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Tomography of a Quark Gluon Plasma by Heavy Quarks :

Tomography of a Quark Gluon Plasma by Heavy Quarks :. P.-B. Gossiaux , V. Guiho & J. Aichelin Subatech/ Nantes/ France. I)Why? II) Approach and ingredients II) Results for R AA III) Results for v 2 IV) Azimuthal correlations V) Conclusions.

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Tomography of a Quark Gluon Plasma by Heavy Quarks :

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  1. Tomography of a Quark Gluon Plasma by Heavy Quarks : P.-B. Gossiaux, V. Guiho & J. Aichelin Subatech/ Nantes/ France I)Why? II) Approach and ingredients II) Results for RAA III) Results for v2 IV) Azimuthal correlations V) Conclusions HEP - Valparaiso 14. december 2004

  2. Schematic view of hidden and open heavy flavor production in AA collision at RHIC and LHC Evolutionof heavyquarks in QGP (thermalization) D/B meson formation at the boundary of QGP through coalescence of c/b and light quark Quarkonia formation in QGP through c+cY+g fusion process (hard) production of heavy quarks in initial NN collisions 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, v2 transfer, back to back correlations HEP - Valparaiso 14. december 2004

  4. t (fm/c) Single trajectories and mean values Evolution of one c quark inside a m=0 -- T=400 MeV QGP. Starting from p=(0,0,10 GeV/c). Evolution time = 30 fm/c True Brownian motion pz py px … looks very smooth when averaged over many trajectories . Relaxation time >> collision time 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 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 Bνμ ~ << (pν- pνf )(pμ- pμf )> > A describes the deceleration of the c-quark B describes the thermalisation HEP - Valparaiso 14. december 2004

  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 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 HEP - Valparaiso 14. december 2004

  8. First results on c-quark evolution E Relaxation of <E>, of and of for c-quarks produced in 200 GeV collisions. Evolution in a m=0 , T=200 MeV QGP. long relaxation times 0 60 100 Time (fm/c) f(E) Approximate scaling for T=0.2  0.5 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 HEP - Valparaiso 14. december 2004

  9. The collisional transport coefficients of charm A (Gev/fm) dE/dx (GeV/fm) T=0.5 T=0.4 T=0.4 T=0.3 T=0.2 p (GeV/c) p (GeV/c) B (GeV^2/fm c) B// (GeV^2/fm c) p (GeV/c) HEP - Valparaiso 14. december 2004 p (GeV/c)

  10. The transport coefficients used in the calculation Two sets parameters: Coefficients deduced by Mustafa, Pal and Srivastava (MPS) for A and B Calculate A and use of the Einstein relation between drift and diffusion coefficient (to get asymptotically a thermal distribution) E <E> B// A=Ath Bth // B Bth HEP - Valparaiso 14. december 2004 pt Time (fm/c)

  11. p-p distribution c-quarks transverse momentum distribution (y=0) Heinz & Kolb’s hydro Just before the hadronisation Plasma will not thermalize the c; It carries information on the QGP MPS kcol=5 k=40 k=20 k=10 Conclusion I: Kcol(coll only) =10-20: Still far away from thermalization !!! HEP - Valparaiso 14. december 2004

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

  13. q ℳqqqg≡ + + Q + + z "Radiative"coefficients « 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). dominant suppresses by 1/Echarm : if evaluated in the large sqrts limit in the lab HEP - Valparaiso 14. december 2004

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

  15. « QED » part of M2 Large at large x and small kt Abelien all masses = 0.001 GeV qt = 0.3 GeV 200 « QED » 0 (abelien) x 0 kt 0.4 2000 0.8 0.4 0 « QCD » 0 « QCD » part of M2 Large at small x and finite kt transverse momentum change x kt 0.8 HEP - Valparaiso 14. december 2004 0.4

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

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

  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 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 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. HEP - Valparaiso 14. december 2004

  21. B (GeV^2/fm c) A (Gev/fm) p (GeV/c) p (GeV/c) As for the collisional energy loss we calculate with these rates Ak = <<Pk – Pkf>> Bkl = < <( Pk-Pkf )(Pl-Plf)>> Radiative energy loss > collisional energy loss 30 Still preliminary T=360 T=260 T=160 MeV 0 8 0 HEP - Valparaiso 14. december 2004

  22. Leptons ( D decay) transverse momentum distribution (y=0) RAA 0-10% 20-40% Col.+(0.5x) Rad Col. (kcol=10 & 20) pt pt • Conclusion II: • One can reproduce theRAA either : • With a high enhancement factor for collisional processes • With « reasonnable » enhancement factor (krad not far away from unity) including radiative processes. Min bias HEP - Valparaiso 14. december 2004 pt

  23. c-quarks D decay e D q c Non-Photonic Electron elliptic-flow at RHIC: comparison with experimental results Collisional (kcol=20) v2 Tagged const q Freezed out according to thermal distribution at "punch" points of c quarks through freeze out surface: pt Collisional + Radiative v2 Conclusion III: One cannot reproduce thev2consistently with the RAA!!! Contribution of light quarks to the elliptic flow of D mesons is small HEP - Valparaiso 14. december 2004 pt

  24. Non-Photonic Electron elliptic-flow at RHIC: Looking into the details v2 (all d/u) const quark tagged by c v2 (d/u met by c) pt pt 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. Bigger enhancement κ helps… a little but RAA becomes worse. HEP - Valparaiso 14. december 2004 pt

  25. Initial correlation (at RHIC); supposed back to back here Azimutal Correlations for Open Charm Transverse plane What can we learn about "thermalization" process from the correlations remaining at the end of QGP ? D c c-bar How does the coalescence - fragmentation mechanism affects the "signature" ? Dbar HEP - Valparaiso 14. december 2004 -

  26. c-quarks Coll (kcol=10) Coll (kcol=20) Coll + rad (kcol=krad=1) D Azimutal Correlations for Open Charm Averagept(1 GeV/c < pt < 4 GeV/c ) No interaction Coll (kcol=1) 0-10% Conclusion IV:Broadening of the correlation due to medium, but still visible. Increasing κ values wash out the correlation jc - jcbar coalescence Azimutal correlations might help identifying better the thermalization process and thus the medium jD - jDbar HEP - Valparaiso 14. december 2004 -

  27. Coll (kcol=10) Coll (kcol=20) Coll + rad (kcol=krad=1) Azimutal Correlations for Open Charm Smallpt(pt < 1GeV/c ) No interaction Coll (kcol=1) 0-10% c-quarks jc - jcbar coalescence Small correlations at small pt,, mostly washed away by coalescence process. D jD - jDbar 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 RAA, at the price of a large rescaling k-factor (especially at large pt), of the order of k=10or by including radiative processes. • Still a lot to do in order to understand for the v2. Possible explanations for discrepencies are: • Role of the spatial distribution of initial c-quarks • Part of the flow is due to the hadronic phase subsequent to QGP • Caveat of Langevin approach • Azimutal correlations could be of great help in order to identify the nature of thermalizing mechanism. HEP - Valparaiso 14. december 2004

  29. Back up HEP - Valparaiso 14. december 2004

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

  31. Results for open charm : rapidity distribution at RHIC Heinz & Kolb’s hydro (boost invariant) Set II Tiny diffusion effect (no E loss, no drag) (Set I) HEP - Valparaiso 14. december 2004

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

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

  34. Other ingredients of the model specific for J/y production (I) • J/y are destroyed via gluon dissociation: J/y + g  c + cbar and can be formed through the reverse mechanism, following the ideas of Thews. Uncorrelated quarks recombination  quadratic dependence in Nc : Question: How much is a ??? HEP - Valparaiso 14. december 2004

  35. Other ingredients of the model specific for J/y production (II) • As sel(J/y) is small, we assume free streaming of J/y through QGP (no thermalization of J/y)... But possible gluo dissociation • Clear cut melting mechanism: J/y 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=1.5Tc). • Up to now: No prompt J/y (supposed to be all melted) HEP - Valparaiso 14. december 2004

  36. Heinz & Kolb’s hydro No radial exp. hydro Results for J/y production at mid-rapidity, central Component stemming out the recombination mechanism: • 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. HEP - Valparaiso 14. december 2004

  37. Results for J/y production vs. rapidity • 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/y stemming out the fusion of uncorrelated c and cbar than for direct J/y. HEP - Valparaiso 14. december 2004

  38. J/y transverse momentum distribution at mid rapidity Tdissoc=180 MeV Tdissoc=180 MeV (no transv. flow) (Heinz & Kolb) Direct J/y(NN scaling) • Clear evidence of the recombination mechanism: • pt anti-broadening in Au-Au • effective temperatures > Tc Direct J/y (NN scaling) 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 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/y direct)=0 assumption / include beauty /find a name. HEP - Valparaiso 14. december 2004

  40. So what should we do ??? • 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) 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 ? HEP - Valparaiso 14. december 2004

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

  43. ℳqrad∝ ℳqelⅹgsⅹ Qq Qqg HEP - Valparaiso 14. december 2004

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

  45. Heavy quarks in QGP (or in strongly interacting matter) • Starting point: For heavy quarks, relaxation time >> collision time ; at large momentum (as for all quarks) but also at low momentum (thanks to inertia) • 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 ? HEP - Valparaiso 14. december 2004

  46. Non-Photonic electron elliptic-flow at RHIC: …and the bites (ouch) strong coupling c D t=1fm/c t 1 2 3 4 5 No coupling t=4fm/c r Spatial transverse-distribution might play some role as c-quarks are not from the beginning "on" the freeze out surface. HEP - Valparaiso 14. december 2004 SQM06

  47. Masses= . 33GeV qt2 =0.3 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 ~ < pμf - pμi > • B ~ < (pνf - pνi )(pμf - pμi ) > • 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 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. 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) HEP - Valparaiso 14. december 2004

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