Thermalization of the quark gluon matter in ultrarelativistic heavy ion collisions
Zhe Xu
Weihai, August 14, 2009
Institut für Theoretische PhysikGoethe-Universität Frankfurt, Germany
Zhe Xu, Weihai 2009 2/44
Outline
• Motivation
• Transport model
• Why 2-3 important
• Initial condition dependence
• Summary
Zhe Xu, Weihai 2009 3/44
Motivation
sonfermion/bo : ,)1(
1),(
/
TEeq epxf
thermal equilibrium
non-equilibrium
),(~),( pxfpxf eq
),(~),( pxfpxf eq
in kinetic equilibrium, but not in chemical equilibrium
not in kinetic equilibrium
Zhe Xu, Weihai 2009 4/44
deviation from thermal equilibrium
1. momentum spectra
2. mometum isotropy, average of angles
3. momentum energy tensor
TTdppdN /
2
2
eq
2
2
2
23
2
2
or ; 3
1 ),,(
1
T
zzzz
p
p
E
ppxf
E
ppd
NE
p
frame moving-co in ),,,( diag
),,(3
PPPeT
TTpxfppE
pdT
eq
eq
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High energy heavy ion collisions
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Momentum space anisotropy:Time dependence
M. Strickland
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P.Huovinen et al., PLB 503, 58 (2001)
nearly perfect fluid
Assumption: full thermalization at 0.6 fm/c
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Thermalization driven by plasma instabilitiesRefs.:
Mrowczynski;
Arnold, Lenaghan, Moore, Yaffe;
Rebhan, Romatschke, Strickland,
Bödeker, Rummukainen;
Dumitru, Nara;
Berges, Scheffler, Sexty
Dumitru, Nara, Strickland, PRD 75, 025016 (2007)
Dumitru, Nara, Schenke, Strickland, arXiv:0710.1223
Zhe Xu, Weihai 2009 9/44
),(),(),( pxCpxCpxfv ggggggggg
BAMPS: Boltzmann Approach of MultiParton Scatterings
A transport algorithm solving the Boltzmann-Equations for on-shell partons with pQCD interactions
new development ggg gg(Z)MPC, VNI/BMS, AMPT, PACIAE
Elastic scatterings are ineffective in thermalization !
Inelastic interactions are needed !
Transport Model
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Stochastic algorithm P.Danielewicz, G.F.Bertsch, Nucl. Phys. A 533, 712(1991)A.Lang et al., J. Comp. Phys. 106, 391(1993)
3x
)''()2(||'2)2(
'
'2)2(
'
!2
1
2)2(2
1
)''()2(||'''2)2(
'
'2)2(
'
!2
1
2)2(2
1 ),(
2121)4(42
'2'112212
32
3
13
13
23
23
1
2121)4(42
12'2'1212
32
3
13
13
23
23
1122
ppppMffE
pd
E
pd
E
pd
E
ppppMffE
pd
E
pd
E
pd
EpxC
collision rate per unit phase space for incomingparticles p1 and p2 with 3p1 and 3p2:
22212
32
3
1133
)2(1
22
22)2(2
1
3
sffE
p
Epxt
Ncoll
j
jj px
Nf
33
)2(1
3
x
tv
NN
Nrel
coll322
21
22
collision probability (Monte Carlo)
Space has to be dividedinto small cells !
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ZX and C. Greiner, PRC 71, 064901 (2005)
Interaction Probability
23321
3232
32323
32222
)(823for
32for
22for
x
t
EEE
IP
x
tvP
x
tvP
rel
rel
)''()2('2)2(
'
'2)2(
'
2
121321
)4(42
'2'11232
32
3
13
13
32 pppppME
pd
E
pdI
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A simple case
2->2 with isotropic differential cross section
p1 p2
p1‘
p2‘
const.cos
22
d
d
225
6
T
shear viscosity
Huovinen and Molnar, PRC 79, 014906 (2009)
Zhe Xu, Weihai 2009 13/44A. El, ZX and C. Greiner, arXiv: 0907.4500 [hep-ph]
225
6
T
),2/,2/,0( diag
eqTT
deviation from equilibrium, relaxation towards equilibrium
2/
PP
PP
T
L
one-dimensional expansion with Bjorken boost invariance
ln4 nns
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Relativistiv shock waves
I. Bouras et al. PRL 103, 032301 (2009)
P0 > P4
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)cosh()(
12
)(2
9
,)(2
9
222
22
222
242
222
242
ykmqkk
qg
mq
sgM
mq
sgM
gLPM
DDggggg
Dgggg
J.F.Gunion, G.F.Bertsch, PRD 25, 746(1982)
screened partonic interactions in leading order pQCD
Incoherent treatment: the formation time g1 cosh
ykg: mean free path
LPM suppression:
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frame CM in axis collision the and between angle :
frame CM to frame comoving the fromvelocity boost :
where,1
costanh1 with
)cosh()(
12
)(2
9
2
222
22
222
242
yL
yLkmqkk
qg
mq
sgM gLPM
DDggggg
Jet-quenching
O. Fochler, ZX and C. Greiner,PRL 102, 202301 (2009)
Zhe Xu, Weihai 2009 17/44
T.S.Biro at el., PRC 48, 1275 (1993) chemical equilibration of quarks and gluons by solving the rate equations
S.M.Wong, NPA 607, 442 (1996) kinetic equilibration within the relaxation time approach
J.Chen, H.Dong, K.Ohnishi, Q.Wang, arXiv:0907.2486 [nucl-th] shear viscosity using the variation method
rel
eq fffC
~][
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mb 0.57
mb 0.82
MeV 400T,3.0 for s
ggggg
gggg
-23 T ~ ,T~ , nvnR rel
ZX and C.Greiner, PRL 100, 172301 (2008)
ggggggggg
collision rates
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What leads to fast thermalization?
• large cross section (collision rate)
• large collision angle -> large momentum deflection -> fast momentum isotropization
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,)(2
9222
242
Dgggg mq
sgM
p1 p2
p1‘
p2‘
qT
small angle scatterings
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J.F.Gunion, G.F.Bertsch, PRD 25, 746(1982)
yyyE
pz small for tanhcos 3
Central plateau in cos3 , thus not small angluar scatterings
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gg gg: small-angle scatterings
gg ggg: large-angle bremsstrahlung
distribution of collision angles
at RHIC energies
central plateau
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2tr sin section cross transportd
dd
trgggg
trggggg
BUT, this is not the full story !
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Transport Rates
trggggg
trggggg
trgggg
trdrift
z
z
eq
RRRRtEp
Epdtd
)(/3/1
/122
22
ZX and C. Greiner, PRC 76, 024911 (2007)
ggggggggggggggi
Ep
n
fCpd
Ep
fCEppd
Rz
iz
iz
tri
,,
,
)31
(
][)2(
][)2(
with
2
2
3
3
2
2
2
2
3
3
• Transport rate is the correct quantity describing kinetic equilibration.
• Transport collision rates have an indirect relationship to the collision-angle distribution.
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53
tr
gggg
trggggg
R
R
Transport Rates for a static gluon gas
222222 )(ln~~: sss
tr RRgggg
01.0for)(ln~~: 222323 sss
tr RRggggg
01.0for)(ln~~ 232323 ssss
tr RR
Large Effect of gg->ggg
ZX and C.Greiner, PRL 100,172301, (2008))1(
2
TE
pff zeq assume:
Zhe Xu, Weihai 2009 26/44
)3(2
2
uu
TTT
zz
zzyyxx
From Navier-Stokes approximation
Cfv From Boltzmann-Eq.
Cpd
vuun
Cvpd
fvvpd
zzz
zz
3
32
23
32
3
3
)2()41()3(
15
2
)2()2(
322323
31
31
1)(
5
1
2
2
2
2
RRR
En
tr
E
p
E
p
z
z
relation between and Rtr
ZX and C.Greiner, PRL 100, 172301, (2008)
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)(7
1)( gggg
sggggg
s
Ratio of shear viscosity to entropy density in 2-3
AdS/CFTRHIC
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A. El, A. Muronga, ZX and C. Greiner, PRC 79, 044914 (2009)
]1)[,(),( with),(
method sGrad' using 2
00
3
0
ppcpxfpxfpxfCppp
pdP
PTc
eq
)1(
2
TE
pff zeq comparing
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Calculating f(x,p) in heavy ion collisionsusing 3+1 dimensional parton cascade
BAMPS
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Initial conditions
dcba
cdab
TbTa
T
jet
td
dpxfxpxfxK
dydydp
d
,;,
2
22
2
11
21
2 ˆ),(),(
ppjetAA
AAjet bTN )0(2
Glauber-type: Woods-Saxon profile, binary nucleon-nucleon collision
700/ dydN gfor a central Au+Au collision at RHICat 200 AGeV using p0=1.4 GeV
minijets production with pt > p0
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total transverse energy per rapidity at midrapidity
b=0 fms=0.3
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3-2 + 2-3: thermalization! Hydrodynamic behavior! 2-2: NO thermalization
simulation pQCD 2-2 + 2-3 + 3-2simulation pQCD, only 2-2
at collision center: xT<1.5 fm, || < 0.2
pT spectra
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time scale of thermalization in heavy ion collisions
eqeq
ZZeq
ZZ tt
E
pt
E
p
E
pt
E
p
0
2
2
02
2
2
2
2
2
exp)()(
eq = time scale of kinetic equilibration.
fm/c 1eq
theoretical result from parton cascadecalculations
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Transport Rates
trggggg
trggggg
trgggg
trdrift
z
z
eq
RRRRtEp
Epdtd
)(/3/1
/122
22
ZX and C. Greiner, PRC 76, 024911 (2007)
ggggggggggggggi
Ep
n
fCpd
Ep
fCEppd
Rz
iz
iz
tri
,,
,
)31
(
][)2(
][)2(
with
2
2
3
3
2
2
2
2
3
3
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5.
22
.32
.23 tr
trtr
R
RR
The drift term is large.
.
.32
.23
.22
trdrift
tr
tr
tr
R
R
R
R
ggggg interactions are essential for kinetic equilibration!
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Elliptic Flow and Shear Viscosity in 2-3 at RHIC 2-3 Parton cascade BAMPS ZX, Greiner, Stöcker, PRL 101, 082302, 2008
viscous hydro.Romatschke, PRL 99, 172301,2007
322323
31
31
1)(
5
1
2
2
2
2
RRR
En
tr
E
p
E
p
z
z
/s at RHIC > 0.08
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Initial condition dependenceof thermalization at RHIC
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Initial Condition – Wounded Nucleons
binaryN PP from Gluons and Quarks
A Afrom Gluons and Quarks
P+P using PYTHIA 6.4
semi-hard partonic collisionswith initial and final radiations
GeV200A of %80E
RHIC at Au Aucentral for 1000N
partons
binary
new work by L.Cheng
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Initial Condition – Color Glass Condensate
max
))(,(),(1
4 22
21
22
2
22
p
BAsc
cg kpxkxkdp
pd
N
N
dyrd
dN
,1min)(
1~),,( KLN
01.0
53.1
)(GeV2),(
),max()(
1~),,( KLN
),(
2
22
,2
,2
22
22
2
kx
s
ss
BApart
BAs
s
s
ss
k
Q
Qrkx
x
rnrxQ
kQ
Q
Qrkx
GeV200A of %80Egluons
Kharzeev, Levin, Nardi, NPA 730, 448 (2004); 747, 609 (2005)Hirano and Nara, NPA 743, 305 (2004)Adil, Drescher, Dumitru, Hayashigaki, Nara, PRC 74, 044905 (2006)
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Wounded nucleons vs Color Glass Condensate
Initial Conditions: I. Only gluons from WNII. Gluons and quarks from WN. Quarks as gluons.III. Color Glass condensate
Formation time: 0.15 fm/c
by L.Cheng and A. El
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Decrease of the transverse energy
3-fmGeV 6.0
15.0/
3.0
c
s
s
QGP from wn needs a larger /s than 0.15.QGP from cgc needs a smaller /s than 0.15.
(RHIC)GeV 33620
using BAMPS
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Kinetic equilibration
fm/c 1.5
exp)()( 02
2
02
2
2
2
2
2
eq
eqeq
ZZeq
ZZ tt
E
pt
E
p
E
pt
E
p
no difference betweenwn and cgc !
0.25 || fm, 5.1x
:region central the within
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Chemical equilibration due to gg ggg
nnn
neq
eq
3/T withT16
where,fugacity 32
wn: gluons system stays in chemical equilibrium.cgc: chemical equilibrium is achieved at the same timesacle, 1.5 fm/c, as the kinetic equilibration.
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Inelastic pQCD interactions (23 + 32) explain:
• Fast Thermalization,
• Large Collective Flow,
• Small shear Viscosity of QCD matter at RHIC,
because the bremsstrahlung favors large-angle radiation.
Summary
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chemical equilibration in a box
gluons and light quarks
eqnn /fugacity
qqqq ,qqqq
gqgq ,
,
qqgg
ggggggggg
gluons and charm quarks
ccgg
by J. Uphoff (diploma thesis)
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more details on elliptic flow at RHIC …
moderate dependence on critical energy density
/s at RHIC: 0.08-0.2
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… looking on transverse momentum distributions
gluons are not simply pions …
need hadronization (and models) to understand
the particle spectra
new work planned with G. Burau et al.
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ZX, C.Greiner, H. Stöcker, PRL 101:082302,2008
Perturbation QCD describes well
• fast thermalization, • low /s,• large v2 at RHIC.
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trireli
tri vnAR
due to the fact that a 2->3 process brings one more particletoward isotropy than a gg->gg process.
ggggggggg AA
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pt-spectra
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Life time of QGP
nT 3/
Tc=175 MeV