1

Quark-Gluon Plasma From Concepts

To “Precision” Science

Berndt Mueller

RHIC Users Meeting BNL - March 28, 2008

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Part 1…

The Quest for the

Quark-Gluon Plasma

3

QCD phase diagram

B

Hadronicmatter

Critical point?

Plasma

Nuclei

Chiral symmetrybroken

Chiral symmetryrestored

Color superconductor

Neutron stars

T

1st order line

Quark-Gluon

RHIC

quark-

gluon plasma

nucleons +

mesons

Melting nuclear matter

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QCD equation of state

( )27f4(2 8)Degrees of fr (2 3 ) 1 ( )eedom : N O gν = × + × × × × −⎡ ⎤⎣ ⎦

quarksgluons

colorcolorspin spin flavor

RHIC

2

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π ν =170 340 510 MeV

Weak or strong coupling?

Lattice QCD

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The QCD EoS (at =0)

The precise value of Tc is still under debate:

Tc = 170 ± 20 MeV with 20 - 30 MeV width.

EoS near Tc is far from ideal ultrarelativistic gas! Sound velocity cs

2 = P/ << 1/3.

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Lattice - susceptibilities

223XS

XS X SC

S S

−=−

−

pQGP

R4,2B =

∂4 lnZ / ∂B4

∂2 lnZ / ∂B2:

ΔB4

ΔB2

Eijiri Karsc

h Redlic

h

QCD matter above Tc may be a highly correlated system, but what is correlated are quarks and not hadrons! (What about gluons?)

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Part 2…

The 6 Stages of the Collision

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A multi-stage reaction

eqPre-equil. phase

Liberation of saturated low-x

glue fields (CGC) init

ial sta

te

pre

-eq

uilib

riu

m

QG

P a

nd

hyd

rod

yn

am

ic e

xp

an

sio

n had

ron

izati

on

had

ron

ic p

hase

an

d f

reeze

-ou

t

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Stage 1

Decoherence of

the initial state

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The initial state

The Color Glass Condensate model is based on a brilliant idea:

None of these components of the baryon wave function are calculable…

Baryon =c1 qqq + c2 qqqg + c3 qqqqq +L + c435 qqqggL g30

1 2 3 +L

…but this one is, because it contains a large scale

What applies to the proton (at high energy!), applies much better to a large nucleus, and at lower energy, because the gluon density per area is enhanced by a factor A1/3.

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Gluon saturation

~ 1/Q2

gluon density × area :

A1/3x−0.3

Qs2

≈1

2s ( , )Q x A⇒

Universal saturated state at small x: Qs >> QCD

Gribov, Levin, Ryskin ’83

Blaizot, A. Mueller ’87

McLerran, Venugopalan ‘94

“Color glass condensate” (CGC)

/sat

x

Evolution in x is described by BK or JIMWLK equations. Location of the onset of saturation is determined by fluctuations (Iancu, Peschanski,…)

p A

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CGC: Gluon production

Fields carried by moving sources interact non-linearly and generate classical spectrum of gluonic modes. This requires numerical solution of YM eqs. with CGC initial cond’s.

Krasnitz-Nara-Venugopalan, Lappi, Gelis

Simulation of Tν(x,t) possible.

QuickTime™ and a decompressor

are needed to see this picture.

Classical 2-particle rapidity correlations(Dumitru et al. ‘08)

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Stage 2

Entropy:

From 0 to (dS/dy=) 5000 in

0.000 000 000 000 000 000 000 002

seconds

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Final entropy

final2

( ) /( )

( ) /

( / )

dN dys

dV dy

dN dy

R

π ≤

:

s( 0 =1 fm/c) ≈33 fm-3 → T( 0 ) ≈300 MeV

Bjorken’s formula

Assuming isentropic expansion up to Tch, averaging over πR2 with R = 7 fm, and using lattice EOS:

dS

dyfinal

=d3rd3p(2π )3dy

−fi ln fi ±(1± fi )ln(1± fi )[ ]∫i∑

=5600 ±500 [for 6% central Au+Au @ 200]

Phase space analysis (Pal & Pratt):

dS

dyfinal

= (S / N)idNi

dyi∑ =5100 ±200 [for same cond.]

Chemical analysis (BM & Rajagopal):

How is this entropy produced?

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Decoherence

Coherent state:

α =e− α 2 /2 α n

n!n∑ n decoherence⏐ →⏐ ⏐ ⏐ ρmn = n α

2δ mn

Sdeco =12

ln 2πN( ) +1+O N−1( )( ) with N =α 2 ≈3

Counting causally disconnected transverse domains:

dSdec

dy≈

Qs2R2αs

2ln2CF ln2αs

2 +1⎛

⎝⎜⎞

⎠⎟≈1500 ≈

13

dSdy

final

for Qs2 =2 GeV2

In D dimensions after equilibration:

Seq(D ) ≈

43

DN

Clearly, fully 3-dimensional equilibration is essential - how and when?

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From 2D to 3D

1/Qs Nielsen-Olesen instability of longitudinal color-magnetic field(Itakura & Fujii, Iwazaki)

∂2φ

∂τ 2+

1

τ

∂φ

∂τ+

(kz − gAη )2

τ 2− gBz

⎛

⎝⎜⎞

⎠⎟φ = 0

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Weibel instability

Br

vr

vr

vr vr

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Color “turbulence”

Exponential growth saturates when

B2 > g2 T4.

Mrowczynski

Rebhan,Romatschke, Strickland

Arnold, Moore, Yaffe

Dumitru, Schenke

Wavelength and growth rate of unstable modes can be calculated perturbatively:

kz ~ gQs , ~ gQs < kz

Turbulent power

spectrum

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Turbulent color fields

Color correlation

lengthTime

Length (z)

Quasi-

abelian

Non-abelian

Noise

M. Strickland, hep-ph/051121

2

Extended domains of coherent color field can create “anomalous” contributions to transport coefficients and accelerate equilibration (as in EM plasmas).

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Stage 3

The (almost) perfect liquid

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Collision Geometry: Elliptic Flow

Elliptic flow (v2):

• Gradients of almond-shape surface will lead to preferential expansion in the reaction plane• Anisotropy of emission is quantified by 2nd Fourier coefficient of angular distribution: v2

prediction of fluid dynamics

Reaction plane

x

z

y

Bulk evolution described by relativistic fluid dynamics,

assumes that the medium is in local thermal equilibrium,

but no details of how equilibrium was reached.

Input: (x,i), P(), (,etc.).

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v2(pT) vs. hydrodynamics

Mass splitting characteristic property of hydrodynamics

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Elliptic flow “measures” QGP

Boost invariant hydrodynamics with T00 ~ 1 requires /s ≤ 0.1

∂Tν =0 with T ν =( + P)uuν −Pgν +Πν

Π

dΠν

d+ uΠνλ +uνΠλ( )

duλ

d⎡

⎣⎢

⎤

⎦⎥= ∂uν + ∂νu −trace( )−Πν

Relativistic viscous hydrodynamics:

13

tr3f

pnpλ

σ≈ =

Small shear viscosity implies:

The QGP is an almost perfect liquid

Romatschke & Romatschke

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String theory weighs in

General argument [Kovtun, Son & Starinets, PRL 94 (2005) 111601] based on duality between thermal QFT and string theory on curved background with the “black-brane” metric:

Dominated by absorption of (thermal) gravitons by the black hole:

σabs ω( ) =

8πGω

dtd3x∫ eiωt Txy t,rx( ),Txy 0,0( )⎡

⎣⎤⎦ ω→ 0⏐ →⏐ ⏐ a (horizon area)

Therefore:

s=σabs(0)16πG

4Ga

⎛

⎝⎜⎞

⎠⎟=

14π

horizon

(3+1)-D world

r0

r

0=

1πT

(t,x)

(0,0)

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An age-old problem solved!

( ) ( )23

and are defined as coefficients in the

expansion of the stress tensor in gradients of the velocity fie

viscosity

ld

Shear b k

:

ul

ik i k i k k i ik ikik i kT u u P u uu u u uε δ ς δη δ= + + ∇ + +∇ − ∇⋅ ∇ ⋅−

Unfortunately, this renders relativistic viscous hydrodynamics a-causal !

Solution, in principle: include time derivatives (Israel,Stewart, Müller - 1960s).

Full second-order expression for shear stress in conformal limit finally given by Baier, Romatschke, Son, Starinets & Stephanov (arXiv:0712.2451):Πν =−σ ν + s uα ∂α σ μν +

σ μν ∂α uα

d −1

⎛

⎝⎜⎞

⎠⎟

+λ1

η 2Π μ

λ Πν λ −λ 2

ηΠ μ

λ Ων λ + λ 3Ωμ

λ Ων λ

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Viscosity of RHIC

PT =Peq +Π+12Φ

PT =Peq +Π−Φ

s

dΦ

dτ=

4η

3τ− 1+

4τ s

3τ⎛⎝⎜

⎞⎠⎟

Φ −λ1

2η 2Φ2

τ b

dΠ

dτ=

ς

τ− Π

/ s, ς / s

from lattice

Lattice EOS

τ s = τ b =2 − ln2

2πT(N = 4 SUSY)

0.1 0.2 0.3 0.4 0.5 0.6

1.0

0.5

2.0

0.2

5.0

0.1

10.0

20.0

T HGeVL

zH1êfm̂ 3L, hH1êfm̂ 3L

Tc

R.J.Fries, BM, A. Schäfer, tbp

PTPL

Φ2 Π

Peq

PL ( 0 ) =0

Tc

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Stage 4

Hadronizing the

Quark-Gluon Plasma

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v2(pT) vs. hydrodynamics

Failure of ideal hydrodynamics tells us how hadrons form

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Quark number scaling of v2

In the recombination regime, meson and baryon v2 can be obtained from the quark v2 :

( ) ( )2 2 2 2v22

v3

v3v Btt

q tM q tp ppp

⎛ ⎞= ⎜

⎛ ⎞= ⎜ ⎟

⎝⎝ ⎠ ⎠⎟

qqq

qqT,,v

Emitting medium is composed of unconfined,

flowing quarks.

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CEP Observables

Observables that are not be sensitive to final state interactions

After Freeze-out, no effect of final state interactions

Chemical Freeze-out • usually assumed to be momentum independent

• but this is not right chemical freeze-out timeis pT (or yT) dependent

Larger pT (or yT), earlier ch. Freeze-out

Critical fluctuations, the primary signature of the CEP, are modifiedDuring expansion until chemical or kinetic freeze-out, in addition to being suppressed near CEP by critical slowing down.

Critical point?

B

T

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Emission Time Distribution

Emission Time

• Larger yT, earlier emission

• To minimize resonance effect, yT is used instead of pT

• No CEP effect (UrQMD)

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Focus on chemistry

Tc

Tc

ratio near CEP falling with pTp / p

Asakawa, Bass, BM, Nonaka ‘08

pT

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STAR PRELIMINARY

RHIC can do it!

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Part 3…

Probing the structure

of the Quark-Gluon Plasma

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qq

Radiative energy loss:

Energy loss in QCD

Density of scattering centers

Range of color forceScattering power of the QCD medium:

dE

dx=−

αsC2

4q̂L

q̂ = q2dq2 dσ

dq2∫ ≡σ kT2 = dx− Fi

+(x−)F +i (0)∫

dE

dx=−

αsC2

4mD

2 1−y

u 2

⎛

⎝⎜⎞

⎠⎟lnc

ETmD

2Nonradiative energy loss:

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Towards q-hat

3-D ideal hydrodynamics withradiative energy loss only

Bass, Majumder, Qin, Renk et al. (tbp)

Numbers change by up to factor of 2, depending on whether q-hat is scaled with T3, s, or 3/4 !

Other unresolved issues: Consistent treatment of virtuality of parton created by hard scattering;Nature of scattering centers

~4~2~10

AMYHTASW

q̂0 (GeV/fm3)

~20 ~4

T3

3/4

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Closing in on q-hat

Zhang et al. (using higher twist energy loss theory + back-to-back coincidences)

RAA vs. reaction plane

Bass et al.

More differential measurements of jet quenching with very high statistics are needed, as well as consistent theories of jet quenching for these observables.

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Collisions + radiation

Qin, Ruppert, Gale, Jeon, Moore & Mustafa, PRL 100,072301 (2007)

collisons

radiationcoll+rad

collisons

radiationcoll+rad

Inclusion of collisional energy loss leads to reduction of αs from 0.33 to 0.27, correspondingto a reduction of extracted valueof q-hat by 33%.

Contributions from collisional and radiative energy loss may be separated due to theirdifferent fluctuations (Poisson vs. intermittent) by comparing singles quenching (RAA) with coincident back-to-back quenching (IAA), and by their different quark mass dependence by comparing with charm RAA.

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Connecting jets with the medium

Hard partons probe the medium via the density of colored scattering centers:

q̂ = q2dq2 dσ / dq2( )∫ : dx− F ⊥+(x−)F⊥

+(0)∫If kinetic theory applies, thermal gluons are quasi-particles that experience the same medium. Then the shear viscosity is:

≈1

3ρ pλ f (p) =

1

3

p

σ tr (p)

In QCD, small angle scattering dominates:

σ tr (p) ≈2q̂

p2

ρ

With p ~ 3T and s 3.6(for gluons) one finds:

s

≈ 1.25T 3

q̂A. Majumder, BM, X-N. Wang, PRL 99 (2007)

192301

From RHIC data:

T0 ≈335 MeV, ̂q0 ≈2.8 GeV2 /fm → ( / s)0 ≈0.10

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An interesting question

How does a fast parton interact with the quark- gluon plasma ?

What happens to the energy and momentum lost by a fast parton on its passage through the hot medium ?

How does the energy and momentum perturbation of the medium propagate ?

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What happens here ?!?

Trigger jet

Back jet

Thanks to: E.

Wenger (PHOBOS

)

Hard scatteri

ng

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Parton-medium coupling

Color field of moving parton interacts with

the quanta of the medium

p

E∂∂x −∇p ⋅D(x, p)⋅∇p

⎡

⎣⎢

⎤

⎦⎥ f0 x, p( ) =C f0[ ]

with

Dij (x, p) = dt'Firx,t( )Fj

rx+

rv(t'−t),t'( )

−∞

t

∫ .

∂∂xμ

T μν = J ν

with T μν = ε + p( )uμuν − pgμν + Tdiss

μν

J ν = dp∫ pν ∇ p ⋅D(x, p) ⋅∇ p f (x, p)

⎧⎨⎪

⎩⎪

Space-time distribution of collisional eneregy loss

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Unscreened source

For an unscreened color charge, an analytical result is obtained in u1 limit:

J (x) = J 0 ,ruJ 0 −

rJ V( ) with

J 0 (x) = f(,z,t) u2 1−z−

z−2 + 2 z−+

u2

2 + 2z−2( )

1/2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

⎛

⎝⎜⎜

⎞

⎠⎟⎟

rJ V (x) = f(,z,t)

rx−

rut

2 + 2z−2

2 2z−2 + (u2 + 2)2

2 + 2z−2( )

1/2 +u44

2 + 2z−2( )

3/2 −2uz−

⎛

⎝⎜⎜

⎞

⎠⎟⎟

f(,z,t) =g2 %Q2mD

2

32π 2 2 + 2 (z−ut)2⎡⎣ ⎤⎦3/2 , z− =z−ut

Spatial integral over deposited energy and momentum distribution equals collisional energy loss; radiated gluons increase effective color charge.

R.B. Neufeld

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Linearized hydro

Linearize hydro eqs. for a weak source: T00 0 + , T0i gi .

∂∂t

δε +∇ ⋅rg = J 0 ∂

∂t

rg + cs

2∇δε +η

ε 0 + p0

4

3∇ ∇ ⋅

rg( ) =

rJ

Solve in Fourier space for longitudinal sound:

=iω + iΓ sk

2( ) J 0 + kJL

ω 2 − cs2k2 + iΓ sωk2

gL = ics

2kJ 0 + ω JL

ω 2 − cs2k2 + iΓ sωk2

… and dissipative transverse perturbation:gT =iJ T

ω + 34 iΓsk

2

See: J. Casalderrey-Solana, E.V. Shuryak and D. Teaney, arXiv:hep-ph/0602183

Use: u =0.99955c, cs2 =

13, Γs =

13πT

for T =350 MeV.

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pQCD vs. N=4 SYM

Chesler & Yaffe

arXiv:0712.0050

Neufeld et al. arXiv:0802.2

254

u = 0.99955 c

u = 0.75 c

(z - ut)

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Mach cone: cs and η

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/ s = 0.13

[Xu & Greiner]

/ s = 0.48

[Arnold, Moore & Yaffe (AMY)]

Plasma behind jet:Correlated flow, not just thermal !

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RHIC data

Away side shape modification

2.5 < pT

trig< 4 GeV/c

1< pT

assoc < 2.5 GeV/c

Technique: Measure 2- and 3- particle correlations on the away-side triggered by “high” pT hadron in central coll’s. Cone-shaped emission should show up in 3-particle correlations as signal on both sides of backward direction.

Central Au+Au 0-12% (STAR)

(⏐ ⏐ 1-⏐ ⏐ 2)/2

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Summary 1

The RHIC program has shown that

• equilibrated matter is rapidly formed in heavy ion collisions;

• new, powerful probes become available at collider energies;

• systematic study of matter properties is possible.

QGP appears to be a strongly coupled, maybe turbulent color liquid with novel and unanticipated transport properties.

Experimental and theoretical surprises have opened a gold mine for theorists:

• extreme opaqueness of matter to colored probes;

• collective flow phenomena;

• collective medium response to jets;

• large enhancement of baryon production;

• connection to string theory and AdS/CFT duality.

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Summary 2

Ultimate success of the RHIC program requires:

• precision data for key (often rare) observables;

• continued progress of our understanding of thermal QCD;

• sustained collaboration between theorists and experimentalists on precision data interpretation.

Superficially different observables (flow, jet quenching, two-particle correlations) are connected at a deep level.

Their exploration in a comprehensive framework will lead to deep insights into how bulk QCD matter behaves and, ultimately, to the fulfillment of the scientific promise of RHIC.

The LHC heavy ion program will help resolve ambiguities, due to its extended kinematic range for critical observables.

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Summary 3

Experimental and theoretical surprises have opened a gold mine for theorists, but to extract the gold, painstaking work will be required in collaboration between theorists and experimentalists.

The first steps have been taken:

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For report and details see:

https://wiki.bnl.gov/TECHQM/index.php/Main_Page

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THE END