Relativistic Heavy Ion Collisions and String Theory

Relativistic Heavy Ion Collisionsand String Theory

Josh FriessPrinceton University

in collaboration with Steve Gubser, Georgios Michalogiorgakis,and Silviu Pufu

KITP String Phenomenology 2006

AdS/CFT - RHIC, Friess, KITP String Phenomenology 2006 2

1. IntroductionThe Relativistic Heavy Ion Collider (RHIC), operating at Brookhaven National Lab-oratory, collides gold on gold.

• Total center of mass energy is about39 TeV.• There is good evidence that a thermalized quark-gluon plasma (QGP) forms with

temperature above the confinement scale,TC ≈ 170MeV ≈ 2× 1012K


The theoretical understanding of RHIC physics is imperfect.

• The QGP is strongly coupled, so perturbative QCD is of limited utility.• Lattice calculations provide good information about static properties, e.g.,TC

for confinement, but not transport properties (like viscosity).• String theory, in particular AdS/CFT, offers an alternative description of strongly

coupled gauge theory.

Two main themes of the AdS/CFT - RHIC connection are

A The viscosity boundη/s ≥ ~/4π.B Jet-quenching and the drag force on hard partons, especially heavy quarks.

A has been under discussion for about 5 years.B is a relatively new development,and the focus of our contribution.


2. What happens at RHIC?RHIC accelerates beams of heavy nuclei (gold, copper, etc.) in opposite directionsaround a large circular ring and collides them.

Gold nuclei are nearly spherical with radius of about7 fm in rest frame; Lorentzcontraction reduces front-to-back length to∼ 0.07fm.

Figure 1: Before-and-After shots of ultra-relativistic dynamics simulation of a gold-gold collision[1].


• The main ring is3.8 km in circumference.• Beam CM energy per nucleon per nucleon is

√sNN = 200 GeV .

• RHIC’s design luminosity is2× 1026 cm−2s−1. Integrated luminosity to date isin the ballpark of4 nb−1.


Experimentalists claim that a thermalized QGP gets formed. Hadron yields follownearly Boltzmann distributions (Kaneta 2004 [2]):

42 BERNDT MULLER and JAMES L. NAGLE

10-3

10-2

10-1

1

Rat

io200 GeV Au+Au, <N

part> = 322

!-

!+K

+

K"

p_

p

(*1)

p_

p

(*2)

#" +

#"

$"

$

%K*0&

%K±&

K-

!"p_

!"

(*1)

p_

!"

(*2)

!+!"

#"

$

#"

%"

#"

&

#"

-4-2024

dev

.

(*1) : feed-down effect is corrected in data(*2) : feed-down effect is included

Tch

=

µq=

µs=

's=

(2/dof=

157±

9.4±

3.1±

1.03±

19.9 /

3 [MeV]

1.2 [MeV]

2.3 [MeV]

0.04

10

Figure 4: Comparison of BRAHMS (circles), PHENIX (triangles), PHOBOS(crosses) and STAR (stars) particle ratios from central gold-gold collisions at√

sNN = 200 GeV at mid-rapidity. The thermal model descriptions from (111)are also shown as lines. Similar results are obtained in (112).

Figure 2: Yellow lines are thermal model predictions, icons represent experimental data.Tch ischemical freeze-out temperature,µq is up/down chemical potential,µs is strange chemical potential,andγs is strangeness saturation.


Furthermore, theoretical predictions from lattice simulations find that deconfinementhappens atTc ≈ 170 MeV, and thatε/T 4 has a plateau at80% of the free field value(Karsch 2001 [3]):

RHIC RESULTS 39

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

T/Tc

!/T4

!SB/T4

3p/T4

3 flavor, N"=4, p4 staggered

m#=770 MeV

(3/4)s/T3

Figure 1: Figure of ε(T )/T 4, P (T )/T 4, and s(T )/T 3 for three light flavors ofquarks on the lattice.

Table 1: Table of RHIC Performance.Run Species Particle Energy Total Delivered Average Store

[GeV/n] Luminosity PolarizationRun-1 2000 Au + Au 27.9 < 0.001µb−1 -

Au + Au 65.2 20 µb−1 -Run-2 2001-2 Au + Au 100.0 258 µb−1 -

Au + Au 9.8 0.4 µb−1 -pol. p + p 100.0 1.4 µb−1 14%

Run-3 2002-3 d + Au 100.0 1.4 pb−1 -pol. p + p 100.0 5.5 pb−1 34%

Run-4 2003-4 Au + Au 100.0 3740 µb−1 -Au + Au 31.2 67 µb−1 -pol. p + p 100.0 7.1 pb−1 45%

Run-5 2004-5 Cu + Cu 100.0 42.1 nb−1 -Cu + Cu 31.2 67 µb−1 -Cu + Cu 11.2 0.02 nb−1 -pol. p + p 100.0 29.5 pb−1 46%pol. p + p 204.9 0.1 pb−1 30%

Figure 3:Lattice results for the equation of state of QCD.


Jet-quenchingrefers to the rapid loss of energy of a hard parton propagating throughthe hot dense matter created in a gold-gold collision. The prima facie evidence forjet quenching is the the suppression of highpT jets (more precisely, highpT hadrons)relative to expectations from “binary collsion scaling.”

• Jet production from proton-proton collisions is well studied, as is photon pro-duction.

• Binary scaling means to multiply yields in proton-proton by the ratio of incidentparton flux of a gold-gold collision to the analogous flux for proton-proton.

• This scaling basically works for high-energy photons(2 GeV/c < pT < 14 GeV/c) (Adler 2005[4]).

• It doesn’t work for highpT hadrons: at mid-rapidity,

RAA ≡dN(gold-gold)/dpTdη

〈Nbinary〉dN(proton-proton)/dpTdη≈ 0.2 (1)

where〈Nbinary〉 is the number of nucleon-nucleon collisions in a “factorized”gold-gold collision.


RHIC RESULTS 45

)c (GeV/Tp0 2 4 6 8 10 12 14 16

AA

R

-110

1

10

Au+Au (central collisions):

γDirect

(STAR)±Inclusive h

(PHENIX Preliminary)0π

ηPHENIX Preliminary

/dy = 1100)g

GLV parton energy loss (dN


γ (PHENIX) Direct

(STAR)±Inclusive h

(PHENIX)0π


/dy = 1100)g


Figure 8: Nuclear modification factor RAA for direct photons and hadrons incentral (0-10% of σAu+Au

inel. ) gold-gold collisions at√

sNN = 200 GeV (149).Figure 4: Nuclear modification factorRAA for photons and hadrons in0 to 10% central gold-goldcollisions.

Can AdS/CFT explain this deficit?

AdS/CFT - RHIC, Friess, KITP String Phenomenology 200610

RHIC RESULTS 45

)c (GeV/Tp0 2 4 6 8 10 12 14 16

AA

R

-110

1

10


γDirect

(STAR)±Inclusive h

(PHENIX Preliminary)0π


/dy = 1100)g



γ (PHENIX) Direct

(STAR)±Inclusive h

(PHENIX)0π


/dy = 1100)g


Figure 8: Nuclear modification factor RAA for direct photons and hadrons incentral (0-10% of σAu+Au

inel. ) gold-gold collisions at√

sNN = 200 GeV (149).Figure 4: Nuclear modification factorRAA for photons and hadrons in0 to 10% central gold-goldcollisions.

Can AdS/CFT explain this deficit?


The entropy and viscosity calculations have drawn the attention of RHIC phenome-nologists, as well as DOE higher-ups:

“The possibility of a connection between string theory and RHIC collisions is unexpected and

exhilarating.” — Ray Orbach, DOE Office of Science Director [5]

But before we proceed, some cautionary statements are worth noting —N = 4gauge theorymisses several essential features of QCD:

• No confinement. Coupling doesn’t run: it’s a parameter you can dial.

But is this so bad? We want to use AdS/CFT at finite temperature to model the QGP above Tc.

Phenomenological studies of RHIC physics routinely set vs = 1/√

3 and ε ∼ 1/t4/3 (both cor-

responding to conformal invariance) for the QGP, e.g. when the energy density ε is significantly

above 1 GeV.

• No chiral condensate.

But is this so bad? The chiral condensate turns off around Tc according to lattice calculations.

• All fundamental matter fields are in adjoint representation:Aµ, four Majoranafermionsλi, six real scalarsXI .

This looks kind of bad. Maybe gauge interactions dominate the dynamics anyway?


3. Jet Quenching in AdS/CFT

5

R3,1

AdS −Schwarzschild

vq

fundamental string

Tmn

mnh

horizon

Figure 5:In blue: the trailing string of an external quark (Herzog et al, 2006 [6]; Gubser 2006[7]).The dashed line shows classical propagation of a graviton from the string to the boundary, where itsbehavior can be translated into the stress-energy tensor〈Tmn〉 of the boundary gauge theory.

An analog of jet-quenching in AdS/CFT should involve a colored probe that we dragthrough the QGP, preferably at relativistic speeds. Readiest at hand are externalquarks: strings with one end on the boundary.


Our background metric is

ds2 =L2

z2Hy2

(−hdt2 + d~x2 + z2

H

dy2

h

) ∣∣∣h ≡ 1− y4∣∣∣ zH =

1

πT(2)

We’re interested in non-zero quark masses, so consider splitting one D3-brane fromthe large stack and putting it at a heighty∗ ≥ 0, and treat it in the test-brane approx-imation. (Alternatively, wrap a D7 on anS3 ⊂ S5, and have it fill three extendeddirections plus the interval0 ≤ y ≤ y∗.) Then a string with one endpoint aty∗,going straight down into the horizon has a mass

mstatic =L2

2πα′

(1

z∗− 1

zH

)=

√g2

Y MN

2T

(zH

z∗− 1

)(3)


In the QGP,u, d, ands quarks are dominated by thermal mass, whereas electroweakcontribution still dominates forc andb. We use

mu = md = ms = 300MeV mc = 1400MeV mb = 4800MeV (4)

(B)

x

yAdS5−Schwarzschild

fundamentalstring ξ(y)

R3,1

vy=yb

b

momentum flow

y=1 horizon

y=0

D3 D3

(A)

Figure 6:(A) A finite mass quark moving at velocityv through the QGP can be represented as a stringhanging from a “flavor brane” (Herzog et al, 2006). This picture is best justified for heavy quarks likec and b. In this figure and below, we use the radial coordinatey = z/zH . (B) At T = 0, flavor branescan be realized by separating one D3-brane from several others. The massiveW boson is similar to aheavy quark. We also show anRB gluon.

AdS/CFT - RHIC, Friess, KITP String Phenomenology 200615 3.1 A drag force computation

3.1. A drag force computation

We need to know the shape of the trailing string and the momentum flow down it.We assume a “co-moving” ansatz, and parameterize the worldsheet as:

(t, x1, x2, x3, y) = (τ, vt + ξ(y), 0, 0, σ) (5)

A “reduced” lagrangian follows from the Nambu-Goto action:

L = − 1

y2

√1 +

hξ′2

z2H

− v2

h(6)

And the solution is

ξ′ = −vzHy2

1− y4ξ = −vzH

4i

(log

1− iy

1 + iy+ i log

1 + y

1− y

). (7)

This is deceptively real, since the argument for the firstlog is just a phase.


flow across

3,1

ξ(y)

Rq

v

horizon momentum flowI

momentummeasure

Figure 7:The drag force is computed by measuring the momentum flux down the string. The positionof I is arbitrary because the energy-momentum current is conserved.

Momentum and energy drains down the string:

∆P1 = −∫Idt√−gP y

x1 =dp1

dt∆t . (8)

dp1/dt is precisely the drag force:

F ≡ dp

dt= −

π√

g2Y MNT 2

2

v√1− v2

. (9)


The expression for the string shapeξ holds for any mass (i.e.,y∗) – just chop offthe string abovey∗. The drag force expression holds for heavy quarks,m � T ,so thaty∗ is near the boundary, and we can use standard relativistic expressions likeE =

√p2 + m2 andp = mv/

√1− v2.

F ≡ dp

dt= −

π√

g2Y MNT 2

2

v√1− v2

≈ −π√

g2Y MNT 2

2mp . (10)

Simply integrate this to find

p(t) = p0e−t/t0 , t0 =

2

π√

g2Y MN

m

T 2. (11)

Plug in T = 318 MeV andλ = 10, we find thatt0 = 0.6 fm/c for charm, andt0 = 1.9 fm/c for bottom, compared totQGP ≈ 6 fm/c for the typical lifetimeof the QGP. This temperature is also a significant overestimate, convenient so thatzH = 1/πT = 1 GeV−1. A more realistic temperature would reduce the quenchingeffect – QGP also cools substantially as it expands.

AdS/CFT - RHIC, Friess, KITP String Phenomenology 200618 3.2 Graviton perturbations

3.2. Graviton perturbations

A good measure of the energy loss is〈Tmn〉 in the boundary gauge theory. Herewe’ll attempt a concise description of the calculation.

〈Tmn〉 is determined by the behavior near the boundary of linearized graviton per-turbations ofAdS5-Schwarzschild:

ds2(0) = G(0)

µν dxµdxν =L2

z2Hy2

(−hdt2 + d~x2 + z2

H

dy2

h

)h ≡ 1− y4 . (12)

Gµν = G(0)µν + hµν , (13)

The Einstein equations are

Rµν − 1

2GµνR− 6

L2Gµν = τµν , (14)

whereτµν is the stress-energy of the trailing string.


A priori, this leaves us with15 equations for15 perturbative modes. The stresstensor involves delta functions at the location of the string, so move to Fourier space.This gives a series of coupled ordinary differential equations iny for the co-movingFourier componentshµν

K . Then:

• Choose “axial gauge,”hµyK = 0. Now there are10 independent quantitieshmn

K ,where0 ≤ m, n ≤ 3.

• We are left with10 second order equations of motions,Emn = 0, and5 firstorder constraints,Eµy = 0.

• The differential equations may be partially decoupled and simplified by makinga series of field redefinitions. They are still complicated—see below.

• Since hep-th/0607022, we have generalized by allowing the trailing string to endon a flavor brane aty = y∗. This is accomplished simply by including a factorof θ(y − y∗) in τµν.

• We take~K = (K1, K⊥, 0) = K(cos θ, sin θ, 0).


Here’s the full problem:

hKµν =

κ2

2πα′1√

1− v2

L

z2Hy2

0BBB@

H00 H01 H02 H03 0H10 H11 H12 H13 0H20 H21 H22 H23 0H30 H31 H32 H33 00 0 0 0 0

1CCCA . (15)

K =q

K21 + K2

⊥ θ = tan−1 K⊥

K1(16)

A =−H11 + 2 cot θH12 − cot2 θH22 + csc2 θH33

2v2(17)

�∂2

y +

�−3

y+

h′

h

�∂y +

K2

h2(v2 cos2 θ − h)

�A =

y

he−iK1ξ/zH ϑ(y − y∗) (18)

B1 =H03

K2vB2 = −H13 + tan θH23

K2v2(19)

"∂2

y +

− 3

y0

0 − 3y

+ h′

h

!∂y +

K2

h2

�−h v2 cos2 θh−1 v2 cos2 θ

�#�B1

B2

�=

�00

�(20)

B′1 − hB′

2 = 0 (21)

C =− sin θH13 + cos θH23

K(22)

�∂2

y +

�−3

y+

h′

h

�+

K2

h2(v2 cos2 θ − h)

�C = 0 (23)

D1 =H01 − cot θH02

2vD2 =

−H11 + 2 cot 2θH12 + H22

2v2(24)

"∂2

y +

− 3

y0

0 − 3y

+ h′

h

!∂y +

K2

h2

�−h v2 cos2 θh−1 v2 cos2 θ

�#�D1

D2

�=

y

he−iK1ξ/zH ϑ(y − y∗)

�11

�(25)


D′1 − hD′

2 =y3

ivK1e−iK1ξ/zH ϑ(y − y∗) (26)

E1 =1

2

�− 3

hH00 + H11 + H22 + H33

�E2 =

H01 + tan θH02

2v

E3 =H11 + H22 + H33

2E4 =

−H11 −H22 + 3 cos 2θ(−H11 + H22) + 2H33 − 6 sin 2θH12

4

(27)

26664∂2

y +

0BBB@− 3

y+ 3h′

2h0 0 0

0 − 3y

0 0

0 0 − 3y

+ h′

2h0

0 0 0 − 3y

+ h′

h

1CCCA ∂y

+K2

3h2

0B@−2h 12v2 cos2 θ 6v2 cos2 θ + 2h 0

0 0 2h h0 0 −2h −h2h −12v2 cos2 θ 0 3v2 cos2 θ + h

1CA3750B@

E1

E2

E3

E4

1CA

=y

he−iK1ξ/zH ϑ(y − y∗)

0BB@

1 + v2

h1

−1 + v2 − v2

h

v2 1+3 cos 2θ2

1CCA

(28)

240@ 0 1 1 0−h 0 −3v2 cos2 θ − h −hh 0 2 0

1A ∂y

+1

6h

0@ 0 −6h′ −3h′ 0−3hh′ 18v2 cos2 θh′ 3(3v2 cos2 θ + h)h′ 02K2yh −12K2v2y cos2 θ −2K2y(3v2 cos2 θ − h) 2K2yh

1A350B@

E1

E2

E3

E4

1CA

=h′

4Kyhe−iK1ξ/zH ϑ(y − y∗)

0@ −ivy sec θ

3ivy cos θ(v2 + h)K(v2 − h)

1A .

(29)


To solve the10 second order equations of motion for specifiedK, we must fix20integration constants.

• Think of 15 as being fixed at the boundary ofAdS5-Schwarzschild (that is,y = 0) and the remaining5 at the horizon to suppress solutions describinggravitons comingout of the black hole. Each set of equations has exactly onenon-vanishing oscillatory mode at the horizon whose frequency must have thecorrect sign.

• Of the15 boundary conditions aty = 0, five come from imposing the first-orderconstraints. This is arbitrary: the constraints can be imposed anywhere.

• The10 remaining boundary conditions come from requiringHµν → 0 asy →0, i.e., the metric in the boundary theory remains Minkowski.


In practice, to proceed we:

• Note that theB andC sets are odd under theZ2 reflection in the(x1, x2) planespanned two comoving momenta, so these functions must be zero.

• For the seven functionsA, D, andE, find solutions to the equations of motionthat are asymptotically exact at the boundary and horizon. Near the boundary,each mode roughly takes the formH ∼ P + Qy4

• Pick some specific~K = (K1, K⊥).

• SetA for each mode to zero at the boundary (n integration constants in each setwith n equations).

• Impose the constraint equations at the boundary, which relate theQ′s in each set– there aren − 1 first-order constraints per set. We’re left with one remainingintegration constantQ per set – call themQA, QD, QE.

• AdjustQX until the one undesirable outgoing mode at the horizon goes away.


0 1 2 3 4K1

0.5

1

1.5

2

Kp

cL KpÈQEHK1,KpLÈ for v=0.95

0 1 2 3 4K1

0.5

1

1.5

2

Kp

dL KpÈQEHK1,KpLÈ for v=0.99

0 1 2 3 4K1

0.5

1

1.5

2

Kp

aL KpÈQEHK1,KpLÈ for v=0.75

0 1 2 3 4K1

0.5

1

1.5

2

Kp

bL KpÈQEHK1,KpLÈ for v=0.9

Figure 8:Contour plots ofK⊥|QKE | for various values ofv. QK

E is proportional to theK-th Fouriercomponent of the energy density after a near-field subtraction. The phase space factorK⊥ arises inFourier transforming back to position space. The green line shows the Mach angle. The red curve

shows whereK⊥|QKE | is maximized for fixedK =

√K2

1 + K2⊥. The blue curves show whereK⊥|QK

E |takes on half its maximum value for fixedK. For T = 318 MeV, momenta axes are in units ofGeV.


Same plot as previous page in polar coordinates:K1 = K cos θ, K⊥ = K sin θ

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.25

0.5

0.75

1

1.25

1.5

1.75

2

K

cL KpÈQEHK,ΘLÈ for v=0.95

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.25

0.5

0.75

1

1.25

1.5

1.75

2

K

dL KpÈQEHK,ΘLÈ for v=0.99

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.25

0.5

0.75

1

1.25

1.5

1.75

2

K

aL KpÈQEHK,ΘLÈ for v=0.75

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.25

0.5

0.75

1

1.25

1.5

1.75

2

K

bL KpÈQEHK,ΘLÈ for v=0.9


Energy density for a charm quark in polar coordinates, without a near-field sub-traction – qualitatively identical to infinite mass case with the near-field subtraction

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.5

1

1.5

2

K

cL KpÈQEHK,ΘLÈ for v=0.95

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.5

1

1.5

2

K

dL KpÈQEHK,ΘLÈ for v=0.99

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.5

1

1.5

2

K

aL KpÈQEHK,ΘLÈ for v=0.75

0 0.2 0.4 0.6 0.8 1 1.2 1.4Θ

0.5

1

1.5

2

K

bL KpÈQEHK,ΘLÈ for v=0.9

Figure 9:K⊥QE(K, θ) for y∗ = 0.26, i.e.,mc = 1400MeV, T = 318MeV, andλ = 10

AdS/CFT - RHIC, Friess, KITP String Phenomenology 200627 3.3 The wake of a quark

3.3. The wake of a quark

A much-discussed aspect of RHIC’s current experimental program hinges on thefollowing picture:look for the jet on the other side

STAR PRL 90, 082302 (2003)

Central Au + Au

Peripheral Au + Au

Medium is opaque!

_ high density

large !interaction

1

Atrigger jet

2

!M

B

C

Figure 1. A schematic picture of flow created by a jet going through the fireball. The triggerjet is going to the right from the origination point (the black circle at point B) from which soundwaves start propagating as spherical waves (the dashed circle). The companion quenched jet ismoving to the left, heating the matter and thus creating a cylinder of additional matter (shadedarea). The head of the jet is a “nonhydrodynamical core” of the QCD gluonic shower, formedby the original hard parton (black dot A). The solid arrow shows a direction of flow normalto the shock cone at the angle θM , the dashed arrows show the direction of the flow after theshocks hit the edge of the fireball.

Elastic energy losses were first studied by Bjorken[1], while those due to “ionization” ofbound states in sQGP were recently considered by Shuryak and Zahed [6]. These mechanismdeposit additional energy, momentum and entropy into the matter. (Like for delta electronsin ordinary matter, this excitation kicks particles mostly orthogonal to the jet direction.) It istheir combined magnitude, dE/dx = 2 − 3GeV/fm, the one we will use below. Even at suchloss rates, a jet passing through the diameter of the fireball, created in central Au-Au collisions,may deposit up to 20-30 GeV, enough to absorb the jets of interest at RHIC.

Let us start our discussion of associated collective effects by recalling the energy scalesinvolved. While the total CM energy in a Au-Au collision at RHIC is very large (about 40TeV) compared to the energy of a jet (typically 5-20 GeV ), the jet energy is transverse. Thetotal transverse energy of all secondaries per one unit of rapidity is dE⊥/dy ∼ 600GeV . Mostof it is thermal, with only about 100 GeV being related to collective motion. Furthermore,the so called elliptic flow is a ∼ 1/10 asymmetry and therefore it carries energy ∼ 10GeVwhich is comparable to that lost by jets. Since elliptic flow was observed and studied in detail,we conclude that conical flow should be observable as well. (In order to separate the two, itis beneficial to focus first on the most central collisions, where the elliptic flow is as small aspossible.)

Fig.1 explains a view of the process in a plane transverse to the beam. Two oppositely movingjets originate from the hard collision point B. Due to strong quenching, the survival of the triggerjet biases it to be produced close to the surface and to move outward. This forces its companionto move inward through matter and to be maximally quenched. The energy deposition startsat point B, thus a spherical sound wave appears (the dashed circle in Fig.1 ). Further energydeposition is along the jet line, and is propagating at the speed of light, till the leading partonis found at point A at the moment of the snapshot.

As is well known, the interference of perturbations from a supersonically moving body (suchas a supersonic jet plane or a meteorite) creates a conical flow behind the shock waves. Similarflow was discussed in Refs[7] for shocks in cold nuclear matter, in which compression up to QGP

Figure 10: Left: A di-jet event with significant away-side jet quenching. (Jacak [8]). Right:The away-side parton may generate a sonic boom, withθM = cos−1(cs/v) the Mach angle. From(Casalderrey-Solana et al, 2004 [9]).

• Two hard partons collide near the surface of the QGP.• One escapes and fragments into the “near side” jet.• The other plows through the QGP and dissipates a lot of energy.


0.02 0.04 0.06 0.08K1

0.02

0.04

0.06

0.08

Kp

cL KpÈQEHK1,KpLÈ for v=0.6

0.02 0.04 0.06 0.08K1

0.02

0.04

0.06

0.08

Kp

dL KpÈQEHK1,KpLÈ for v=0.7

0.02 0.04 0.06 0.08K1

0.02

0.04

0.06

0.08

Kp

aL KpÈQEHK1,KpLÈ for v=0.4

0.02 0.04 0.06 0.08K1

0.02

0.04

0.06

0.08

Kp

bL KpÈQEHK1,KpLÈ for v=0.5

Figure 11: Numerical data for stress tensor at lowK for various velocities. Speed of sound iscs = 1/

√3 ≈ 0.577. Green lines indicate the Mach angle, as usual. Red lines are peak of distributions,

and blue lines are locations of the half-maximum.


The sonic boom can also be immediately identified in the graviton calculation via asmallK expansion for the stress tensor, which can be found by matching boundaryand horizon asymptotic solutions that areK-exact to smallK solutions for ally.

〈TK00〉 ∝

3iv(1 + v2) cos θ

2K (1− 3v2 cos2 θ)− 3v2 cos2 θ [2 + v2 (1− 3 cos2 θ)]

2 (1− 3v2 cos2 θ)2 + O(K)

=3iv(1 + v2) cos θ

2K

1

(1− 3v2 cos2 θ)(1− ivK cos θ

1+v2

)− ivK cos θ

+ O(K) .

(30)

First expression is clearly singular at each order inK at the angle given bycos θ = 1v√

3,which is the Mach angle. Second expression is equal, up toO(K) terms, but it isregular and sharply peaked at the Mach angle, and appears to be a better fit to thenumerics.


The sonic boom picture and related theoretical proposals suggest that high-angleemission carries away a lot of the energy. And data seems to confirm this:

5

)!

"(d

i-je

t)/d

(A

B d

NA

pairs p

er

trig

ger:

1/N

(rad)! "0 0.5 1 1.5 2 2.5 3

00.05

0.10.15

0.20.25

0.30.35

0.4(e) 40-60%

00.05

0.10.15

0.20.25

0.30.35

0.4(c) 10-20%

00.05

0.10.15

0.20.25

0.30.35

0.40.45

(a) 0-5%

(rad)! "0.5 1 1.5 2 2.5 3

(f) 60-90%

(d) 20-40%

(b) 5-10%

FIG. 2: Jet-pair distributions dNAB

(Di−)Jet/d(∆φ) for differ-ent centralities, normalized per trigger particle. The shadedbands indicate the systematic error associated with the de-termination of ∆φMin. The dashed (solid) curves are thedistributions that would result from increasing (decreasing)〈vA

2 vB2 〉 by one unit of the systematic error; the dotted curve

would result from decreasing by two units.

The existence of these local minima per se is not signif-icant once we take the systematic errors on 〈vA

2 vB2 〉 into

account (see below), but it is clear that the away-sidepeaks in all the more central samples have a very differ-ent shape than in the most peripheral sample.

Given the dramatic results for the away-side peaks seenin Fig. 2, it is important to establish that they are notsimply artifacts created by our method for backgroundpair subtraction. If we relax the ZYAM assumption andlower b0 slightly, the effect on any (di-)jet pair distribu-tion would essentially be to raise it by a constant, whichwould not change the presence of the local minima at∆φ = π.

Changes to our estimate for 〈vA2 vB

2 〉 can alter the shapeof the (di-)jet distribution for some centrality samples,but the result of away-side broadening with centralityremains robust. The curves in Fig. 2 show the distribu-tions that would result if the 〈vA

2 vB2 〉 products were arbi-

trarily lowered by one and two units of their systematicerror. With a two-unit shift the shape in the mid-centralwould no longer show significant local minima at ∆φ = π.However, the widths of the away-side peaks are clearlystill much greater than in the peripheral sample and thedistributions in the two most central samples are hardlychanged at all in shape. Even lower values of 〈vA

2 vB2 〉

could be contemplated, but they would still not changethe qualitative result of away-side broadening. And, such

low 〈vA2 vB

2 〉 values would also require a severe breakdownof the assumption 〈vA

2 vB2 〉 = 〈vA

2 〉〈vB2 〉, indicating that

these background pairs have a large, hitherto-unknownsource of azimuthal anti-correlation.

Convoluting the jet fragments’ angles with respect totheir parent partons and the acoplanarity between thetwo partons [23] would yield a Gaussian-like shape in∆φ, possibly broadened through jet quenching[13, 25].The observed shapes in the away-side peaks cannot resultfrom such a convolution.

We define the part of the ∆φ distribution in|∆φ| < ∆φMin as the “near-side” peak and |∆φ| > ∆φMin

as the “away-side” peak. Each peak is characterized byits yield of associated partners per trigger, and by itsRMS width. We measure these for the full peak in thedistribution over all values of ∆φ; the folded distributionsover 0 < ∆φ < π shown here contain only half of eachfull peak’s shape. These yields and widths are plotted inFig. 3 for the different Au+Au centrality samples, alongwith the same quantities for 0–20% central d+Au colli-sions at

√s

NN=200 GeV [23]. The yields and widths for

the near- and away-side peaks in peripheral Au+Au col-lisions are consistent with those in d+Au collisions. Theyields of both the near- and away-side peaks increasefrom peripheral to mid-central collisions, and then de-crease for the most central collisions. The near-side widthis unchanged with centrality, while the away-side widthincreases substantially from the 60–90% sample to the40–60% sample and then remains constant with central-ity.

A(d

i-je

t)/N

AB

N RM

S (

rad

)

>part

<N0 100 200 300 400

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 (a)

>part

<N100 200 300 400

< 2.5 GeV/cT

B(b) 1 < p

< 4 GeV/cT

A 2.5 < p

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Au+Au near side

Au+Au away side

d+Au near side

d+Au away side

FIG. 3: (a) Associated yields for near- and away-side peaks inthe jet pair distribution, and (b) widths (RMS) of the peaks inthe full 0–2π distributions; plotted versus the mean number ofparticipating nucleons for each event sample. Triangles showresults from 0–20% central d+Au collisions at the same

√s

NN

[23]. Bars show statistical errors, shaded bands systematic.

In summary, we have presented correlations of highmomentum charged hadron pairs as a function of col-lision centrality in Au+Au collisions. Utilizing a noveltechnique we extract the jet-induced hadron pair dis-

5

)!

"(d

i-je

t)/d

(A

B d

NA

pairs p

er

trig

ger:

1/N

(rad)! "0 0.5 1 1.5 2 2.5 3

00.05

0.10.15

0.20.25

0.30.35

0.4(e) 40-60%

00.05

0.10.15

0.20.25

0.30.35

0.4(c) 10-20%

00.05

0.10.15

0.20.25

0.30.35

0.40.45

(a) 0-5%

(rad)! "0.5 1 1.5 2 2.5 3

(f) 60-90%

(d) 20-40%

(b) 5-10%






2 vB2 〉 into







2 vB2 〉


low 〈vA2 vB


2 vB2 〉 = 〈vA






√s



A(d

i-je

t)/N

AB

N RM

S (

rad

)

>part

<N0 100 200 300 400

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 (a)

>part

<N100 200 300 400

< 2.5 GeV/cT

B(b) 1 < p

< 4 GeV/cT

A 2.5 < p

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Au+Au near side

Au+Au away side

d+Au near side

d+Au away side


√s

NN



5

)!

"(d

i-je

t)/d

(A

B d

NA

pairs p

er

trig

ger:

1/N

(rad)! "0 0.5 1 1.5 2 2.5 3

00.05

0.10.15

0.20.25

0.30.35

0.4(e) 40-60%

00.05

0.10.15

0.20.25

0.30.35

0.4(c) 10-20%

00.05

0.10.15

0.20.25

0.30.35

0.40.45

(a) 0-5%

(rad)! "0.5 1 1.5 2 2.5 3

(f) 60-90%

(d) 20-40%

(b) 5-10%






2 vB2 〉 into







2 vB2 〉


low 〈vA2 vB


2 vB2 〉 = 〈vA






√s



A(d

i-je

t)/N

AB

N RM

S (

rad

)

>part

<N0 100 200 300 400

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 (a)

>part

<N100 200 300 400

< 2.5 GeV/cT

B(b) 1 < p

< 4 GeV/cT

A 2.5 < p

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Au+Au near side

Au+Au away side

d+Au near side

d+Au away side


√s

NN



Figure 12:Histograms of the azimuthal angle between the trigger hadron (with2.5 GeV/c < pT <4 GeV/c) and the partner hadron (with1 GeV/c < pT < 2.5 GeV/c). Away-side jet splitting, illustratedby the broad peak around∆φ = 2, is evidence for high-angle emission in the QGP. (Adler 2005 [10]).


The numerical data from AdS/CFT agrees (at least) qualitatively with the RHICdata:

1.8 2.2 2.4 2.6 2.8 3 DΦ

0.5

1

1.5

2

cL KpÈQEHK1,KpLÈ for K = 1.4

1.8 2.2 2.4 2.6 2.8 3 DΦ

0.5

1

1.5

2

dL KpÈQEHK1,KpLÈ for K = 2.

1.8 2.2 2.4 2.6 2.8 3 DΦ

5

10

15

20

25

aL KpÈQEHK1,KpLÈ for K = 0.08

1.8 2.2 2.4 2.6 2.8 3 DΦ

0.5

1

1.5

2

2.5

3

bL KpÈQEHK1,KpLÈ for K = 0.8

Figure 13: K⊥|QKE | at fixed K as a function of angle, forv = 0.95. ∆φ = π − θ where

θ = tan−1 K⊥/K1. The dashed lines are from an analytic estimate, and the solid lines are from nu-merics. The green line is the Mach angle; the red dot is the peak; and the blue dots are at half the peakheight. Plots (c) and (d) are in the ballpark of the experimental study summarized in the previous figure.K is the total momentum, in units ofGeV if T = 318 MeV.


4. Conclusions• A simple type IIB string configuration helps elucidate the physics of jet quench-

ing at RHIC.

• Broadly directional peaks agree qualitatively with observed splitting of the away-side jet.

• The string theory setup involves significant idealizations of the experimentalsetup, notably replacing QCD byN = 4 super-Yang-Mills.

• Nevertheless, we hope that further improvements may lead to more precise com-parisons of string theory predictions with data.


ptReferences[1] The full MPEG can be found at

http://www.bnl.gov/RHIC/images/movies/Au-Au200GeV.mpeg, where it isattributed to the UrQMD group at Frankfurt, seehttp://www.physik.uni-frankfurt.de/∼urqmd/.

[2] M. Kaneta and N. Xu, “Centrality dependence of chemical freeze-out in Au +Au collisions at RHIC,”nucl-th/0405068 .

[3] F. Karsch, “Lattice QCD at high temperature and density,”Lect. Notes Phys.583(2002) 209–249,hep-lat/0106019 .

[4] PHENIX Collaboration, S. S. Adleret. al., “Centrality dependence of directphoton production in s(NN)**(1/2) = 200-GeV Au + Au collisions,”Phys.Rev. Lett.94 (2005) 232301,nucl-ex/0503003 .

[5] http://www.bnl.gov/bnlweb/pubaf/pr/PRdisplay.asp?prID=05-38.

[6] C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G. Yaffe, “Energy lossof a heavy quark moving through N = 4 supersymmetric Yang-Mills plasma,”hep-th/0605158 .


[7] S. S. Gubser, “Drag force in AdS/CFT,”hep-th/0605182 .

[8] Talk by B. Jacak, ”Plasma physics of the quark gluon plasma”, Boulder, May2006. Available at http://www4.rcf.bnl.gov/∼steinber/boulder2006/.

[9] J. Casalderrey-Solana, E. V. Shuryak, and D. Teaney, “Conical flow inducedby quenched QCD jets,”J. Phys. Conf. Ser.27 (2005) 22–31,hep-ph/0411315 .

[10] PHENIX Collaboration, S. S. Adleret. al., “Modifications to di-jet hadronpair correlations in Au + Au collisions at s(NN)**(1/2) = 200-GeV,”nucl-ex/0507004 .

Documents

Relativistic Heavy Ion Collisions and String Theory