Towards an understanding of the single electron spectra or

24th Winter Workshop Nuclear Dynamics

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Towards an understanding of the single electron spectra

or

what can we learn from heavy quarks of a plasma?

P.-B. Gossiaux , A. Peshier & J. Aichelin

Subatech/ Nantes/ France

arXiv: 0802.2525


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Present situation:a) Multiplicity of stable hadrons made of (u,d,s) is described by thermal modelsb) Multiplicity of unstable hadrons can be understood in terms of hadronic final state interactionsc) Slopes difficult to interpret due to the many hadronic interactions (however the successful coalescence models hints towards a v2 production in the plasma)d) Electromagnetic probes from plasma and hadrons rather similar

If one wants to have direct information of the plasma one has to find other probes:

Good candidate: hadrons with a c or b quarkHere we concentrate on open charm mesons for which indirect experimental data are available (single electrons)


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Why Heavy Quarks probe the QGP

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 radiate and therefore change their momentum.

If the relaxation time is larger than the time they spent in the plasma their 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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Individual heavy quarks follow Brownian motion: we can describe the time evolution of their distribution by a

Fokker – Planck equation:

fBfAtf

pp

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Input reduced to Drift (A) and Diffusion (B) coefficient.

Much less complex than a parton cascade which has to follow the light particles and their thermalization as well.

Can be combined with adequate models like hydro for the dynamics of light quarks


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The drift and diffusion coefficients

Strategy: take the elementary cross sections for charm scattering (Qq and Qg) and calculate the coefficients (g = thermal distribution of the collision partners)

and then introduce an overall K factor to studythe physics

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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c-quarks transverse momentum distribution (y=0)

Distribution just before hadronisation

p-p distribution

Plasma will notthermalize the c:It carries informationon the QGP

Heinz & Kolb’s hydro


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There is a generic problem ! Van Hees and Rapp:Charmed resonances andexpanding fireball. Communicates more efficientlyv2 to the c- quarks than hydro

Moore and Teaney:Even choice of the EOS which gives the largest v2 possibledoes not predict non charmedhadron data.

Only ‘exotic hadronization mechanisms’ may explain the

large v2

K15


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Can one improve and, if yes, how?Boltzmann vs Fokker-Planck

T=400 MeV

as=0.3

collisions with thermal quarks & gluons

2fm/c

10fm/c

Fokker-Planck does not givea good description of theenergy loss at intermediatetimes.Reason: Qg->Qg

pt

pt

FP

Boltzm

-> Boltzmann equation


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Problems of the existing approaches: RAA or energy loss is determined by the elementary elastic scattering cross section

α(t) =g2/4 is taken as constant [0.2 < α < 0.6]

mD gT

But α(t) is running and mD is not well determined

Is there a way to get a handle on them ?

These large artificial K (multiplication) factors or new physical processes are needed to describe the data if


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Information on the strong coupling constantA ) Brodsky et al. PRD 67 055008

- Nonstrange hadronic decay (OPAL)- e+e--> hadrons

Provide ‘physical’ coupling constants (determined by fits to theexperimental data in the time like sector) :

R = R0 (1+(m2

)/) Re = (e+e--> h)/(e+e-->+ -) = R0(1+R(s)/)

They remain finite down to very low values of s. .

The physical coupling constant is a summation of all orders of perturbationswhich nature has done for us.

V (m

2) = eI=1 (s)

s(s) and s(t) connected by generalized Crewther relations


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The detailed form very close to Q2 =0 is not important does not contribute to the energy loss

b) Dokshitzer NPB 469 (96) 93:

Observable = time like effective coupling * Process dependent fct Effective coupling is infrared save

This approach we use for the actual calculation


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presently used:

But prefactor not well determined taken between 1 and 1/3

How to improve on the Debye mass ?

Regulates the long range behaviour of the interaction


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In reality the situation is more complicated:

Loops are formed

If t is small (<<T) : Born has to be replaced by a hard thermal loop (HTL) approach (Braaten and Thoma PRD44 (91) 1298,2625)

For t>T Born approximation is ok

In QED the energy loss does not depend on t*

which separates the two regimes and which is artificial)

= E-E’


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This concept has been extended to QCD

HTL in QCD cross sections is too complicated

Idea: - Use HTL (t<t*) and Born (t>t*) amplitude to calculate dE/dx make sure that result does not depend on t*

- determine which gives the same enegry loss if one uses a cross section of the form

In reality a bit more complicated: with Born matching region of t*

outside the range of validity of HTL -> add to Born a constant

Constant coupling constant -> Analytical formula -> arXiv: 0802.2525Running -> numerically


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dE/dx does not dependon t*

HTL+semihard neededto have the transitionin the range of validityof HTL

The resulting values aresmaller than those used up tonow.


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The matching gives 0.15 mD for running S for theDebye mass and 0.22 mD not running!

standard

new

new

c+q->c+q

c+g_>c+g

Large enhancement of bothcross sections at small t

Difference between runningcoupling and fixed coupling for the Debye mass negligible Little change at large t

Qq->Qq

Qg_>Qg

Standard: (2T), =mD


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Consequences for the energy lossProbability P(w) that a c-quark looses the energy win a collision with a quark and a gluon, T= 400 MeV

c+q -> c+q c+g->c+g

wP(w) ~ dE/dxdw contribution to theenergy loss


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Input quantities for our calculations 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 Cronin effect included.

• 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) taken from hydrodynamical evolution (Heinz & Kolb)

•D meson produced via coalescence mechanism. (at the transition temperature we pick a u/d quark with the a thermal distribution) but other scenarios possible.


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Consequences for RAA including Cronin effectb and c separated (Cacciari PRL 95 (05) 122001)

OldK=12

NewK=1.5-2

b

c

central

The new approach reducesthe K- factor

K=12 -> K=1,5-2

No radiative energy loss yet

pT > 2 bottom dominated!!more difficult to stopcompatible with experiment


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minimum bias

NewK=1,5-2

OldK=12

Central and Minimum bias events

described by the same parameter

Different between b and c becomes smaller


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Out of plane distribution V2

NewK=1,5-2

OldK=20-40

Even with very large K factors the data are out of range

Cronin influences little

v2 heavy mesons depends on where fragmentation/ coalescence takes place

end of mixed phase beginning of mixed phase


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Back to back correlations of c and cbar pairare unfortunatelly not able to distinguish between thedifferent cross sections

running = 0.15

(2T)


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Conclusions

• Experimental data point towards a significant (although not complete) thermalization of c quarks in QGP.

• Using a running coupling constant, determined by experiment, and an infrared regulator which approximates hard thermal loop results we are almost able to reproduce the experimental RAA and v2. The missing factor of two can be due to radiative processes or due uncertainties:

1) spatial distribution of initial c-quarks

2) Part of the flow is due to the hadronic phase subsequent to QGP

3) Dependence on the fragmentation scale in a thermal environment

Azimutal correlations could be of great help in order to identify the nature of thermalizing mechanism but does not distinguish between the ’sdiscussed here

Single electron data are compatible with pQCD


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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 formation at the boundary of QGP through coalescence of c/b and light quark

Schematic view of our model for hidden and open heavy flavors production in AA collision at RHIC and LHC


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« radiative » coefficients deduced using the elementary cross section for cQ cQ+g and for cg cg +g in t-channel (u & s-channels are suppressed at high energy).

"Radiative« coefficients

dominant suppresses by Eq/Echarm

ℳqcqg ≡

c

Q+ ++ +

:if evaluated in the large pi

c+ limit in the lab (Bertsch-Gunion)

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Towards an understanding of the single electron spectra or

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