1. 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) Results for v 2 IV) Azimuthal correlations V) Conclusions P.-B. Gossiaux, V. Guiho & J. Aichelin Subatech/ Nantes/ France

2. HEP - Valparaiso 14. december 2004 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. HEP - Valparaiso 14. december 2004 3 Heavy quarks in QGP (or in strongly interacting matter) Idea: Heavy quarks are produced in hard processes with a known initial 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 using pt distribution, v 2 transfer, back to back correlations

4. HEP - Valparaiso 14. december 2004 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) pzpz pxpx pypy

5. HEP - Valparaiso 14. december 2004 5 When individual heavy quarks follow Brownian motion we can describe the time evolution of their distribution by a Fokker – Planck equation: Input reduced to a Drift (A) and a Diffusion (B) coefficient. Much less complex than a parton cascade which has to follow the light particles and their thermalization as well. Can be calculated using adequate models like hydro for the dynamics of light quarks

6. HEP - Valparaiso 14. december 2004 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 B νμ ~ > A describes the deceleration of the c-quark B describes the thermalisation

7. HEP - Valparaiso 14. december 2004 7 Energy loss and A,B are related (Walton and Rafelski) p i A i + p dE/dx = - > which gives easy relations for E c >>m c and E c <<m c In case of collisions (2  2 processes): Pioneering work of Cleymans (1985), Svetitsky (1987), extended later by Mustafa, Pal & Srivastava (1997). Teany and Moore Rapp and Hees similar approach but plasma treatment is different For radiation: Numerous works on energy loss; very little has been done on drift and diffusion coefficients

8. HEP - Valparaiso 14. december 2004 8 First results on c-quark evolution Relaxation of, 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 (E-m)/T f(E) Approximate scaling for T=0.2  0.5 E Time (fm/c) 601000

9. HEP - Valparaiso 14. december 2004 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) B  (GeV^2/fm c) B// (GeV^2/fm c) T=0.4

10. HEP - Valparaiso 14. december 2004 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: B th // B // B th  BB A=A th pt Time (fm/c) E The transport coefficients used in the calculation

11. HEP - Valparaiso 14. december 2004 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 not thermalize the c; It carries information on the QGP

12. HEP - Valparaiso 14. december 2004 12 Leptons (  D decay) transverse momentum distribution (y=0) R AA B=0 (Just deceleration) Langevin A and B finite κ = 20, κ=10 0-10% Transition from pure deceleration (high E) towards thermalization regime (intermediate E) pt Comparison to B=0 calculation

13. HEP - Valparaiso 14. december 2004 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/E charm z ℳ q qqg ≡ q Q + + + + : if evaluated in the large sqrts limit in the lab

14. HEP - Valparaiso 14. december 2004 14 q k x=long. mom. fraction In the limit of vanishing masses: Gunion + Bertsch PRD 25, 746 But: Masses change the radiation substantially Evaluated in scalar QCD and in the limit of E charm >> masses and >>qt Factorization of radiation and elastic scattering

15. HEP - Valparaiso 14. december 2004 15 « QCD » part of M 2 Large at small x and finite kt transverse momentum change « QED » part of M 2 Large at large x and small kt « QCD » « QED » kt x x 0 0.4 0.8 0 0.4 0 0 0.8 200 2000 Abelien all masses = 0.001 GeV qt = 0.3 GeV (abelien) 0.4

16. HEP - Valparaiso 14. december 2004 16 Influence of finite masses on the radiation kt 1 0 0 0.8 x Thermal masses M gluon = M quark = 0.3 GeV Masses : M gluon = M quark = 0.01 GeV 1 0 0.8 0 1 kt x

17. HEP - Valparaiso 14. december 2004 17 charm kt 1 0 0 0.5 x bottom 0 0 0.5 1 kt x The larger the quark mass the more the gluons have small kt and x

18. HEP - Valparaiso 14. december 2004 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 qt 2  dead cone

19. HEP - Valparaiso 14. december 2004 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. HEP - Valparaiso 14. december 2004 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. HEP - Valparaiso 14. december 2004 21 As for the collisional energy loss we calculate with these rates A k = > B kl = > A (Gev/fm) 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. HEP - Valparaiso 14. december 2004 22 R AA 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 R AA either : With a high enhancement factor for collisional processes With « reasonnable » enhancement factor (  rad not far away from unity) including radiative processes. pt

23. HEP - Valparaiso 14. december 2004 23 Non-Photonic Electron elliptic-flow at RHIC: comparison with experimental results Collisional (  col = 20 ) Collisional + Radiative c-quarksD decay e Tagged const q D c q Conclusion III: One cannot reproduce the v 2 consistently with the R AA !!! 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: v2v2 v 2 pt

24. HEP - Valparaiso 14. december 2004 24 Non-Photonic Electron elliptic-flow at RHIC: Looking into the details const quark tagged by c Bigger enhancement κ helps… a little but R AA becomes worse. Reason: the (fast) u/d quarks which carry large v 2 values never meet the (slow) c quarks. Hence in collisions at hadronisation and at coalescence little v 2 transfer. v 2 (d/u met by c) v 2 (all d/u) pt

25. HEP - Valparaiso 14. december 2004 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. HEP - Valparaiso 14. december 2004 26 Azimutal Correlations for Open Charm - c-quarks Conclusion IV: Broadening of the correlation due to medium, but still visible. Increasing κ values wash out the correlation D Coll (  col = 10 ) Coll (  col = 20 ) Coll (  col = 1 ) Coll + rad (  col =  rad = 1 ) No interaction Average p t (1 GeV/c < p t < 4 GeV/c ) coalescence Azimutal correlations might help identifying better the thermalization process and thus the medium  c -  cbar  D -  Dbar 0-10%

27. HEP - Valparaiso 14. december 2004 27 Azimutal Correlations for Open Charm - c-quarks Small correlations at small p t,, mostly washed away by coalescence process. D Coll (  col = 10 ) Coll (  col = 20 ) Coll (  col = 1 ) Coll + rad (  col =  rad = 1 ) No interaction Small p t (p t < 1GeV/c ) coalescence  c -  cbar  D -  Dbar 0-10%

28. HEP - Valparaiso 14. december 2004 28 Conclusions Experimental data point towards a significant (although not complete) thermalization of c quarks in QGP. The model seems able to reproduce experimental R AA, at the price of a large rescaling  -factor (especially at large p t ), of the order of  or by including radiative processes. Still a lot to do in order to understand for the v 2. 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. HEP - Valparaiso 14. december 2004 29 Back up

30. HEP - Valparaiso 14. december 2004 30 Total emission from quark lines (Mpro+Mpost) 2

31. HEP - Valparaiso 14. december 2004 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. HEP - Valparaiso 14. december 2004 32 Strong correlation of y vs. Y (spatial rapidity) Why so tiny ? y Y

33. HEP - Valparaiso 14. december 2004 33 J/  ’s

34. HEP - Valparaiso 14. december 2004 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 N c : Question: How much is  ??? Other ingredients of the model specific for J/  production (I)

35. HEP - Valparaiso 14. december 2004 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 > T dissoc (considered as a free parameter, taken between T c and 300 MeV; conservative choice according to lattice calculations: T dissoc =1.5  T c ). Up to now: No prompt J/  (supposed to be all melted) Other ingredients of the model specific for J/  production (II)

36. HEP - Valparaiso 14. december 2004 36 Results for J/  production at mid-rapidity, central Component stemming out the recombination mechanism: Heinz & Kolb’s hydro No radial exp. hydro N c and T dissoc : 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. HEP - Valparaiso 14. december 2004 37 Results for J/  production vs. rapidity Scaling like (dN c /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. HEP - Valparaiso 14. december 2004 38 T dissoc =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: p t anti-broadening in Au-Au effective temperatures > T c

39. HEP - Valparaiso 14. december 2004 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 T dissoc 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. HEP - Valparaiso 14. december 2004 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. HEP - Valparaiso 14. december 2004 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. HEP - Valparaiso 14. december 2004 42 Gunion & Bertsch ‘82 ℳ q rad / 2 ≅ + + + k l q(E) q g(  ) Soft gluons Gunion & Bertsch  << E k ⊥ << l ⊥  << E k ⊥ >< l ⊥

43. HEP - Valparaiso 14. december 2004 43 ℳ q rad ∝ ℳ q el ⅹ g s ⅹ Qq Qqg

44. HEP - Valparaiso 14. december 2004 44 Total spectrum : Qq Qqg

45. HEP - Valparaiso 14. december 2004 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 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. HEP - Valparaiso 14. december 2004 46 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. rr t=1fm/c t=4fm/c strong coupling No coupling c D t 12345 SQM06

47. HEP - Valparaiso 14. december 2004 47 Masses=. 33GeV qt 2 =0.3

48. HEP - Valparaiso 14. december 2004 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 ~ B ~ 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. HEP - Valparaiso 14. december 2004 49 The transport coefficients (II) with 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.

50. HEP - Valparaiso 14. december 2004 50 The transport coefficients with drift coefficient is proportional to momentum loss per unit of time (Walton and Rafelski) At high momenta, one has (assuming f is peaked):  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)