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![Page 1: 1 Quark-Gluon Plasma From Concepts To “Precision” Science Berndt Mueller RHIC Users Meeting BNL - March 28, 2008.](https://reader036.fdocuments.in/reader036/viewer/2022062314/56649ef55503460f94c08e9c/html5/thumbnails/1.jpg)
1
Quark-Gluon Plasma From Concepts
To “Precision” Science
Berndt Mueller
RHIC Users Meeting BNL - March 28, 2008
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2
Part 1…
The Quest for the
Quark-Gluon Plasma
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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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4
QCD equation of state
( )27f4(2 8)Degrees of fr (2 3 ) 1 ( )eedom : N O gν = × + × × × × −⎡ ⎤⎣ ⎦
quarksgluons
colorcolorspin spin flavor
RHIC
2
30
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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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 ?
QuickTime™ and a decompressor
are needed to see this picture.
QuickTime™ and a decompressor
are needed to see this picture.
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