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Transport Theory for the Transport Theory for the Quark-Gluon PlasmaQuark-Gluon Plasma
V. GrecoV. Greco UNIVERSITY of CATANIAUNIVERSITY of CATANIAINFN-LNSINFN-LNS
Quark-Gluon Plasma and Heavy-Ion Collisions – Turin (Italy), 7-12 March 2011 Quark-Gluon Plasma and Heavy-Ion Collisions – Turin (Italy), 7-12 March 2011
x
yz
0)(
0)(
xj
xT
B
HydrodynamicsHydrodynamicsNo microscopic descriptions (mean free path -> 0, =0)
implying f=feq
+ EoS P()
All the observables are in a way or the otherrelated with the evolution of the phase spacedensity :
What happens if we drop such assumptions?There is a more “general” transport theoryvalid also in non-equilibrium?Is there any motivation to look for it?
pdxd
dNtpxf
33),,(
Picking-up four main results at RHIC Picking-up four main results at RHIC
Nearly Perfect FluidNearly Perfect Fluid,, Large Collective FlowsLarge Collective Flows:: Hydrodynamics good describes dN/dpT + v2(pT) with mass
ordering
BUT VISCOSITY EFFECTS SIGNIFICANT (finite and f ≠feq) High OpacityHigh Opacity, Strong, Strong Jet-quenchingJet-quenching:: RAA(pT) <<1 flat in pT - Angular correlation triggered by jets pt<4 GeV
STRONG BULK-JET TALK: Hydro+Jet model non applicable at pt<8-10 GeV
Hadronization modifiedHadronization modified, Coalescence, Coalescence: B/M anomalous ratio + v2(pT) quark number scaling (QNS) MICROSCOPIC MECHANISM RELEVANT
Heavy quarks strongly interactingHeavy quarks strongly interacting:: small RAA large v2 (hard to get both) pQCD fails: large scattering
rates NO FULL THERMALIZATION ->Transport Regime
BULK BULK (p(pTT~T)~T)
MINIJETS MINIJETS (p(pTT>>T,>>T,QCDQCD))
CGC (x<<1)Gluon saturation
Heavy Quarks Heavy Quarks (m(mqq>>T,>>T,QCDQCD))
Microscopic Microscopic MechanismMechanism
Matters!Matters!
Initial Conditions Quark-Gluon Plasma Hadronization
ppTT>> T , intermediate p>> T , intermediate pTT
m >> T , heavy quarksm >> T , heavy quarks /s >>0 , high viscosity/s >>0 , high viscosity Initial time studies of thermalizationsInitial time studies of thermalizations Microscopic mechanism for HadronizationHadronization can modify QGP observable
Non-equilibrium + microscopic scale are relevant in all the subfields
Plan for the LecturesPlan for the Lectures Classical and Quantum Transport Theory - Relation to Hydrodynamics and dissipative
effects - density matrix and Wigner Function Relativistic Quantum Transport Theory - Derivation for NJL dynamics - Application to HIC at RHIC and LHC
Transport Theory for Heavy Quarks - Specific features of Heavy Quarks - Fokker-Planck Equation - Application to c,b dynamics
),()(
),(),(
0
pxfp
xFxm
p
pxfpd
dp
xd
dx
d
pxdf
pdxd
dPptpxtxtprf
N
iii 33
1
33 ))(())((),,(
For a classical relativistic system of N particles
Gives the probability to find a particle in phase-space
pdxd
dPppxxpxf
N
iii 44
1
44 ))(())((),(
f(x,p) is a Lorentz scalar &P0=(p2+m2)1/2
Classical Transport TheoryClassical Transport Theory
If one is interested to the collective behavior or to the behavior of a typical particleknowledge of f(x,p) is equivalent to the full solution … to study the correlations among particles one should go to f(x1,x2,p1,p2) and so on…
Liouville Theorem: if there are only conservative forces -> phase-space density is a constant o motion
Force
),()(1
0),()( pxfxFpm
pxfp
xFxm
ppx
0
fFfvt
fpr
The non-relativistic reduction
Relativistic Vlasov Equation
Liouville -> Vlasov -> No dissipation + Collision= Boltzmann-Vlasov
DissipationEntropy production
Allowing for scatteringsparticles go in and out phase space
(d/dt) f(x,p)≠0
),]([),()(1
pxfCpxfxFpm px
Collision term
The Collision TermIt can be derived formally from the reduction of the 2-body distribution Function in the N-body BBGKY hierarchy.The usual assumption in the most simple and used case:1) Only two-body collisions2) f(x1,x2,p1.p2)=f(x1,p1) f(x2,p2)
The collision term describe the change in f (x,p) because:a) particle of momentum p scatter with p2 populating the phase space in (p’1,p’2)
),,(),,,( 21]2[
212123
13
23 ppxfppppwpdpdpdCloss
probability finding 2 particles in p e p2
and space x
Probability to makethe transition
Sum over all themomenta the kick-out
The particle in (x,p)
dvpdpdw 1223
13)2112(
),,(),,,( 21]2[
212123
13
23 ppxfppppwpdpdpdCgain
Collision Rate
In a more explicit form and covariant version:
gain
loss
At equilibrium in each phase-space region Cgain =Closs 0)],([ 0 pxfC
0][
fffC
vv
1
Relaxation timetime between 2 collisions
When one is close to equilibrium or when the mfp is very small One can linearize the collision integral in f=f-f0 <<f
What is the f0(x,p)=0?
Local Equilibrium SolutionLocal Equilibrium Solution
The necessary and sufficient condition to have C[f]=0 is
),(),(),(),( 2121 pxfpxfpxfpxf
),(),(),(),(),,,(][ 2121212123
13
23 pxfpxfpxfpxfppppwpdpdpdfC
Noticing that p1+p2=p’1+p’2 such a condition is satisfied by the relativistic extension of the Boltzmann distribution:
It is an equilibrium solution also with LOCAL VALUES of T(x), u(x), m(x)
=1/T temperatureu collective four velocity chemical potential
The Vlasov part gives the constraint and the relation among T,u, locally
Main points:• Boltzmann-Vlasov equation gives the right equilibrium distributions• Close to equilibrium there can be many collisions with vanishing net effect
Relation to HydrodynamicsRelation to Hydrodynamics
0)(
0)(
xj
xT
Ideal Hydro
),(
),(
4
4
pxfpppdT
pxfppdj
)()( 04044 ffpd
mffmfxmFpdfppdj p
InsertingVlasov Eq.
Integral of a divergency
We can see that ideal Hydro can be satisfied only if f=feq , on the other hand the underlying hypothesis of Hydro is that the mean free path is so small that the f(x,p)is always at equilibrium during the evolution.
Similarly ∂T, for f≠feq and one can do the expansion in terms of transportcoefficients: shear and bulk viscosity , heat conductivity [P. Romatschke]
At the same time f≠feq is associated to the entropy production ->
General definitions
Notice in Hydro appear only p-integrated quantities
),(ln1),()( 4 pxfpxfppdxS
0
0444 ln)(...ln)()ln1(
f
fffpd
mffppdffppdS xx
0,0ln)1( xxx
Approach to thermal equlibrium is always associated to entropy production
Entropy Production <-> Thermal EquilibriumEntropy Production <-> Thermal Equilibrium
All these results are always valid and do not rely on the relaxation time approx. more generally:
fffffCpdst
sS ln'2'12
S=0 <-> C[f]=0Collision integral is associated to entropy production but if a local equilibrium
is reached there are many collisions without dissipations!
Boltzmann-Vlasov Eq.
Does such an approach can make sense for a quantum system?
One can account also for the quantum effect of Pauli-Blocking in the collision integral
)1)(1()1)(1( 212121212121 ffffffffffff
does not allow scattering if the final momenta have occupation number =1-> Boltzmann-Nordheim Collision integral
This can appear quite simplistic, but notice that C[f]=0 now is
So one gets the correct quantum equilbrium distribution, but what isF(x,p) for a quantum system?
Quantum Transport TheoryQuantum Transport TheoryIn quantum theory the evolution of a system can be described in terms ofthe density matrix operator:
2/ˆ2
),(~
yxxxAxedy
pxAipy
For any operator one can define the Weyl transform of any operator:
which has the property
The Weyl transform of the density operator is called Wigner function
)()(2
ˆ2
),(
xxe
dyxxe
dypxf
ipyipy
W
fW plays in many respectsthe same role of the distribution function in statistical mechanics
and any expectation value can be calculated as
and by (*)
(*)
Properties of the Wigner Function
)()()(),(
)()()(),(
ppppxfdx
xxxpxfdp
W
W
However for pure state fW can be negative so it cannot be a probabilityOn the other hand if we interpret its absolute value as a probabilty it doesnot violate the uncertainty principle because one can show:
1
),( pxfW 1211
2
1 pxN
dxdp
dN
So if we go in a phase space smaller thanxp<h/2 one can never locate a particleIn agreement with the uncertainty principle
Quantum Transport EquationQuantum Transport Equation
Ht
ˆ,ˆˆ
0ˆ,ˆ
ˆ
xHt
x 0ˆ
2
ˆ,ˆ
ˆ
2
2
xUm
p
txe
dy pyi
One can Wigner transform this or the Schr. Equation
After some calculations one gets the following equation
0),())((2)!12(
1
0
122
kw
kpx
k
WW pxfxUkx
f
m
p
t
f I
This exactly equivalent to the Equation for the denity matrix or the Schr. Eq.NO APPROXIMATION but allows an approximation where h does not appear explicitly and still accounting for quantum evolution when the gradient of the potential are not too strong :
0...),()(12
),()( 332
pxfxUpxfxU WpxWpx
0
wpxWW fUx
f
m
p
t
f
This has the same form of the classicaltransport equation, but it is for exampleexact for an harmonic potential
See : W.B. Case, Am. J. Phys. 76 (2008) 937
Transport Theory in Field TheoryTransport Theory in Field TheoryOne can extend the Wigner function (4x4 matrix):
2/,:)()(:)2(
),(4
4
yxxxxeyd
pxFpy
i
It can be decomposed in 16 indipendent components (Clifford Algebra)
TPVS FFiFFFF
2
1555
For example the vector current
),(4),( 44 pxFpdpxFTrpdj V
In a similar way to what done in Quantum mechanicsone can start from the Dirac equation for the fermionic field
See : Vasak-Gyulassy- Elze, Ann. Phys. 173(1987) 462 Elze and Heinz, Phys. Rep. 183 (1989) 81 Blaizot and Iancu, Phys. Rep. 359 (2002) 355
Just for simplicity lets consider the case with only a scalar field
0:)(::)()(:)2(
),(2 4
4
xxxeRd
pxFmi
p Rip
For the NJL G
...)(2
)()( xR
xx x
This is the semiclassical approximation. If one include higher order derivatives getsan expansion in terms of higher order derivatives of the field and of the Wigner function
1 WF PX
The validity of such an expansion is based on the assumption ħ∂x∂pFW >>1Again the point is to have not too large gradients:
XF typical length scale of the fieldPW typical momentum scale of the system
A very rough estimate for the QGPXF ~ RN ~ 4-5 fm , PW ~ T ~ 1-3 fm-1 -> XF·PW ~ 5-15 >> 1
better for larger and hotter systems
0),()(2
)]([2
pxFx
ixmp
iW
px
Substituting the semiclassical approximation one gets:
There is a real and an imaginary part
0),(ˆ* pxFMp W Which contains the in medium mass-shell
Including more terms in the gradient expansion would have brougth termsbreaking the mass-shell constraint
VSW FFF
Decomposing, using both real and imaginary part and taking the trace
This substitute the force term mF(x) of classical transport
Vlasov Transport Equation in QFTVlasov Transport Equation in QFT
Quantum effects encoded in the fields while f(x,p) evolution appears as the classical one.
Transport solved on latticeTransport solved on lattice
...22 Cfp
Solved discretizing the space in x, ycells
See: Z. Xhu, C. Greiner, PRC71(04)
t0
3x0exact solutio
n3x
Putting massless partons at equilibrium in a boxthan the collision rate is
Rate of collisionsper unit of phase
space
Approaching equilibrium in a box
where the temperature is
Highly non-equilibrated distributions
F.Scardina
anisotropy in p-space
Transport vs Viscous Hydrodynamics in 0+1D
LK
s
TK
/5
1
Knudsen number-1
000 2.14
2 T
sK
Huovinen and Molnar, PRC79(2009)
Transport TheoryTransport Theory
valid also at intermediate pvalid also at intermediate pTT out of equilibrium: out of equilibrium: region of modified hadronization at RHICregion of modified hadronization at RHIC
valid also at high valid also at high /s/sLHC and/or hadronic phaseLHC and/or hadronic phase
Relevant at LHC due to large amount of minijet production Relevant at LHC due to large amount of minijet production
Appropriate for heavy quark dynamics Appropriate for heavy quark dynamics can follow exotic can follow exotic non-equilibriumnon-equilibrium phase CGC: phase CGC:
A unified framework against a separate modelling with a wider range of validity in pT + microscopic level.
Applications of transport approach to the QGP Physics- Collective flows & shear viscosity- dynamics of Heavy Quarks & Quarkonia
0)(
0)(
xj
xT
B
HydrodynamicsHydrodynamicsNo microscopic details
(mean free path -> 0, =0)
+ EoS P()
Parton cascadeParton cascade
v2 saturation pattern reproduced
First stage of RHICFirst stage of RHIC
22 Cfp
Parton elastic 22 interactions
- P=/3)
Information from non-equilibrium: Elliptic FlowInformation from non-equilibrium: Elliptic Flow
xy z
px
py
22
22
xy
xyx
cc22ss=dP/d=dP/d
v2/ measures the efficiencyof the convertion of the anisotropy
from CoordinateCoordinate to Momentum spaceMomentum space
...)2cos(v21 2 TT dp
dN
ddp
dNFourier expansion in p-space
||viscosityviscosity
EEoSoS
Massless gas =3P -> c2s=1/3
Bhalerao et al., PLB627(2005)
More generally one can distinguish:
-Short range: collisions -> viscosityShort range: collisions -> viscosity-Long range: field interactionLong range: field interaction -> -> ≠ ≠ 3P3P
D. Molnar & M. Gyulassy, NPA 697 (02)
2v
time
c2s= 0.6
c2s= 0.1
Measure of Measure of P gradientsP gradients
Hydrodynamics
=0
c2s= 1/3
Parton Cascade
If v2 is very large
To balance the minimum vv44 >0 require >0 require
v4 ~ 4% if v2= 20%
222
4224
4 )(
6)4cos(
yx
yyxx
pp
ppppv
At RHIC a finite vAt RHIC a finite v44 observed observed
for the first time !for the first time !
More harmonics needed to describe an elliptic deformation -> v4
P. Kolb
Viscosity cannot be neglectedViscosity cannot be neglected
but it violates causality, it violates causality, IIII00 order expansion needed -> Israel- order expansion needed -> Israel-Stewart tensor based on entropy Stewart tensor based on entropy increase ∂increase ∂ss
P. Romatschke, PRL99 (07)
y
v
A
F x
yz
x
dissipidealTT
Relativistic Navier-Stokes
two parameters appears +
f ~ feq reduce the pT validity range
Transport approachTransport approach
Collisions -> Collisions -> ≠0≠0Field Interaction -> ≠3PFree streaming
C23 better not to show…
Discriminate short and long range interaction:Collisions (≠0) + Medium Interaction ( Ex. Chiral symmetry breaking)
decrease
We simulate a constant shear viscosityWe simulate a constant shear viscosity
sTn
pTr trtr /
1
415
4)),(( ,
=cell index in the r-space
Neglecting and inserting in (*)
4
1
s3
2
45
24 T
g
T
Pns
2
1
Ttr At T=200 MeVAt T=200 MeV
trtr10 mb10 mb
Time-Space dependent cross Time-Space dependent cross
section evaluated locallysection evaluated locally
V. Greco at al., PPNP 62 (09)G. Ferini et al., PLB670 (09)
(*)cost.)4(15
4
Tn
p
s tr Relativistic Kinetic theory Cascade code
The viscosity is kept constant varying
A rough estimate of A rough estimate of (T) (T)
=cell index in the r-space
a)collisions switched off
for <c=0.7 GeV/fm3
b) b) /s increases in the cross-over /s increases in the cross-over region, faking the smooth region, faking the smooth transition between the QGP and transition between the QGP and the hadronic phasethe hadronic phase
Two kinetic freeze-out schemeTwo kinetic freeze-out scheme
Finite lifetime for the QGP small /s fluid!
At 4/s ~ 8 viscous hydrodynamics is not applicable!
No f.o.
sn
ptr /
1
15
1
This gives also automatically a kind of core-corona effect
4/s >3 too low v2(pT) at pT1.5 GeV/c even with coalescence
4/s =1 not small enough to get the large v2(pT) at pT2 GeV/c
without coalescence
Agreement with Hydro at low pT
Parton Cascade at fixed shear viscosity
Role of ReCo for /s estimate
Hadronic Hadronic /s included /s included
shape for vshape for v22(p(pTT) )
consistent with that consistent with that
needed needed
by coalescenceby coalescenceA quantitative estimate needs an EoS with ≠ 3P : cs
2(T) < 1/3 -> v2 suppression ~~ 30%
-> /s ~ 0.1 may be in ~ 0.1 may be in
agreement agreement with coalescencewith coalescence
Short Reminder from coalescence…Short Reminder from coalescence…
Quark Number ScalingQuark Number Scaling
n
p
nT
2V1
Molnar and Voloshin, PRL91 (03)Greco-Ko-Levai, PRC68 (03)Fries-Nonaka-Muller-Bass, PRC68(03)
2
22)2()(
T
T
q
T
T
M ppd
dNαp
pd
dN
3
22)3()(
T
T
q
T
T
B ppd
dNp
pd
dN
)2cos(v21φ 2q
TT
q
TT
q
dpp
dN
ddpp
dN
Is it reasonable the vIs it reasonable the v2q 2q ~0.08~0.08 needed by needed by Coalescence scaling ?Coalescence scaling ?
Is it compatible with a Is it compatible with a fluid fluid /s /s ~ 0.1-0.2~ 0.1-0.2 ? ?
I° Hot Quark
Effect of Effect of /s of the hadronic phase/s of the hadronic phase
Hydro evolution at /s(QGP) down to thermal f.o. ~50%Error in the evaluation of h/s
Uncertain hadronic /s is less relevant
Effect of Effect of /s of the hadronic phase at LHC/s of the hadronic phase at LHC
RHIC – 4/s=1 + f.o.
RHIC – 4/s=2 +No f.o.
Suppression of v2 respect the ideal 4/s=1
LHC – 4/s=1 + f.o.
At LHC the contamination of mixed and hadronic phase becomes negligibleLonger lifetime of QGP -> v2 completely developed in the QGP phase
S. Plumari, Scardina, Greco in preparation
Impact of the Mean Field and/or Impact of the Mean Field and/or
of the Chiral phase transitionof the Chiral phase transition
- Cascade Boltzmann-Vlasov Transport
- Impact of an NJL mean field dynamics
- Toward a transport calculation with a lQCD-EoS
NJL Mean FieldNJL Mean Field
Two effects:Two effects:
≠ ≠ 3p no longer a massless free gas, c3p no longer a massless free gas, css <1/3 <1/3
Chiral phase transitionChiral phase transition
)()(1)2(
)(4)(3
3
TfTfE
pdTMNgNmTM
pcf
Associated Gap Equation
free gas scalar field interaction
Fodo
r, JE
TP(2
006)NJL
gas
Boltzmann-Vlasov equation for the NJLBoltzmann-Vlasov equation for the NJL
Contribution of the NJL Contribution of the NJL
mean fieldmean field
Numerical solution with Numerical solution with -function test particles-function test particles
Test in a Box with equilibrium Test in a Box with equilibrium ff distribution distribution
Simulating a constant Simulating a constant /s with a NJL mean field/s with a NJL mean field
np15
4
Massive gas in relaxation time approximation
The viscosity is kept modifying locally the cross-section
=cell index in the r-spaceM=0
TheoryCode
=10 mb
Au+Au @ 200 AGeV for central collision, b=0 fm.Au+Au @ 200 AGeV for central collision, b=0 fm.
Dynamical evolution with NJLDynamical evolution with NJL
Does the NJL chiral phase transition affect the elliptic flow of a fluid at fixed /s?
S. Plumari et al., PLB689(2010)
- NJL mean field reduce the vNJL mean field reduce the v22 : attractive field : attractive field
- If If /s is fixed effect of NJL compensated by cross section increase/s is fixed effect of NJL compensated by cross section increase
- vv22 /s not modified by NJL mean field dynamics!/s not modified by NJL mean field dynamics!
Next stepNext step - use a quasiparticle model - use a quasiparticle model
with a realistic EoS [vwith a realistic EoS [vss(T)](T)]
for a quantitative estimate of for a quantitative estimate of /s /s
to compare with Hydro…to compare with Hydro…
WB=0 guarantees Thermodynamicaly consistency
Using the QP-model of Heinz-Levai Using the QP-model of Heinz-Levai U.Heinz and P. Levai, PRC (1998)
M(T), B(T) fitted to lQCD [A. Bazavov et al. 0903.4379 ]data on and P
NJL
QP
lQC
D-F
odor
° A. Bazavov et al. 0903.4379 hep-lat
P
Transport approach can pave the way for a Transport approach can pave the way for a
consistency among known vconsistency among known v2,42,4 properties: properties:
breaking of v2(pT)/ & persistence of
v2(pT)/<v2> scaling
vv22(p(pTT), v), v44(p(pTT) at ) at /s~0.1-0.2 can agree with /s~0.1-0.2 can agree with
what needed what needed
by coalescence by coalescence (QNS)(QNS) NJL chiral phase transition do not modify NJL chiral phase transition do not modify
vv22 /s/s
Signature of /s(T): large v4/(v2)2
Summary for ligth QGPSummary for ligth QGP
Next Steps for a quantitative estimate:Next Steps for a quantitative estimate:
Include the effect of an EoS fitted to lQCD Implement a Coalescence + Fragmentation mechanism
2Tf
* vm2
1TT
0)(
0)(
xj
xT
B
A Nearly Perfect FluidA Nearly Perfect Fluid*
),( T
m
T
upE
eq
T
egegpxf
TTf f ~ 120 MeV~ 120 MeV
<<TT> ~ 0.5 > ~ 0.5
For the first time very closeFor the first time very close to ideal Hydrodynamicsto ideal Hydrodynamics
Finite viscosity is not negligible
No microscopic description (->0) Blue shift of dN/dpT hadron spectra Large v2/ Mass ordering of v2(pT)
Jet QuenchingJet Quenching
Nuclear Modification Nuclear Modification FactorFactor
How much modification respect to pp?
Jet gluon radiation observedJet gluon radiation observed:
all hadrons RAA <<1 and flat in pT
photons not quenched -> suppression due to QCD
away
near
Medium
Jet triggered angular Jet triggered angular correl.correl.
Surprises…
In vacuum p/ ~ 0.3 due to Jet fragmentation
Hadronization has been modifiedHadronization has been modified ppTT < 4-6GeV !? < 4-6GeV !?
PHENIX, PRL89(2003)
Baryon/MesonsBaryon/Mesons
Protons not suppressed
QuenchingQuenching
Au+Au
p+p
Jet quenching should affect both
suppression: evidence of jet quenching before fragmentation
Hadronization in Heavy-Ion CollisionsInitial state: no partons in the vacuum but a thermal ensemble of partons -> Use in mediumUse in medium quarksNo direct QCD factorization scale for the bulk: dynamics much less violent (t ~ 4 fm/c)
Parton spectrum
H
Baryon
Meson
Coal.
Fragmentation
V. Greco et al./ R.J. Fries et al., PRL 90(2003)
Fragmentation: energy needed to create quarks from vacuum hadrons from higher pT
partons are already there $ to be close in phase space $
ph= n pT ,, n = 2 , 3 baryons from lower momenta (denser)
Coalescence:
ReCo pushes out soft physics by factors x2 and x3 !ReCo pushes out soft physics by factors x2 and x3 !
More easy to More easy to produce baryons!produce baryons!
HqqMqqH Dfff
Pd
Nd3
3
Hadronization ModifiedHadronization Modified
Baryon/MesonsBaryon/Mesons
Au+Au
p+p
PHENIX, PRL89(2003)
Quark number scalingQuark number scaling
n
TT
qT
T
H nppd
dNp
pd
dN
)()(
22
n
p
nT
2V1
Dynamical quarks are visibleDynamical quarks are visibleCollective flowsCollective flows
)2cos(v21φ 2
TT
q
TT
q
dpp
dN
ddpp
dN
/3)(p3v)(pv
/2)(p2v)(pv
Tq2,TB2,
Tq2,TM2,
Enhancement of vEnhancement of v22
v2q fitted from v2
GKL
Coalescence scalingCoalescence scaling
Heavy QuarksHeavy Quarks
mmc,bc,b >> >> QCDQCD produced by pQCD processes (out of
equilibrium)
eqeq > > QGPQGP with standard pQCD cross section (and also
with
non standard pQCD)non standard pQCD)
Hydrodynamics does not apply to heavy quark dynamics
(f≠feq)
pQCD
“D”QGP- RHIC
Equilibration time
npQCD
v4 more sensitive to both /s and f.o.
v4(pT) at 4s could also be consistent with
coalescence
vv44 generated later than v generated later than v22 : more sensitive to properties at : more sensitive to properties at
TTTTcc
What about v4 ?
Relevance of time scale !Relevance of time scale !
Effect of EOS on vEffect of EOS on v22
Decrease in v2 of about 40%H. Song and U.Heinz
Very Large v4/(v2)2 ratio
Ratio v4/v22 not very much depending on not very much depending on /s/s
and not on the initial eccentricity and not on the initial eccentricity
and not on particle species and not on particle species ……
see also M. Luzum, C. Gombeaud, O. Ollitrault, arxiv:1004.2024
Same Hydro with
the good dN/dpT and v2
/
s
1
1
T/Tc
QGP
2
2
V2 develops earlier at higher /s
V4 develops later at lower /s
-> v-> v44/(v/(v22))2 2 larger larger
Effect of Effect of /s(T) on the anisotropies/s(T) on the anisotropies
Hydrodynamics Effect of finite /s+f.o.
Effect of/s(T) + f.o.
Au+Au@200AGeV-b=8fm |y|<1
vv44/(v/(v22))2 2 ~~ 0.8 signature of 0.8 signature of //ss
close to phaseclose to phase transition!transition!
If the system if very dense one can derive and add the three-body collision that make the transition from the dilute to the dense system:
See: Zhu and Greiner PRC71 (2004)
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