hep - valparaiso 14. december 2004 1 tomography of a quark gluon plasma by heavy quarks : i)why? ii)...
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HEP - Valparaiso 14. december 2004
1
Tomography of a Quark Gluon Plasma
by Heavy Quarks :
I)Why?
II) Approach and ingredients
II) Results for RAA
III) Results for v2
IV) Azimuthal correlations
V) Conclusions
P.-B. Gossiaux , V. Guiho & J. Aichelin
Subatech/ Nantes/ France
HEP - Valparaiso 14. december 2004
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(hard) production of heavy quarks in initial NN collisions
Evolution of heavy quarks in QGP (thermalization)
Quarkonia formation in QGP through c+c+g fusion process
D/B meson formation at the boundary of QGP through coalescence of c/b and light quark
Schematic view of hidden and open heavy flavor production in AA collision at RHIC and LHC
HEP - Valparaiso 14. december 2004
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Heavy quarks in QGP (or in strongly interacting matter)
Idea: Heavy quarks are produced in hard processes with a knowninitial momentum distribution (from pp).
If the heavy quarks pass through a QGP they collide and radiateand therefore change their momentum.
If the relaxation time is larger than the time they spent in the plasmatheir final momentum distribution carries information on the plasma
This may allow for studying plasma properties usingpt distribution, v2 transfer, back to back correlations
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Single trajectories and mean values
Evolution of one c quark inside a =0 -- T=400 MeV QGP.
Starting from p=(0,0,10 GeV/c). Evolution time = 30 fm/c
True Brownian motion
… looks very smooth when averaged over many trajectories .Relaxation time >> collision time
t (fm/c)
pz f
pz
px
py
HEP - Valparaiso 14. december 2004
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When individual heavy quarks follow Brownian motion we can describe the time evolution of their distribution by a
Fokker – Planck equation:
fBfAtf
pp
�
Input reduced to a Drift (A) and a Diffusion (B) coefficient.
Much less complex than a parton cascade which has to followthe light particles and their thermalization as well.
Can be calculated using adequate models like hydro for the dynamics of light quarks
HEP - Valparaiso 14. december 2004
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The drift and diffusion coefficients
Strategy: take the elementary cross sections for charm/bottom elastic scattering and use a Vlasov equation to calculate the coefficients (g = thermal distribution of the collision partners)
and the introduce an overall κ factor Similar for the diffusion coefficient Bνμ ~ << (pν
- pνf )(pμ
- pμf )> >
A describes the deceleration of the c-quark B describes the thermalisation
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Energy loss and A,B are related (Walton and Rafelski)
pi Ai + p dE/dx = - << (pμ – pμf)2
>>
which gives easy relations for Ec>>mc and Ec<<mc
In case of collisions (2 2 processes): Pioneering work of Cleymans (1985), Svetitsky (1987), extended later by Mustafa, Pal & Srivastava (1997). Teany and MooreRapp and Hees similar approach but plasma treatmentis different
• For radiation: Numerous works on energy loss; very little has been done on drift and diffusion coefficients
HEP - Valparaiso 14. december 2004
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First results on c-quark evolution
Relaxation of <E>, of and of for c-quarks produced in 200 GeV collisions.
Evolution in a =0 , T=200 MeV QGP.
long relaxation times
Typical times 60 fm/c
Asymptotic energy distribution: not Boltzmann; more like a Tsallis
Walton & Rafelski (1999)
Too much diffusion at large momentum
p f
2 pf//
2
(E-m)/T
pf//
2
p f
2
f(E)
Approximate scaling for T=0.2 0.5
E
Time (fm/c)
60 1000
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The collisional transport coefficients of charm
p (GeV/c)
A (Gev/fm)
T=0.3
T=0.4
T=0.5
T=0.2
p (GeV/c)
dE/dx (GeV/fm)
p (GeV/c)p (GeV/c)
B (GeV^2/fm c) B// (GeV^2/fm c)
T=0.4
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1. Coefficients deduced by Mustafa, Pal and Srivastava (MPS) for A and B
1. Calculate A and use of the Einstein relation between drift and diffusion coefficient (to get asymptotically a thermal distribution)
Two sets parameters:
pf//
2
<E>
p f
2
Bth //
B//
Bth
B
A=Ath
pt Time (fm/c)
E
The transport coefficients used in the calculation
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c-quarks transverse momentum distribution (y=0)
col
PS
Heinz & Kolb’s hydro
Just before the hadronisation
p-p distribution
Conclusion I:
col(coll only)10-20: Still far away from
thermalization !!!
Plasma will notthermalize the c;It carries informationon the QGP
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Leptons ( D decay) transverse momentum distribution (y=0)
RAA
1 2 3 4 5
0.2
0.4
0.6
0.8
1
B=0 (Just deceleration)
Langevin A and B finite
κ = 20, κ=100-10%
Transition from pure deceleration (high E) towards thermalization regime (intermediate E)
pt
Comparison to B=0 calculation
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« radiative » coefficients deduced using the elementary cross section for cQ cQ+g and its equivalent for cg cg +g in t-channel (u & s-channels are suppressed at high energy).
"Radiative"coefficients
dominant suppresses by 1/Echarm
z
ℳqqqg ≡
q
Q+ ++ +
:if evaluated in the large sqrts limit in the lab
sss
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q
k
x=long. mom. fraction
In the limit of vanishing masses:Gunion + Bertsch PRD 25, 746
But:
Masses change the radiationsubstantially
Evaluated in scalar QCD and in the limit of Echarm >> masses and >>qt
Factorization of radiation and elastic scattering
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« QCD » part of M2
Large at small x and finite kttransverse momentum change
« QED » part of M2
Large at large x and small kt
« QCD »
« QED »
ktx
kt
x
0
0.40.8
0
0.4
0
0
0.8
200
2000
Abelien
all masses = 0.001 GeV qt = 0.3 GeV
(abelien)
0.4
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Influence of finite masses on the radiation
kt1
0
0
0.8
x
Thermal masses
Mgluon = Mquark = 0.3 GeV
Masses : Mgluon = Mquark = 0.01 GeV
1
0
0.8
0
1
kt
x
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charm
kt1
0
0
0.5
x
bottom
0
0
0.5
1kt
x
The larger the quark mass the more the gluons have small kt and x
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Dead cone effect: Dokshitzer and Kharzeev PLB 519, 199
Masses suppress the gluon emission at small kt
If one uses the full matrix element the formula is more complicated
but
F<1 for realistic masses and finite qt2 dead cone
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Input quantities for the calculation
• Au – Au collision at 200 AGeV.
• c-quark transverse-space distribution according to Glauber
• c-quark transverse momentum distribution as in d-Au (STAR)… seems very similar to p-p No Cronin effect included; too be improved.
• c-quark rapidity distribution according to R.Vogt (Int.J.Mod.Phys. E12 (2003) 211-270).
• Medium evolution: 4D / Need local quantities such as T(x,t) Bjorken (boost invariant with no transverse flow) for tests realistic hydrodynamical evolution (Heinz & Kolb) for comparison
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Input quantities for the calculation (II)
• Langevin force on c-quarks inside QGP and no force on charmed « mesons » during and after hadronisation.
• D & B meson produced via coalescence mechanism. (at the transition temperature we pick a u/d quark with the a thermal distribution) but other scenarios possible.
• No beauty up to now; will be included.
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As for the collisional energy loss we calculate with these rates Ak = <<Pk – Pk
f>>
Bkl = < <( Pk-Pkf )(Pl-Pl
f)>>
A (Gev/fm)
p (GeV/c)
p (GeV/c)
B (GeV^2/fm c)
Still preliminary
Radiative energy loss > collisional energy loss
T=360
T=160 MeV
0 8
30
0
T=260
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2 4 6 8
0.2
0.4
0.6
0.8
1
1.2
1.4
2 4 6 8
0.2
0.4
0.6
0.8
1
1.2
1.4
RAA
2 4 6 8
0.2
0.4
0.6
0.8
1
1.2
1.4
Leptons ( D decay) transverse momentum distribution (y=0)
0-10% 20-40%
Min bias
Col. (col=10 & 20)
Col.+(0.5x) Rad
Conclusion II:
One can reproduce the RAA either :
• With a high enhancement factor for collisional processes
• With « reasonnable » enhancement
factor (rad not far away from unity)
including radiative processes.
pt
pt
pt
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Non-Photonic Electron elliptic-flow at RHIC: comparison with experimental results
0.5 1 1.5 2 2.5 3 3.5 4
0.05
0.05
0.1
0.5 1 1.5 2 2.5 3 3.5 4
0.05
0.05
0.1
Collisional
(col= 20)
Collisional + Radiative
c-quarks D
decay eTagged const q
D
cq
Conclusion III:
One cannot reproduce the v2
consistently with the RAA!!! Contribution of light quarks to the elliptic flow of D mesons is small
Freezed out according to thermal distribution at "punch" points of c quarks through freeze out surface:
v2
v2
pt
pt
HEP - Valparaiso 14. december 2004
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Non-Photonic Electron elliptic-flow at RHIC: Looking into the details
0.5 1 1.5 2 2.5 3 3.5 4
0.05
0.05
0.1
0.5 1 1.5 2 2.5 3
0.025
0.05
0.075
0.10.125
0.15const quark tagged by c
Bigger enhancement κ helps… a
little but RAA becomes worse.
Reason: the (fast) u/d quarks which carry large v2 values never meet the (slow) c quarks.
Hence in collisions at hadronisation and at coalescence little v2 transfer.
v2 (d/u met by c)
v2 (all d/u)
ptpt
pt
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Azimutal Correlations for Open Charm
What can we learn about "thermalization" process from the
correlations remaining at the end of QGP ?c
D
c-bar
Dbar
Transverse plane
Initial correlation (at RHIC); supposed back to back here
How does the coalescence - fragmentation mechanism affects
the "signature" ?
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Azimutal Correlations for Open Charm
1 2 3 4 5 6
1
2
3
4
5
6
7
8
c-quarks
Conclusion IV: Broadening of the correlation due to medium, but still visible. Increasing κ values wash out the correlation
1 2 3 4 5 6
1
2
3
4
5
6
7
8
D
Coll (col= 10)
Coll (col= 20)
Coll (col= 1)
Coll + rad (col= rad = 1)
No interactionAverage pt (1 GeV/c < pt < 4 GeV/c )
coalescence
Azimutal correlations might help identifying better the thermalization
process and thus the medium
c - cbar
D - Dbar
0-10%
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1 2 3 4 5 6
1
2
3
4
5
6
7
8
1 2 3 4 5 6
1
2
3
4
5
6
7
8
Azimutal Correlations for Open Charm
c-quarks
Small correlations at small pt,, mostly
washed away by coalescence process. D
Coll (col= 10)
Coll (col= 20)
Coll (col= 1)
Coll + rad (col= rad = 1)
No interactionSmall pt (pt < 1GeV/c )
coalescence c - cbar
D - Dbar
0-10%
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Conclusions
• Experimental data point towards a significant (although not complete) thermalization of c quarks in QGP.
• The model seems able to reproduce experimental RAA, at the
price of a large rescaling -factor (especially at large pt), of the
order of or by including radiative processes.
• Still a lot to do in order to understand for the v2. Possible
explanations for discrepencies are:1) Role of the spatial distribution of initial c-quarks
2) Part of the flow is due to the hadronic phase subsequent to QGP
3) Caveat of Langevin approach
• Azimutal correlations could be of great help in order to identify the nature of thermalizing mechanism.
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Back up
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Total emission from quark lines
(Mpro+Mpost)2
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Tiny diffusion effect (no E loss, no drag)
Results for open charm : rapidity distribution at RHIC
Heinz & Kolb’s hydro (boost
invariant)
(Set I)
Set II
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Strong correlation of y vs. Y (spatial rapidity)
Why so tiny ?
y
Y
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J/’s
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• J/ are destroyed via gluon dissociation: J/ + g c + cbar and can be formed through the reverse mechanism, following the ideas of Thews. Uncorrelated quarks recombination quadratic dependence in Nc :
NNN ccJ
ch
2
/
Question: How much is ???
Other ingredients of the model specific for J/ production (I)
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• As el(J/) is small, we assume free streaming of J/ through QGP (no thermalization of J/)... But possible gluo dissociation
• Clear cut melting mechanism: J/cannot exist / be formed if T > Tdissoc (considered as a free parameter, taken
between Tc and 300 MeV; conservative choice according
to lattice calculations: Tdissoc=1.5Tc).
• Up to now: No prompt J/(supposed to be all melted)
Other ingredients of the model specific for J/ production (II)
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10 20 30 40K0.001
0.0015
0.002
0.003
0.005
0.007
0.01
0.015
dNJy 0
dy
NN scaling
T dissoc 180 MeV
T dissoc 200 MeV
T dissoc 250 MeV
T dissoc 300 MeV
Ncc10conservative NLO
Results for J/ production at mid-rapidity, centralComponent stemming out the recombination mechanism:
10 20 30 40K
0.01
0.015
0.02
0.03
0.05
0.07
dNJy 0
dy
nucleonnucleon scaling
T dissoc 180 MeV
T dissoc 200 MeV
T dissoc 250 MeV
T dissoc 300 MeV
Ncc20STAR
Heinz & Kolb’s hydro
No radial exp. hydro
• Nc and Tdissoc : key parameters as far as the total numbers are considered
• Thermalization increases production rates, but only mildly.
• Radial expansion of QGP has some influence for a very specific set of parameters (cf. )
• Firm conclusions can only be drawn when the initial number of c-cbar pairs is known more precisely.
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Results for J/ production vs. rapidity
3 2 1 0 1 2 3y0.0002
0.0005
0.001
0.002
dNJdydNc
dy
dNcdy
T dissoc 180 MeV
T dissoc 200 MeV
T dissoc 250 MeV
• Scaling like (dNc/dy)^2
• A way to test the uncorrelated c-cbar recombination hypothesis.
• Grain of salt: boost invariant dynamics for the QGP assumed.
Rapidity distribution is somewhat narrower for J/ stemming out the fusion of uncorrelated c and cbar than for direct J/.
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Tdissoc=180 MeV Tdissoc=180 MeV
J/ transverse momentum distribution at mid rapidity
(no transv. flow)
(Heinz & Kolb)
Direct J/ (NN scaling)
Direct J/scaling
Clear evidence of the recombination mechanism:
• pt anti-broadening in Au-Au
• effective temperatures > Tc
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Other conclusions & Perspectives
• Heavy quark physics could be of great help in the metrology of QGP transport coefficients, especially at low momentum… Go for the differential !
• Recombination mechanism should be there if one believes the large value of Tdissoc found on the lattice.
• The Fokker Planck equation: a useful unifying phenomenological transport equation that makes the gap between fundamental theory & experimental observables. Permits to generate input configuration for mixed-phase and hadronic-phase evolution.
• Mandatory & To be done soon: Cronin effect / relax the N(J/ direct)=0 assumption / include beauty /find a name.
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No time for thermalization anyhow. Then take these FP coefficients as they are, period (at least, it comes from some microscopic model).
Add some more KM coefficients in your game (we are not that far from Boltzmann after all). Some more ? In fact 6 th order
Do Boltzmann (or whatever microscopic).
Change your point of view : Assume physics of c-quark is closer to Fokker Planck (long relaxation time) then to Boltzmann collision term (QGP, diluted ?), PCM, fixed collision centers,… Construct some phenomenological A and B (until lattice can calculate them) and see if you can fit (a lot of) experimental data. (In other field of physics, one measures the A and B)
So what should we do ???
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So What ???
A) « since the drag and the diffusion coefficients are not evaluated exactly but in some valid approximation, typically applying a perturbative expansion,… » (Walton & Rafelski)
And later (last sentence of the paper):
B) « … only a major change in the transport coefficients from the results of the microscopic calculations will lead to a Boltzmann / Jüttner equilibrium distribution. »
My personnal comments
• Wrt A) : Boltzmann collision-integral can (at least formally) be rewritten as a power series implying derivatives of f of higher and higher degree (Kramers – Moyal expansion). FP coefficients ARE the 2 first two coefficients and are perfectly defined.
• Wrt A) & B) : If the approximation (truncation of KM series) is valid, why should it be necessary to perform a major change on the coeff ?
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Gunion & Bertsch ‘82
ℳqrad/2 ≅ +
+
+
kl
q(E)
q
g()
Soft gluons Gunion & Bertsch
<< Ek⊥ << l⊥
<< Ek⊥ >< l⊥
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ℳqrad∝ℳq
elⅹgsⅹ
dc
T
T
TT
TT
T
T Fkk
klkl
kk
222
Tdc k
EMF
;;1 0
1
2
20
Qq Qqg
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dc
TTT
TTT
TTT
dc
T
sA
TF
klkklk
klkF
kC
kdyddn 1.211
22
22
2
222
Total spectrum :
Qq Qqg
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Heavy quarks in QGP (or in strongly interacting matter)
• Heavy quarks behave according to Brownian motion / Langevin forces c quarks distribution evolves according to Fokker – Planck equation
fBfAtf
pp
�
N.B.: What is the best model (if any) ? FP or Boltzmann equation ?
• Starting point: For heavy quarks, relaxation time >> collision time ; at large momentum (as for all quarks) but also at low momentum (thanks to inertia)
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1 2 3 4 5t
0.1
10
#
Non-Photonic electron elliptic-flow at RHIC: …and the bites (ouch)
Spatial transverse-distribution might play some role as c-quarks are not from the beginning "on" the freeze out surface.
1 2 3 4 5 6 7 8
1
2
3
4
5
6
7
r
t=1fm/c
t=4fm/c
strong coupling
No coupling
c
D
t1 2 3 4 5
SQM06
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Masses= . 33GeVqt2 =0.3
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The transport coefficients (III)
How precisely do we know these transport coefficients (in the case of heavy quarks) ?
Start from a more « fundamental » theory
Two body collisions with thermal distribution of the collision partner.
Moments A ~ < pμf - pμ
i >
B ~ < (pνf - pν
i )(pμf - pμ
i ) >
• In case of collisions (2 2 processes): Pioneering work of Cleymans (1985), Svetitsky (1987), extended later by Mustafa, Pal & Srivastava (1997).
• For radiation: Numerous works on energy loss; very little seems to have been done on diffusion coefficients
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The transport coefficients (II)
with pppf
: Bpp
dtd
ijji ff 2
• Diffusion (in momentum space); (not to be confused with diffusion in "normal" space (D) thermalisation
• In isotropic media: decomposition of into longitudinal and transverse contribution only 2 independent coefficients.
B�
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The transport coefficients
with Ap ffdtd
• drift coefficient is proportional to momentum loss per unit of time (Walton and Rafelski)
• At high momenta, one has (assuming f is peaked):
ppApA
)(~
)(
pAβAβdtpdβ
dtpdβ
dtEd
~
- -
A(p) and the energy loss per unit of length are the same quantities
• At low momenta, not true anymore: On the average, particles can gain/loose energy without gaining or loosing momentum (thermalisation)