1. Guang-You Qin and Steffen A. Bass
Dynamical Evolution, Hadronization and Angular De-correlation of Heavy Flavor in Hot and Dense QCD Matter
Shanshan Cao
In collaboration with
Guang-You Qin and Steffen A. Bass
Duke University

2. Outline Heavy flavor initial production: pQCD + shadowing
Heavy flavor evolution inside QGP : Improved Langevin approach incorporating gluon radiation
Heavy flavor hadronization: a hybrid coalescence plus fragmentation model
Results of heavy flavor suppression and flow and comparison with LHC and RHIC data
Angular correlation of heavy flavor pairs
Summary

3. Heavy Flavor Initial Production
Initial production: MC-Glauber for the position space and leading-order pQCD calculation for the momentum space
Shadowing effect: different PDF’s of nucleon and nuclei lead to different production rate of HQ in pp and AA collisions
K. Eskola, et al., JHEP 0807 (2008) 102
Significant shadowing effect for charm quark production at low pT (especially at the LHC energy)  impact on RAA

4. Heavy Flavor Evolution inside QGP (Energy Loss Mechanisms)
Two ways for heavy quarks to lose energy:
Unless in the ultra-relativistic limit (γv >> 1/g), gluon radiation is suppressed by the “dead cone effect”
Heavy quark inside QGP medium: Brownian motion
Description: Langevin equation c q g
medium θ
collisional
radiative
What shall we modify to go from RHIC to LHC?
HQ becomes ultra-relativistic – cannot ignore gluon radiation

5. Heavy Flavor Evolution inside QGP ( Improved Langevin Approach )
Modified Langevin Equation:
Fluctuation-dissipation relation between drag and thermal random force:
Force from gluon radiation:
Gluon distribution taken from Higher Twist calculation:
Guo and Wang, PRL 85, 3591; Majumder, PRD 85, ; Zhang, Wang and Wang, PRL 93,
Transport Coefficients:

6. Heavy Quark Energy Loss
Collisional energy loss dominates low energy region, while radiative dominates high energy region.
Crossing point: 6 GeV for c and 16 GeV for b quark.
Collisional energy loss alone may work well to describe previous RHIC data but is insufficient for LHC.

7. Heavy Flavor Hadronization: Fragmentation + Recombination
Most high momentum heavy quarks fragment into heavy mesons: use PYTHIA 6.4
Most low momentum heavy quarks hadronize to heavy mesons via recombination (coalescence) mechanism: use the sudden recombination model
Y. Oh, et al., PRC 79, (2009)

8. The Sudden Recombination Model
Two-particle recombination:
Distribution of the i th kind of particle
Light quark: fermi-dirac distri. in the l.r.f of the hydro cell
Heavy quark: the distribution at Tc after Langevin evolution
Probability for two particles to recombine
Variables on the R.H.S. are defined in the c.m. frame of the two-particle system.

9. The Hybrid Coal. + Frag. Model
Use f W to calculate Pcoal.(pHQ) for all channels: D/B Λ Σ Ξ Ω
Normalization: Pcoal.(pHQ=0) =1
Use Monte-Carlo to determine the hadronization channel of each HQ: frag. or recomb.? recomb. to D/B or a baryon?
Fragmentation dominates D meson production at high pT.
Recombination significantly enhances the D meson spectrum at intermediate pT.

10. Sum up: Heavy Flavor inside QGP
Generation of QGP medium: 2D viscous hydro from OSU group (thanks to Qiu, Shen, Song, and Heinz)
Initialization of heavy quarks: MC-Glauber for position space and pQCD calculation for momentum space
Simulation of heavy quark evolution: the improved Langevin algorithm in the local rest frame of the medium
Hadronization of HQ into heavy meson: frag. + recomb.
D=6/(2πT), i.e., qhat around 2~3 GeV2/fm at initial temperature (around 350~400 MeV)
outside the medium
(below Tc)
hadronize

11. RAA of LHC D meson
Collisional dominates low pT, radiative dominates high pT.
The combination of the two mechanisms provides a good description of experimental data.

12. RAA of LHC D meson
Shadowing effect reduce RAA significantly at low pT.
Recombination mechanism raise RAA at medium pT.

13. v2 of LHC D meson
frag. only: force fragmentation, i.e., fW(q)=0 for any q.
recomb. only: force combination, i.e., fW(q)=1 for any q.
Recombination mechanism provides larger v2 than fragmentation. However, due to the momentum dependence of the Wigner function, our combined mechanism only slightly raises the D v2 at medium pT.

14. RAA and v2 of RHIC D Meson
Recombination mechanism has a significant contribution to RAA at RHIC energy.
With the incorporation of the low energy effect (shadowing and recombination), our improved Langevin framework provides good descriptions of the RHIC data.

15. Angular Correlation of QQbarΔθ
At LO: Back-to-back production of initial QQbar with the same magnitude of momentum

16. Angular De-correlation of CCbar
Though each energy loss mechanism alone can fit RAA with certain accuracy, they display very different behaviors of angular de-correlation.
Pure radiative energy loss does not influence the angular correlation significantly; pure collisional leads to peak at collinear distribution because of the QGP flow.

17. More Realistic Analysis
MCNLO + Herwig radiation for HQ initial production
Angular correlation function of final state ccbar pairs
Within each event, loop each D with all Dbar’s
Similar shape as ccbar pairs, but on top of a large background
Experimental observations will help distinguish the energy loss mechanisms of heavy quark inside QGP.

18. Summary
Study the heavy flavor evolution and hadronization in QGP in the framework of the Langevin approach
Improve the Langevin algorithm to incorporate both collisional and radiative energy loss of heavy quark
Reveal the significant effect of gluon radiation at high energies and recombination at medium energies, and provide good descriptions of D meson suppression and flow measured at both RHIC and LHC
Discussed about a potentially measurable quantity – angular correlation function of HF pairs – that may help distinguish the HQ energy loss mechanisms

19. Thank you!

20. Heavy Flavor Evolution inside QGP ( Improved Langevin Approach )
Numerical Implementation (Ito Discretization)
Drag force:
Thermal random force:
Momentum of gluon radiated during Δt:
Lower cut for gluon radiation: πT
Balance between gluon radiation and absorption
Guarantee equilibrium after sufficiently long evolution

21. Charm Quark Evolution in Static Medium
T = 300 MeV, D=6/(2πT), i.e., qhat ~ 1.3 GeV2/fm
Einit = 15 GeV z
Evolution of E distribution
Before 2 fm/c, collisional energy loss dominates; after 2 fm/c, radiative dominates;
Collisional energy loss leads to Gaussian distribution, while radiative generates long tail.

22. Prediction for B Meson Measurements
The shadowing effect for b-quark is not as significant as c-quark, but still non-negligible. “Anti-shadowing” at RHIC energy.

23. Uncertainties of v2
Different geometries and flow behaviors of the QGP medium influence heavy flavor v2 while does not significantly impact the overall suppression (SC, Qin and Bass, J. Phys. G40 (2013) )
Currently use smooth initial condition for hydro evolution, an event-by-event study of HQ may result in different final state spectrum
No hadronic interaction yet after the QGP phase

24. The Sudden Recombination Model
N: normalization factor
gM: statistics factor
e.g. D ground state: 1/(2*3*2*3)=1/36 – spin and color
D*: 3/(2*3*2*3)=1/12 – spin of D* is 1
ΦM: meson wave function – approximated by S.H.O.
Integrating over the position space leads to
μ: reduced mass of the 2-particle system
ω: S.H.O frequency – calculated by meson radius
0.106 GeV for c, and GeV for b
Can be generalized to 3-particle recombination (baryon)

25. Questions to be Discussed
How to quantify such distribution functions
How to subtract the large background of non-correlated heavy meson pairs (Zero Yield at Minimum [ZYAM]? Will over-subtract the background for heavy flavor.)
Theoretical uncertainties – initial production of QQbar pairs: similar single particle spectra but different angular distribution

26. pT Dependence of Angular De-correlationΔθ
π-Δθ
For the limit of uniform distribution between [0,π]:
The de-correlation behavior strongly depends on HQ pT.
Collisional: charm quarks below 2 GeV are close to thermal, but above 2 GeV are off-equilibrium.
Radiative: CCbar pair remains strongly correlated.

27. Prediction for LHC B Meson
The shadowing effect for b-quark is not as significant as c-quark, but still non-negligible.
The dip in the B meson v2 results from the transition behavior from the collisional dominating region to the radiative dominating region. (SC, Qin and Bass, Nucl. Phys. A (2013) 653c-656c)

28. Prediction for RHIC B Meson
Instead of “shadowing”, RHIC b-quark has “anti-shadowing” effect at low pT.

29. Check of Detail Balance
Modified Langevin Equation:
Gluon radiation only, may break the detail balance
Cut off gluon radiation at low energies where collisional energy loss dominates and detail balance is preserved.
Large enough cut reproduces charm quark thermalization behavior.
More rigorous solution: include gluon absorption term into the higher-twist formalism directly and recalculate term.

30. Back up: the dip of B meson v2
Glauber for hydro
frag. only
During the transition from the collisional dominating region the radiative dominating region, a dip may occur in the v2 spectrum, might be more significant for KLN initial condition than Glauber because the former leads to higher peaks of v2.

31. RAA and v2 of RHIC Heavy-decay Electron
The b/c ratio in the initial production is tuned to be 0.3% in order to describe the measured RAA – slightly smaller than the usually adopted value (0.5 ~ 1%).

32. Recombination Probability
Use the Wigner function to calculate Pcoal.(pHQ) for all channels: D/B Λ Σ Ξ Ω
Fix N: Pcoal.(pHQ=0) =1
Normalization is tuned in the lab frame with Teff = 175 MeV to mimic the flow (0.6c) effect of 165 MeV (Tc) hydro medium
Generate random number: frag. or recomb.? recomb. to D/B or baryons?
If recombine to D/B, pick up a light quark in the cell frame (Monte Carlo according to the Wigner function), boost into the lab frame and combine with the heavy quark.

33. RAA of D meson
Consistency between our calculation and the latest ALICE data from Quark Matter 2012

34. v2 of D meson
Similar trend of competition between the two energy loss mechanisms as revealed by RAA
KLN provides larger eccentricity for the QGP profile than Glauber and therefore larger D meson v2
Different geometries and flow behaviors of the QGP medium may have impact heavy flavor v2 while does not significantly influence the overall suppression (SC, Qin and Bass, arXiv: )
Influence of coalescence mechanism and hadronic interaction

35. Prediction for B Meson
Similar behaviors as D meson – collisional energy loss dominates low pT region, while radiative dominates high pT.
The crossing point is significantly higher because of the much larger mass of bottom quark than charm quark.
B meson has larger RAA and smaller v2 than D meson.