Thermalization of Heavy Quarks and Consequences on Charmonia Production

1. Thermalization of heavy quarks and consequences on charmonia production P.-B. Gossiaux (with V. Guiho & J. Aichelin) • I) Heavy quarks in QGP • Model • Results for RAA • Results for v2 • First conclusion • II) J/Psi’s in QGP • Model • Results • III) Conclusion & Perspectives

2. 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.: Why not Boltzmann equation ? One first answer: not efficient.

3. Heavy quarks in QGP (or in strongly interacting matter) • The physics : a couple of heavy quarks, a medium (+ transport coefficients) and some hidden charm and beauty mesons. • Ideally : measure the transport coefficients and compare their value with theory (Lattice).

4. A few words on the transport coefficients with • drag coefficient is thus the proportion of momentum lost per unit of time • At high energy, one has (assuming f is peaked):  A(p) and the energy loss are the same quantities • At low energy, not true any more: On the average, particles can gain/loose energy without gaining or loosing momentum.

5. A few words on the transport coefficients (II) with • Diffusion (in momentum space); not to be confused with diffusion in « normal" space. • In isotropic media,admits a (longitudinal,transverse) decomposition  only 2 independent coefficients. • Essentially a markovian process

6. A few words on the transport coefficients (III) How well / precisely do we know these transport coefficients (in the case of heavy quarks) ? • Start from a more « fundamental » theory • (for instance, Boltzmann equation, assuming – although not mandatory – that surrounding medium is thermalized) • In case of collisions (2 2 processes): Pioneering work of Cleymans (1985), Svetitsky (1987), extended later by Mustafa, Pal & Srivastava (1997). The FP coefficients are expressed as moments of the differential cross section. • In case of radiation: Numerous works on energy loss; very little seems to have been done on other coefficients

7. A few words on the transport coefficients: The case 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) p (GeV/c)

8. t (fm/c) From Fokker-Planck coefficients  Langevin forces pz 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 py px …looks a little less « erratic » when considered on the average: Relaxation time >> collision time : self consistent

9. First results on c-quark evolution Relaxation of <E>, of and of for c-quarks produced in 200 GeV p-p collisions (Dy=2). Evolution in a m=0 --T=400 MeV QGP. Once again : long relaxation times f(E) Approximate scaling for T=0.2  0.5 Asymptotic energy distribution: no Boltzmann; more like a Tsallis Walton & Rafelski (1999) Too much diffusion at large momentum (E-m)/T

10. 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)

11. So what do we do ??? Three sets: FP coefficients deduced by Mustafa, Pal and Srivastava (MPS) Adapt (A,B)  (Ath =A,Bth) such then the associated fasympt is a Boltzmann distribution, and then  (kcolA, kcolBth) with kcolvarying from 0   in order to span from free streaming  instantaneous thermalization. <E> B// A=Ath Bth // B Bth

12. B (GeV^2/fm c) p (GeV/c) So what do we do ??? Three sets (cont): MPS + « radiative » coefficients deduced using the Gunion and Bertsch elementary cross section for qQ qQ+g and its equivalent for gQ gQ+g in t-channel (u & s-channels are suppressed at high energy). No LPM for the time MPS +krad x RAD A (Gev/fm) p (GeV/c) Should still be tuned !!!

13. Schematic view of our model for hidden and open heavy flavors production in AA collision at RHIC and LHC Evolution of heavy quarks in QGP (thermalization) D/B 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

14. Other hypothesis / ingredients of c-evolution and D production • Au – Au collision at 200 AGeV. • c-quark transverse-space distribution according to Glauber • c-quarktransverse 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) or various more realistic hydrodynamical evolution (Heinz & Kolb, Huovinen) • Evolution according to Bjorken time

15. Other hypothesis / ingredients of c evolution and D production • No force on the c-quarks before thermalization,Langevin force on c-quarks inside QGP and no force on charmed « mesons » during and after transition. • D & B produced via Coalescence vs. Fragmentation mechanism. • In fact, no beauty up to now; should be included.

16. 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)

17. Why so tiny ? Y Strong correlation of y vs. Y (spatial rapidity) y

18. Leptons ( D decay) transverse momentum distribution (y=0) RAA 0-10% 20-40% Col.+(0.5x) Rad Col. (kcol=10 & 20) • Conclusion I: • One can reproduce theRAA either : • With cranked up collisional processes • With « reasonnable » (krad not far away from unity) use of radiative processes. Min bias

19. Leptons ( D decay) transverse momentum distribution (y=0) RAA Coll. Langevin 0-10% B=0 (Just Coll. E-loss) Transition from pure E-loss (high E) towards thermalization regime (intermediate E)

20. p-p distribution c-quarks transverse momentum distribution (y=0) Heinz & Kolb’s hydro (boost invariant) Just before the transition MPS kcol=5 k=40 k=20 k=10 Conclusion II: kcol =10-20:Still ways to go before thermalization !!!

21. c-quarks transverse-momentum spectra Finer Effects: • Effect of radial flow (of QGP) on c-quarks (y=0) • pt« antibroadening »: • … Might be masked by initial-state interactions pt « anti-broadening » due to thermalization

22. c-quarks D Decay electrons Tagged const quarks NP-Electron elliptic flow at RHIC: comparison with experimental results Collisional (kcol=20) Conclusion III: One cannot reproduce thev2consistently with theRAA!!! Contribution of light quarks to the elliptic flow of D mesons is small Collisional + Radiative

23. NP-Electron elliptic flow at RHIC: Looking into the bits… v2 (all p) const quark tagged by c v2 (tagged p) C-quark does not see the « average » const quark… Why ? Bigger coupling helps… a little but at the cost of RAA

24. NP-electron elliptic flow at RHIC: …and the bites (ouch) strong coupling t=1fm/c No coupling t=4fm/c r Spatial transverse-distribution might play some role as c-quarks are not from the beginning on the surface.

25. Results for open charm : First conclusions • Experimental data point towards a significant (although not complete) thermalization of c quarks in QGP, which should result in some pt anti-broadening (beware of Cronin, however) • 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 including radiative processes. • Still a lot to do in order to understand for the v2. Possible explanations are: • Bug • Caveat of Langevin approach • Part of the flow is due to the hadronic phase subsequent to QGP

26. J/y’s

27. 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 ???

28. 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)

29. 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.

30. 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.

31. 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)

32. 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.

33. Some Results for LHC

34. c-quarks transverse momentum distribution (y=0) at LHC Tin=500 MeV Tin=700 MeV

35. J/y’s at LHC Effects of thermalization increase with initial temperature Still a « window over Tc »