 4 0.1 2 " " 0 0.00139363 0 0.0807396 -15 -10 -5 0 5 10 15 -0.4 -0.2 0 0.2 0.4

X1 Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08X1 23 5 Jet-splitting? Sp for v!0.75 S for v!0.75 14 0.0010351 14 12 12 10 10 p p 8 8 X X 6 6 4 4 2 2 "0.000821425 0 0 -15 -10 -5 0 5 10 15 -15 -10 -5 0 5 10 15 X1 X1 Rescaled, subtracted Poynting vector generated by a quark in an inﬁnite, static medium. Green shows the Mach angle, and blue shows the parabolic boundary of the diffusion wake. For T ≈ 318 MeV, X =5is a distance 1 fm from the quark. From [Gubser et al. 2007]. | |

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 24 5 Jet-splitting? A phenomenological comparison [Betz et al. 2008] including Cooper- Frye hadronization shows that AdS/CFT does lead to jet-splitting at p 5 GeV. T ≈ But the reason is unexpected: it’s not the hydro region that does it, it’s the “neck” region with x < 1 fm. | | ∼ Puzzles / problems remain:

Pseudo-Mach angle is smaller than • data, and gets smaller as v 1. → This was for heavy quarks! • Cooper-Frye isn’t perfect. • Interpretation of experimental phe- • nomenon isn’t universally agreed upon. E for v!0.75 E for v!0.75 0.2 14 0.00296125 0.285598 12 0.15 10 p 8 p 0.1 X 6 X 4 0.05 2 " " 0 0.00273632 0 0.092848 -15 -10 -5 0 5 10 15 -0.2 -0.1 0 0.1 0.2 X1 X1 S1 for v!0.75 S1 for v!0.75 0.5 14 0.00221565 0.161937 12 0.4 10 0.3 p 8 p X 6 X 0.2 4 0.1 2 " " 0 0.00139363 0 0.0807396 -15 -10 -5 0 5 10 15 -0.4 -0.2 0 0.2 0.4

Pseudo-Mach angle is smaller than • data, and gets smaller as v 1. → This was for heavy quarks! • Cooper-Frye isn’t perfect. • Interpretation of experimental phe- • nomenon isn’t universally agreed upon. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 25 6 Jet quenching 6. Jet quenching According to pQCD (e.g. [Baier et al. 1997; Zakharov 1997; Wiedemann 2000]), radiative energy loss by light quarks and gluons is 1 ∆E = α C qˆ(∆x)2 , (31) 4 s R where the jet-quenching parameter describes how fast momentum broadens as a function of path length ∆x: p2 qˆ = ⊥ . (32) ∆x

Authors including [Kovner and Wiedemann 2003; t Liu et al. 2006] prefer a deﬁnition in terms of a partially light-like Wilson loop with L ∆x: x 1 W adjoint( ) exp qLˆ 2∆x . (33) z C ≈ −4 A gauge-string calculation of W fundamental leads to C L π3/2Γ(3/4) qˆ = √λ T 3 . (34) Γ(5/4) !x

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 26 6 Jet quenching

A correction factor sQCD/sSYM is advocated in [Liu et al. 2007] to correct for fewer degrees of freedom. Including this factor and using λ =6π, as they prefer, I calculate GeV2 qˆ 2.3 at T = 280 MeV, (35) ≈ fm signiﬁcantly above pQCD’s qˆ 0.77 GeV2/fm and almost big enough to agree ≈ with experiment (more later). But some puzzles remain:

Argyres and collaborators criticize the choice of saddle point [Argyres et al. • 2007 2008] and find log W A( ) L not L2. C ∼ qˆ as defined through Wilson loop may not be directly related to energy loss or • momentum diffusion in strongly coupled gauge theories. 2 Independent calculations of qˆT p /∆x for heavy quarks [Herzog et al. • ≡⊥ 2006; Casalderrey-Solana and Teaney 2006 2007; Gubser 2006b] lead to larger values than (34): larger by √γ as v 1. ∼ → Gubser, AdS/CFT and QCD at finite T , 8-21-08 25 6 Jet quenching 6. Jet quenching According to pQCD (e.g. [Baier et al. 1997; Zakharov 1997; Wiedemann 2000]), radiative energy loss by light quarks and gluons is 1 ∆E = α C qˆ(∆x)2 , (31) 4 s R where the jet-quenching parameter describes how fast momentum broadens as a function of path length ∆x: p2 qˆ = ⊥ . (32) ∆x

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 26 6 Jet quenching

From [Adare et al. 2008], with minor additions. Pre- dictions of PQM model [Dainese et al. 2005] versus PHENIX data. All values of q 2.3 qˆ are in GeV2/fm. q 7 Best ﬁt curve (red) has 2 q 13.2 qˆ = 13.2 GeV /fm. q 28 3σ range is 7 < q<ˆ 28 GeV2/fm. Parton Quenching Model is based on many soft momentum transfers between medium and hard partons. Other formalisms exist (see e.g. [Gyulassy et al. 2001; Arnold et al. 2002; Wang 2004]) for connecting pQCD to data.

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 28 7 Falling strings 7. Falling strings Can we calculate ab initio the energy loss of a gluon in strongly coupled =4? N We propose [Gubser et al. 2008a] to regard an off-shell gluon as a doubled string with both ends passing through the horizon.

energetic off!shell gluon

on!shell gluon ? D3 D3’s T=0 T=0

energetic off!shell gluon

on!shell gluon ? D3 D3’s T=0 T=0

At zero temperature, results of [Alday and Maldacena 2007] show that gluon scattering produces approximately this type of string configuration. At finite temperature, something funny happens: where the string crosses the horizon, it can’t move! (Infinite red-shifting wrt Killing time t.) Gubser, AdS/CFT and QCD at finite T , 8-21-08 29 7 Falling strings A doubled string starts at x1 R3,1 y=0 t =0with some total v(y) y=y energy and virtuality, then y UV AdS !Schwarzschild falls into the horizon over a 5 distance ∆x. t=0 t=1

y=1 horizon ! x

Given initial E, what is ∆x? • Answer must depend on virtuality y , so what is maximum ∆x? • ↔ UV How do we roughly convert the answer to qˆ? • We made estimates based on assuming the shape of the falling string quickly approaches a segment of the trailing string; conﬁrmed numerically in [Chesler et al. 2008]. For E T , we found ∆xˆ Eˆ1/3 (see also [Hatta et al. 2008]), where ≈ 1 E xˆ = πTx Eˆ = . 2 (36) gYMN T

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 30 7 Falling strings This is not too different from pQCD prediction ∆x E/qˆ. So let’s convert to a ∝ rough prediction of qˆ: 4E qˆrough 2 . (37) ≡ 3αs(∆x)

alternative scheme ! PHENIX 3! range q !GeV2#fm" $ $spacetime, UV% 30! ! ! ! ! ! ! ! ! ! ! " " " " " " " " " " " " 25 # $spacetime, trailing, yIR 0.9% $ $ $ 20 $ $ $ $ $ $ $ $ ! 1 # # # # # # # # # # # $spacetime, fixed x % ! ! ! ! ! ! ! ! ! ! ! 15 " ! $POND, trailing, yIR 0.9, cutoff% 10

5 LRW " $ 1% scaled LRW POND, fixed x pQCD ! " 6 8 10 12 14 E GeV

Estimates of the jet-quenching parameter, from (37), comparing at ﬁxed energy density, with λ =5.5. Different symbols correspond to varying assumptions about shape of falling string. From [Gubser et al. 2008a]. LRW is from [Liu et al. 2006] at 280 MeV, for SYM; scaled LRW is for

QCD at 280 MeV, including the sQCD/sSYM factor from [Liu et al. 2007]. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 29 7 Falling strings A doubled string starts at x1 R3,1 y=0 t =0with some total v(y) y=y energy and virtuality, then y UV AdS !Schwarzschild falls into the horizon over a 5 distance ∆x. t=0 t=1

y=1 horizon ! x

5 LRW " $ 1% scaled LRW POND, fixed x pQCD ! " 6 8 10 12 14 E GeV

QCD at 280 MeV, including the sQCD/sSYM factor from [Liu et al. 2007]. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 31 7 Falling strings The overall picture on jet-quenching is, in my view, somewhat muddled at present:

Good that we’re within 3σ range, or close. • Good that we can accommodate gluons that start off signiﬁcantly virtual. • Questionable to compare qˆ from falling strings to a value in PQM model, where • underlying assumptions are different. Bad that we don’t understand relation among jet-quenching calculations, plus • heavy quark drag / diffusion. Interesting to consider including ﬂuctuations or graviton response, starting either • from [Liu et al. 2006] or [Gubser et al. 2008a]. Maybe good that numerical study [Chesler et al. 2008] shows larger ∆x (so • smaller qˆ) for falling strings; or was that due to initial conditions?

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 32 8 Total multiplicity 8. Total multiplicity Central RHIC collision: N 2 197 = 394 nucleons in part ≈ × N 5000 charged particles out. ch ≈ A reasonable estimate of the entropy produced is

S 7.5N 38000 , (38) ≈ charged ≈ (E.g. consider a gas of free hadrons at T and compute S/N c charged Charged tracks measured by STAR in a gold-gold col- starting from partition function.) lision [STA]. For multiplicitly estimates, see e.g. PHO- BOS’s [Back et al. 2005]. How well can we estimate S from the gauge-string duality? Strategy of [Gubser et al. 2008d]:

Replace QCD by a conformal theory with /T 4 = 11, as lattice predicts for • QCD for T > 1.2Tc. (Remarkably slow rise thereafter.) ∼ Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 31 7 Falling strings The overall picture on jet-quenching is, in my view, somewhat muddled at present:

Replace QCD by a conformal theory with /T 4 = 11, as lattice predicts for • QCD for T > 1.2Tc. (Remarkably slow rise thereafter.) ∼ Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 33 8 Total multiplicity Replace a heavy ion with a boosted “conformal soliton,” dual to a point-sourced • 0 3 gravitational shock wave in AdS : if x− = x x , then 5 − 2EL T = δ(x−) , (39) −− π [(x1)2 +(x2)2 + L2]3

(Power law tails are not a good thing, but at least they’re a big power: 1/x6 .) ⊥

R3,1 x 3 x1,2 z A standard but non-rigorous z=L lower bound is SS 1 2 S S ≥ trapped A /4G . ≡ trapped 5 H3 C H3

Earlier related work is Trapped surface is typically on past light-like trajectory reviewed in [Nastase 2008]. of shocks; shown here is projection to t =0.

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 34 8 Total multiplicity The ﬁnal result is

3 1/3 2/3 L √sNN S π (2EL)2/3 35000 . (40) trapped ≈ G ≈ 200 GeV 5

I set L =4.3 fm to match the rms transverse radius of a gold nucleus. • E 19.7 GeV is beam energy; √sNN = 200 GeV is cm energy of a pair of • ≈ nucleons (NN).

2/3 R3,1 E scaling is faster than Landau (E1/2)[Landau 1953] and faster than pQCD discard this region data ( Landau). ≈ I think it’s because strong-coupling conformal window covers only a confinement discard this region range of scales. A crude solution [Gubser et al.]: Assume that most entropy is generated within this range, above conﬁnement and below pQCD. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 33 8 Total multiplicity Replace a heavy ion with a boosted “conformal soliton,” dual to a point-sourced • 0 3 gravitational shock wave in AdS : if x− = x x , then 5 − 2EL T = δ(x−) , (39) −− π [(x1)2 +(x2)2 + L2]3

(Power law tails are not a good thing, but at least they’re a big power: 1/x6 .) ⊥

R3,1 x 3 x1,2 z A standard but non-rigorous z=L lower bound is SS 1 2 S S ≥ trapped A /4G . ≡ trapped 5 H3 C H3

Earlier related work is Trapped surface is typically on past light-like trajectory reviewed in [Nastase 2008]. of shocks; shown here is projection to t =0.

Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 34 8 Total multiplicity The ﬁnal result is

3 1/3 2/3 L √sNN S π (2EL)2/3 35000 . (40) trapped ≈ G ≈ 200 GeV 5

I set L =4.3 fm to match the rms transverse radius of a gold nucleus. • E 19.7 GeV is beam energy; √sNN = 200 GeV is cm energy of a pair of • ≈ nucleons (NN).

2/3 R3,1 E scaling is faster than Landau (E1/2)[Landau 1953] and faster than pQCD discard this region data ( Landau). ≈ I think it’s because strong-coupling conformal window covers only a confinement discard this region range of scales. A crude solution [Gubser et al.]: Assume that most entropy is generated within this range, above conﬁnement and below pQCD. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 35 8 Total multiplicity UV cutoff changes scaling from S E2/3 to E1/3 at large E. So anticipate trapped ∼ N E1/3. Maybe even for protons? charged ∼ Roll-over from Landau’s E1/2 to slower growth might just be starting at top RHIC energies:Landau Hydrodynamics & RHIC Phenomenology 3

) 40 ! pp(pp) Data @ s/2 + - /2 e e Data Fit to e+e- part 30 PHOBOS N

" PHOBOS interp.

(/ NA49 ! E895

ch 20 Landau + µB N

" Landau 10

0 1 10 102 s(GeV)

TotalFig. multiplicity1. Charged perpa participantrticle mult asipl aic functionities for ofA+ energy.A, p+p From(wit [hSteinbergleading p2005artic].le eﬀect removed) and e+e−. Theoretical curves are pQCD (dotted line) [ 12], baryon-free Landau (solid line) [ 6], and Landau including the baryochemical potential eﬀect (dot-dashed) [ 19].

Gubser, AdS/CFTregio andn QCDof ph ata finitese sTp,a 8-21-08ce) in all collision36s involving nuclei, from p + A to Au +9Au Outlook 9. Outlookcollisions [ 13, 14]. Gauge-string3. Therm /a Heavy-ionl Phenom connectionenology a isnd theHa closestdroche interfacemistry we have between • modernIn th stringe Land theoryau scena andrio, modernfreezeout experiment.is not expected to occur immediately, as Fermi assumed, but rather when the temperature reaches the limit of the pion Compton Many comparisons are successful at a semi-quantitative level. (Many more than • wavelength T = mπ. This was based on a suggestion by Pomeranchuk [ 15] to avoid I haveFer summarizedmi’s prediction here...)that nucleons would outnumber pions by virtue of their larger statistical weight. This assumption leads to predictions for the relative population Comparisonsof various p areartic invariablyle states sim plaguedilar to tho byse m theade difficultyin the Hage ofdor translatingn approach [ from16, 17AdS]. • calculationsA+A tocol real-worldlisions clearly QCD.deviate from the Fermi-Landau formula at low energies. An obvious suspect is the phenomenon of baryon stopping, which is absent in p+p We maycollisi oftenons but beis s measuringubstantial in ourA+A successes[ 18]. If one againstputs bac prevailingk the µBN interpretationsB term into the of • first law of thermodynamics, we immediately see how the pr−esence of a conserved dataq ratheruantity thanassoc dataiated itself.with a substantial mass (i.e. the proton mass) will naturally suppress the total entropy: S = (E + pV µ N )/T . Using an existing thermal At the least, gauge-string calculations show− B whatB happens in a truly strongly • model code, Cleymans and Stankiewicz [ 19] calculated the entropy density as a coupledfunct thermalion of √s. plasma.It rises to limiting value where µB 0 and T T0, the Hagedorn temperature. If we then assume that the total mul→tiplicity sca→les linearly with the Insights from AdS/CFT complement pQCD intuitions and may sometimes be • closer to capturing the true dynamics. Gubser, AdS/CFT and QCD at finite T , 8-21-08 35 8 Total multiplicity UV cutoff changes scaling from S E2/3 to E1/3 at large E. So anticipate trapped ∼ N E1/3. Maybe even for protons? charged ∼ Roll-over from Landau’s E1/2 to slower growth might just be starting at top RHIC energies:Landau Hydrodynamics & RHIC Phenomenology 3

) 40 ! pp(pp) Data @ s/2 + - /2 e e Data Fit to e+e- part 30 PHOBOS N

" PHOBOS interp.

(/ NA49 ! E895

ch 20 Landau + µB N

" Landau 10

0 1 10 102 s(GeV)

Gubser, AdS/CFTregio andn QCDof ph ata finitese sTp,a 8-21-08ce) in all collision36s involving nuclei, from p + A to Au +9Au Outlook 9. Outlookcollisions [ 13, 14]. Gauge-string3. Therm /a Heavy-ionl Phenom connectionenology a isnd theHa closestdroche interfacemistry we have between • modernIn th stringe Land theoryau scena andrio, modernfreezeout experiment.is not expected to occur immediately, as Fermi assumed, but rather when the temperature reaches the limit of the pion Compton Many comparisons are successful at a semi-quantitative level. (Many more than • wavelength T = mπ. This was based on a suggestion by Pomeranchuk [ 15] to avoid I haveFer summarizedmi’s prediction here...)that nucleons would outnumber pions by virtue of their larger statistical weight. This assumption leads to predictions for the relative population Comparisonsof various p areartic invariablyle states sim plaguedilar to tho byse m theade difficultyin the Hage ofdor translatingn approach [ from16, 17AdS]. • calculationsA+A tocol real-worldlisions clearly QCD.deviate from the Fermi-Landau formula at low energies. An obvious suspect is the phenomenon of baryon stopping, which is absent in p+p We maycollisi oftenons but beis s measuringubstantial in ourA+A successes[ 18]. If one againstputs bac prevailingk the µBN interpretationsB term into the of • first law of thermodynamics, we immediately see how the pr−esence of a conserved dataq ratheruantity thanassoc dataiated itself.with a substantial mass (i.e. the proton mass) will naturally suppress the total entropy: S = (E + pV µ N )/T . Using an existing thermal At the least, gauge-string calculations show− B whatB happens in a truly strongly • model code, Cleymans and Stankiewicz [ 19] calculated the entropy density as a coupledfunct thermalion of √s. plasma.It rises to limiting value where µB 0 and T T0, the Hagedorn temperature. If we then assume that the total mul→tiplicity sca→les linearly with the Insights from AdS/CFT complement pQCD intuitions and may sometimes be • closer to capturing the true dynamics. Gubser, AdS/CFT and QCD at finite T , 8-21-08 37 9 Outlook ptReferences STAR collision image, from http://www.bnl.gov/RHIC/full en images.htm. J. Adams et al. “Distributions of charged hadrons associated with high transverse momentum particles in p p and Au + Au collisions at s(NN)**(1/2) = 200-GeV”. Phys. Rev. Lett., 95:152301, 2005, nucl-ex/0501016.

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Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 38 9 Outlook Peter Arnold, Guy D. Moore, and Laurence G. Yaffe. “Photon and gluon emission in relativistic plasmas”. JHEP, 06:030, 2002, hep-ph/0204343. B. B. Back et al. “The PHOBOS perspective on discoveries at RHIC”. Nucl. Phys., A757:28–101, 2005, nucl-ex/0410022. R. Baier, Yuri L. Dokshitzer, Alfred H. Mueller, S. Peigne, and D. Schiff. “Radiative energy loss and p(T)-broadening of high energy partons in nuclei”. Nucl. Phys., B484:265–282, 1997, hep-ph/9608322. Dongsu Bak, Andreas Karch, and Laurence G. Yaffe. “Debye screening in strongly coupled N = 4 supersymmetric Yang-Mills plasma”. 2007, arXiv:0705.0994 [hep-th]. Mark D. Baker. “The latest Results from PHOBOS: Systematics of Charged Parti-

cle Production through √sNN = 200 GeV”, 2001. PowerPoint talk, available at http://www.phobos.bnl.gov/Presentations/paris01/. Barbara Betz, Miklos Gyulassy, Jorge Noronha, and Giorgio Torrieri. “Anomalous Conical Di-jet Correlations in pQCD vs AdS/CFT”. 2008, 0807.4526. J. P. Blaizot, E. Iancu, U. Kraemmer, and A. Rebhan. “Hard thermal loops and the entropy of supersymmetric Yang- Mills theories”. 2006, hep-ph/0611393. Mauro Brigante, Hong Liu, Robert C. Myers, Stephen Shenker, and Sho Yaida. “Viscosity Bound Violation in Higher Derivative Gravity”. Phys. Rev., D77:126006, 2008, 0712.0805. doi: 10.1103/ PhysRevD.77.126006. Gubser, AdS/CFT and QCD at ﬁnite T , 8-21-08 39 9 Outlook Alex Buchel. “Transport properties of cascading gauge theories”. Phys. Rev., D72:106002, 2005, hep-th/0509083. Alex Buchel. “Bulk viscosity of gauge theory plasma at strong coupling”. 2007, arXiv:0708.3459 [hep-th]. Alex Buchel and James T. Liu. “Universality of the shear viscosity in supergravity”. Phys. Rev. Lett., 93:090602, 2004, hep-th/0311175. doi: 10.1103/PhysRevLett.93.090602. Alex Buchel, James T. Liu, and Andrei O. Starinets. “Coupling constant dependence of the shear viscosity in N = 4 supersymmetric Yang-Mills theory”. Nucl. Phys., B707:56–68, 2005, hep-th/0406264. Jorge Casalderrey-Solana and Derek Teaney. “Heavy quark diffusion in strongly coupled N = 4 Yang Mills”. 2006, hep-ph/0605199. Jorge Casalderrey-Solana and Derek Teaney. “Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma”. 2007, hep-th/0701123. Paul M. Chesler and Laurence G. Yaffe. “The stress-energy tensor of a quark moving through a strongly- coupled N=4 supersymmetric Yang-Mills plasma: comparing hydrodynamics and AdS/CFT”. 2007, 0712.0050. Paul M. Chesler, Kristan Jensen, and Andreas Karch. “Jets in strongly-coupled N = 4 super Yang-Mills theory”. 2008, 0804.3110. A. Dainese, C. Loizides, and G. Paic. “Leading-particle suppression in high energy nucleus nucleus collisions”. Eur. Phys. J., C38:461–474, 2005, hep-ph/0406201.

Gubser, AdS/CFT and QCD at finite T , 8-21-08 40 9 Outlook A. Fotopoulos and T. R. Taylor. “Comment on two-loop free energy in N = 4 supersymmetric Yang- Mills theory at finite temperature”. Phys. Rev., D59:061701, 1999, hep-th/9811224. S. S. Gubser, Igor R. Klebanov, and A. W. Peet. “Entropy and Temperature of Black 3-Branes”. Phys. Rev., D54:3915–3919, 1996, hep-th/9602135. S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov. “Gauge theory correlators from non- critical string theory”. Phys. Lett., B428:105–114, 1998a, hep-th/9802109. Steven S. Gubser. “Drag force in AdS/CFT”. 2006a, hep-th/0605182. Steven S. Gubser. “Momentum fluctuations of heavy quarks in the gauge-string duality”. 2006b, hep-th/0612143. Steven S. Gubser. “Comparing the drag force on heavy quarks in N = 4 super- Yang-Mills theory and QCD”. 2006c, hep-th/0611272. Steven S. Gubser and Abhinav Nellore. “Mimicking the QCD equation of state with a dual black hole”. 2008, 0804.0434. Steven S. Gubser, Silviu S. Pufu, and Amos Yarom. Forthcoming. Steven S. Gubser, Igor R. Klebanov, and Arkady A. Tseytlin. “Coupling constant dependence in the thermodynamics of N = 4 supersymmetric Yang-Mills theory”. Nucl. Phys., B534:202–222, 1998b, hep-th/9805156. Steven S. Gubser, Silviu S. Pufu, and Amos Yarom. “Sonic booms and diffusion wakes generated by a heavy quark in thermal AdS/CFT”. 2007, arXiv:0706.4307 [hep-th]. Gubser, AdS/CFT and QCD at finite T , 8-21-08 39 9 Outlook Alex Buchel. “Transport properties of cascading gauge theories”. Phys. Rev., D72:106002, 2005, hep-th/0509083. Alex Buchel. “Bulk viscosity of gauge theory plasma at strong coupling”. 2007, arXiv:0708.3459 [hep-th]. Alex Buchel and James T. Liu. “Universality of the shear viscosity in supergravity”. Phys. Rev. Lett., 93:090602, 2004, hep-th/0311175. doi: 10.1103/PhysRevLett.93.090602. Alex Buchel, James T. Liu, and Andrei O. Starinets. “Coupling constant dependence of the shear viscosity in N = 4 supersymmetric Yang-Mills theory”. Nucl. Phys., B707:56–68, 2005, hep-th/0406264. Jorge Casalderrey-Solana and Derek Teaney. “Heavy quark diffusion in strongly coupled N = 4 Yang Mills”. 2006, hep-ph/0605199. Jorge Casalderrey-Solana and Derek Teaney. “Transverse Momentum Broadening of a Fast Quark in a N=4 Yang Mills Plasma”. 2007, hep-th/0701123. Paul M. Chesler and Laurence G. Yaffe. “The stress-energy tensor of a quark moving through a strongly- coupled N=4 supersymmetric Yang-Mills plasma: comparing hydrodynamics and AdS/CFT”. 2007, 0712.0050. Paul M. Chesler, Kristan Jensen, and Andreas Karch. “Jets in strongly-coupled N = 4 super Yang-Mills theory”. 2008, 0804.3110. A. Dainese, C. Loizides, and G. Paic. “Leading-particle suppression in high energy nucleus nucleus collisions”. Eur. Phys. J., C38:461–474, 2005, hep-ph/0406201.

Gubser, AdS/CFT and QCD at finite T , 8-21-08 40 9 Outlook A. Fotopoulos and T. R. Taylor. “Comment on two-loop free energy in N = 4 supersymmetric Yang- Mills theory at finite temperature”. Phys. Rev., D59:061701, 1999, hep-th/9811224. S. S. Gubser, Igor R. Klebanov, and A. W. Peet. “Entropy and Temperature of Black 3-Branes”. Phys. Rev., D54:3915–3919, 1996, hep-th/9602135. S. S. Gubser, Igor R. Klebanov, and Alexander M. Polyakov. “Gauge theory correlators from non- critical string theory”. Phys. Lett., B428:105–114, 1998a, hep-th/9802109. Steven S. Gubser. “Drag force in AdS/CFT”. 2006a, hep-th/0605182. Steven S. Gubser. “Momentum fluctuations of heavy quarks in the gauge-string duality”. 2006b, hep-th/0612143. Steven S. Gubser. “Comparing the drag force on heavy quarks in N = 4 super- Yang-Mills theory and QCD”. 2006c, hep-th/0611272. Steven S. Gubser and Abhinav Nellore. “Mimicking the QCD equation of state with a dual black hole”. 2008, 0804.0434. Steven S. Gubser, Silviu S. Pufu, and Amos Yarom. Forthcoming. Steven S. Gubser, Igor R. Klebanov, and Arkady A. Tseytlin. “Coupling constant dependence in the thermodynamics of N = 4 supersymmetric Yang-Mills theory”. Nucl. Phys., B534:202–222, 1998b, hep-th/9805156. Steven S. Gubser, Silviu S. Pufu, and Amos Yarom. “Sonic booms and diffusion wakes generated by a heavy quark in thermal AdS/CFT”. 2007, arXiv:0706.4307 [hep-th]. Gubser, AdS/CFT and QCD at finite T , 8-21-08 41 9 Outlook Steven S. Gubser, Daniel R. Gulotta, Silviu S. Pufu, and Fabio D. Rocha. “Gluon energy loss in the gauge-string duality”. 2008a, 0803.1470. Steven S. Gubser, Abhinav Nellore, Silviu S. Pufu, and Fabio D. Rocha. “Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics”. 2008b, 0804.1950. Steven S. Gubser, Silviu S. Pufu, and Fabio D. Rocha. “Bulk viscosity of strongly coupled plasmas with holographic duals”. 2008c, 0806.0407. Steven S. Gubser, Silviu S. Pufu, and Amos Yarom. “Entropy production in collisions of gravitational shock waves and of heavy ions”. 2008d, 0805.1551. U. Gursoy and E. Kiritsis. “Exploring improved holographic theories for QCD: Part I”. JHEP, 02:032, 2008, 0707.1324. doi: 10.1088/1126-6708/2008/02/032. U. Gursoy, E. Kiritsis, and F. Nitti. “Exploring improved holographic theories for QCD: Part II”. JHEP, 02:019, 2008a, 0707.1349. doi: 10.1088/1126-6708/2008/02/019. Umut Gursoy, Elias Kiritsis, Liuba Mazzanti, and Francesco Nitti. “Deconfinement and Gluon Plasma Dynamics in Improved Holographic QCD”. 2008b, 0804.0899. M. Gyulassy, P. Levai, and I. Vitev. “Reaction operator approach to non-Abelian energy loss”. Nucl. Phys., B594:371–419, 2001, nucl-th/0006010. doi: 10.1016/S0550-3213(00)00652-0. Y. Hatta, E. Iancu, and A. H. Mueller. “Jet evolution in the N=4 SYM plasma at strong coupling”. JHEP, 05:037, 2008, 0803.2481. doi: 10.1088/1126-6708/2008/05/037.

Gubser, AdS/CFT and QCD at finite T , 8-21-08 42 9 Outlook C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G. Yaffe. “Energy loss of a heavy quark moving through N = 4 supersymmetric Yang-Mills plasma”. 2006, hep-th/0605158. W. A. Horowitz and M. Gyulassy. “Testing AdS/CFT Deviations from pQCD Heavy Quark Energy Loss with Pb+Pb at LHC”. 2007, arXiv:0706.2336 [nucl-th]. Barbara Jacak, 2006. ”Plasma physics of the quark gluon plasma”, Boulder, May 2006. Available at http://www4.rcf.bnl.gov/ steinber/boulder2006/. ∼ Jiangyong Jia. “Mapping out the Jet correlation landscape: Jet quenching and Medium response”. 2007, arXiv:0705.3060 [nucl-ex]. Olaf Kaczmarek and Felix Zantow. “Static quark anti-quark interactions in zero and finite temperature QCD. I: Heavy quark free energies, running coupling and quarkonium binding”. Phys. Rev., D71: 114510, 2005, hep-lat/0503017. Frithjof Karsch. “Lattice QCD at high temperature and density”. Lect. Notes Phys., 583:209–249, 2002, hep-lat/0106019. Frithjof Karsch, Dmitri Kharzeev, and Kirill Tuchin. “Universal properties of bulk viscosity near the QCD phase transition”. 2007, arXiv:0711.0914 [hep-ph]. Yevgeny Kats and Pavel Petrov. “Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory”. 2007, 0712.0743. Dmitri Kharzeev. “QCD-gravity correspondence”. Lectures given at Quark Matter School, Jaipur, India, February 2008. Gubser, AdS/CFT and QCD at finite T , 8-21-08 41 9 Outlook Steven S. Gubser, Daniel R. Gulotta, Silviu S. Pufu, and Fabio D. Rocha. “Gluon energy loss in the gauge-string duality”. 2008a, 0803.1470. Steven S. Gubser, Abhinav Nellore, Silviu S. Pufu, and Fabio D. Rocha. “Thermodynamics and bulk viscosity of approximate black hole duals to finite temperature quantum chromodynamics”. 2008b, 0804.1950. Steven S. Gubser, Silviu S. Pufu, and Fabio D. Rocha. “Bulk viscosity of strongly coupled plasmas with holographic duals”. 2008c, 0806.0407. Steven S. Gubser, Silviu S. Pufu, and Amos Yarom. “Entropy production in collisions of gravitational shock waves and of heavy ions”. 2008d, 0805.1551. U. Gursoy and E. Kiritsis. “Exploring improved holographic theories for QCD: Part I”. JHEP, 02:032, 2008, 0707.1324. doi: 10.1088/1126-6708/2008/02/032. U. Gursoy, E. Kiritsis, and F. Nitti. “Exploring improved holographic theories for QCD: Part II”. JHEP, 02:019, 2008a, 0707.1349. doi: 10.1088/1126-6708/2008/02/019. Umut Gursoy, Elias Kiritsis, Liuba Mazzanti, and Francesco Nitti. “Deconfinement and Gluon Plasma Dynamics in Improved Holographic QCD”. 2008b, 0804.0899. M. Gyulassy, P. Levai, and I. Vitev. “Reaction operator approach to non-Abelian energy loss”. Nucl. Phys., B594:371–419, 2001, nucl-th/0006010. doi: 10.1016/S0550-3213(00)00652-0. Y. Hatta, E. Iancu, and A. H. Mueller. “Jet evolution in the N=4 SYM plasma at strong coupling”. JHEP, 05:037, 2008, 0803.2481. doi: 10.1088/1126-6708/2008/05/037.

Gubser, AdS/CFT and QCD at finite T , 8-21-08 42 9 Outlook C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G. Yaffe. “Energy loss of a heavy quark moving through N = 4 supersymmetric Yang-Mills plasma”. 2006, hep-th/0605158. W. A. Horowitz and M. Gyulassy. “Testing AdS/CFT Deviations from pQCD Heavy Quark Energy Loss with Pb+Pb at LHC”. 2007, arXiv:0706.2336 [nucl-th]. Barbara Jacak, 2006. ”Plasma physics of the quark gluon plasma”, Boulder, May 2006. Available at http://www4.rcf.bnl.gov/ steinber/boulder2006/. ∼ Jiangyong Jia. “Mapping out the Jet correlation landscape: Jet quenching and Medium response”. 2007, arXiv:0705.3060 [nucl-ex]. Olaf Kaczmarek and Felix Zantow. “Static quark anti-quark interactions in zero and finite temperature QCD. I: Heavy quark free energies, running coupling and quarkonium binding”. Phys. Rev., D71: 114510, 2005, hep-lat/0503017. Frithjof Karsch. “Lattice QCD at high temperature and density”. Lect. Notes Phys., 583:209–249, 2002, hep-lat/0106019. Frithjof Karsch, Dmitri Kharzeev, and Kirill Tuchin. “Universal properties of bulk viscosity near the QCD phase transition”. 2007, arXiv:0711.0914 [hep-ph]. Yevgeny Kats and Pavel Petrov. “Effect of curvature squared corrections in AdS on the viscosity of the dual gauge theory”. 2007, 0712.0743. Dmitri Kharzeev. “QCD-gravity correspondence”. Lectures given at Quark Matter School, Jaipur, India, February 2008. Gubser, AdS/CFT and QCD at finite T , 8-21-08 43 9 Outlook Dmitri Kharzeev and Kirill Tuchin. “Bulk viscosity of QCD matter near the critical temperature”. 2007, arXiv:0705.4280 [hep-ph]. Chan-ju Kim and Soo-Jong Rey. “Thermodynamics of large-N super Yang-Mills theory and AdS/CFT correspondence”. Nucl. Phys., B564:430–440, 2000, hep-th/9905205. doi: 10.1016/ S0550-3213(99)00532-5. Alexander Kovner and Urs Achim Wiedemann. “Gluon radiation and parton energy loss”. 2003, hep-ph/0304151. P. Kovtun, D. T. Son, and A. O. Starinets. “Viscosity in strongly interacting quantum field theories from black hole physics”. Phys. Rev. Lett., 94:111601, 2005, hep-th/0405231. L. D. Landau. “On the multiparticle production in high-energy collisions”. Izv. Akad. Nauk SSSR Ser. Fiz., 17:51–64, 1953. Hong Liu, Krishna Rajagopal, and Urs Achim Wiedemann. “Calculating the jet quenching parameter from AdS/CFT”. 2006, hep-ph/0605178. Hong Liu, Krishna Rajagopal, and Urs Achim Wiedemann. “Wilson loops in heavy ion collisions and their calculation in AdS/CFT”. JHEP, 03:066, 2007, hep-ph/0612168. Juan M. Maldacena. “The large N limit of superconformal field theories and supergravity”. Adv. Theor. Math. Phys., 2:231–252, 1998, hep-th/9711200. Harvey B. Meyer. “A calculation of the shear viscosity in SU(3) gluodynamics”. Phys. Rev., D76: 101701, 2007, 0704.1801. doi: 10.1103/PhysRevD.76.101701.

Gubser, AdS/CFT and QCD at finite T , 8-21-08 44 9 Outlook Horatiu Nastase. “AdS-CFT and the RHIC fireball”. 2008, 0805.3579. Agustin Nieto and Michel H. G. Tytgat. “Effective field theory approach to N = 4 supersymmetric Yang-Mills at finite temperature”. 1999, hep-th/9906147. Andrei Parnachev and Andrei Starinets. “The silence of the little strings”. JHEP, 10:027, 2005, hep-th/0506144. G. Policastro, D. T. Son, and A. O. Starinets. “The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma”. Phys. Rev. Lett., 87:081601, 2001, hep-th/0104066. Sang-Jin Sin and Ismail Zahed. “Ampere’s law and energy loss in AdS/CFT duality”. Phys. Lett., B648: 318–322, 2007, hep-ph/0606049. Peter Steinberg. “Landau hydrodynamics and RHIC phenomena”. Acta Phys. Hung., A24:51–57, 2005, nucl-ex/0405022. Derek Teaney. “Effect of shear viscosity on spectra, elliptic flow, and Hanbury Brown-Twiss radii”. Phys. Rev., C68:034913, 2003, nucl-th/0301099. doi: 10.1103/PhysRevC.68.034913. Derek Teaney, 2008. “Noise in AdS/CFT,” talk delivered at Hard Probes 2008. M. A. Vazquez-Mozo. “A note on supersymmetric Yang-Mills thermodynamics”. Phys. Rev., D60: 106010, 1999, hep-th/9905030. doi: 10.1103/PhysRevD.60.106010. Xin-Nian Wang. “High p(T) hadron spectra, azimuthal anisotropy and back- to-back correlations in high-energy heavy-ion collisions”. Phys. Lett., B595:165–170, 2004, nucl-th/0305010. doi: 10.1016/j.physletb.2004.05.021. Gubser, AdS/CFT and QCD at finite T , 8-21-08 43 9 Outlook Dmitri Kharzeev and Kirill Tuchin. “Bulk viscosity of QCD matter near the critical temperature”. 2007, arXiv:0705.4280 [hep-ph]. Chan-ju Kim and Soo-Jong Rey. “Thermodynamics of large-N super Yang-Mills theory and AdS/CFT correspondence”. Nucl. Phys., B564:430–440, 2000, hep-th/9905205. doi: 10.1016/ S0550-3213(99)00532-5. Alexander Kovner and Urs Achim Wiedemann. “Gluon radiation and parton energy loss”. 2003, hep-ph/0304151. P. Kovtun, D. T. Son, and A. O. Starinets. “Viscosity in strongly interacting quantum field theories from black hole physics”. Phys. Rev. Lett., 94:111601, 2005, hep-th/0405231. L. D. Landau. “On the multiparticle production in high-energy collisions”. Izv. Akad. Nauk SSSR Ser. Fiz., 17:51–64, 1953. Hong Liu, Krishna Rajagopal, and Urs Achim Wiedemann. “Calculating the jet quenching parameter from AdS/CFT”. 2006, hep-ph/0605178. Hong Liu, Krishna Rajagopal, and Urs Achim Wiedemann. “Wilson loops in heavy ion collisions and their calculation in AdS/CFT”. JHEP, 03:066, 2007, hep-ph/0612168. Juan M. Maldacena. “The large N limit of superconformal field theories and supergravity”. Adv. Theor. Math. Phys., 2:231–252, 1998, hep-th/9711200. Harvey B. Meyer. “A calculation of the shear viscosity in SU(3) gluodynamics”. Phys. Rev., D76: 101701, 2007, 0704.1801. doi: 10.1103/PhysRevD.76.101701.

Gubser, AdS/CFT and QCD at finite T , 8-21-08 44 9 Outlook Horatiu Nastase. “AdS-CFT and the RHIC fireball”. 2008, 0805.3579. Agustin Nieto and Michel H. G. Tytgat. “Effective field theory approach to N = 4 supersymmetric Yang-Mills at finite temperature”. 1999, hep-th/9906147. Andrei Parnachev and Andrei Starinets. “The silence of the little strings”. JHEP, 10:027, 2005, hep-th/0506144. G. Policastro, D. T. Son, and A. O. Starinets. “The shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma”. Phys. Rev. Lett., 87:081601, 2001, hep-th/0104066. Sang-Jin Sin and Ismail Zahed. “Ampere’s law and energy loss in AdS/CFT duality”. Phys. Lett., B648: 318–322, 2007, hep-ph/0606049. Peter Steinberg. “Landau hydrodynamics and RHIC phenomena”. Acta Phys. Hung., A24:51–57, 2005, nucl-ex/0405022. Derek Teaney. “Effect of shear viscosity on spectra, elliptic flow, and Hanbury Brown-Twiss radii”. Phys. Rev., C68:034913, 2003, nucl-th/0301099. doi: 10.1103/PhysRevC.68.034913. Derek Teaney, 2008. “Noise in AdS/CFT,” talk delivered at Hard Probes 2008. M. A. Vazquez-Mozo. “A note on supersymmetric Yang-Mills thermodynamics”. Phys. Rev., D60: 106010, 1999, hep-th/9905030. doi: 10.1103/PhysRevD.60.106010. Xin-Nian Wang. “High p(T) hadron spectra, azimuthal anisotropy and back- to-back correlations in high-energy heavy-ion collisions”. Phys. Lett., B595:165–170, 2004, nucl-th/0305010. doi: 10.1016/j.physletb.2004.05.021. Gubser, AdS/CFT and QCD at finite T , 8-21-08 45 9 Outlook Urs Achim Wiedemann. “Gluon radiation off hard quarks in a nuclear environment: Opacity expansion”. Nucl. Phys., B588:303–344, 2000, hep-ph/0005129. Edward Witten. “Anti-de Sitter space and holography”. Adv. Theor. Math. Phys., 2:253–291, 1998, hep-th/9802150. Amos Yarom. “The high momentum behavior of a quark wake”. 2007, hep-th/0702164. B. G. Zakharov. “Radiative energy loss of high energy quarks in finite-size nuclear matter and quark- gluon plasma”. JETP Lett., 65:615–620, 1997, hep-ph/9704255. Gubser, AdS/CFT and QCD at finite T , 8-21-08 45 9 Outlook Urs Achim Wiedemann. “Gluon radiation off hard quarks in a nuclear environment: Opacity expansion”. Nucl. Phys., B588:303–344, 2000, hep-ph/0005129. Edward Witten. “Anti-de Sitter space and holography”. Adv. Theor. Math. Phys., 2:253–291, 1998, hep-th/9802150. Amos Yarom. “The high momentum behavior of a quark wake”. 2007, hep-th/0702164. B. G. Zakharov. “Radiative energy loss of high energy quarks in finite-size nuclear matter and quark- gluon plasma”. JETP Lett., 65:615–620, 1997, hep-ph/9704255.