HEP - Valparaiso 14. december 2004 1 Tomography of a Quark Gluon Plasma by Heavy Quarks : I)Why? II) Approach and ingredients II) Results for R AA III)

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


2

(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


3

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


4

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


5

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


6

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


7

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


8

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


9

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


10

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


11

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


12

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


13

« 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


14

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


15

« 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


16

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


17

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


18

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


19

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


20

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.


21

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


22

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


23

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


24

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


25

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" ?


26


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%


27

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


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%


28

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.


29

Back up


30

Total emission from quark lines

(Mpro+Mpost)2


31

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


32

Strong correlation of y vs. Y (spatial rapidity)

Why so tiny ?

y

Y


33

J/’s


34

• 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)


35

• 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)


36

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.


37

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/.


38

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


39

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.


40

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 ???


41

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 ?


42

Gunion & Bertsch ‘82

ℳqrad/2 ≅ +

+

+

kl

q(E)

q

g()

Soft gluons Gunion & Bertsch

<< Ek⊥ << l⊥

<< Ek⊥ >< l⊥


43

ℳqrad∝ℳq

elⅹgsⅹ

dc

T

T

TT

TT

T

T Fkk

klkl

kk

222

Tdc k

EMF

;;1 0

1

2

20

Qq Qqg


44

dc

TTT

TTT

TTT

dc

T

sA

TF

klkklk

klkF

kC

kdyddn 1.211

22

22

2

222

Total spectrum :

Qq Qqg


45

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)


46

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


47

Masses= . 33GeVqt2 =0.3


48

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


49

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�


50

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)

Documents

HEP - Valparaiso 14. december 2004 1 Tomography of a Quark Gluon Plasma by Heavy Quarks : I)Why? II) Approach and ingredients II) Results for R AA III)